(MBS I publi- Rer#ence

MBSIR 82-240

A

Heatmg Values of Natural Gas

and Its Components

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Chemical Physics

Chemical Thermodynamics Division

Washington, DC 20234

May 1982

Technical Report

Issued August 1982

red by e International des Importateurs

Gaz natural Liquifie (GIIGNAL)

NBSIR 82-2401

aXTIOBAlf BUHUAU

O# enrAJta>AKva tlMBAMT

HEATING VALUES OF NATURAL GAS

AND ITS COMPONENTS

AUG 1 9 1982

IC>-_

. Slo

eGeorge T. Armstrong

Thomas L. Jobe, Jr.

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Chemical Physics

Chemical Thermodynamics Division

Washington, DC 20234

May 1982

Technical Report

Issued August 1 982

Sponsored by

Groupe International des Importateurs

de Gaz Natural Liquifie (GlIGNAL)

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
(

f

'i

I

i

''i

ji

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Preface

The Office of Standard Reference Data of the National Bureau of

Standards is responsible for a broad-based program to provide reliable

physical and chemical reference data to the U.S. technical community.

Under this program a number of data evaluation centers both at N3S and at

universities and other private institutions are supported and coordinated;

these activities are collectively known as the National Standard Reference

Data System (NSRDS). Important areas of the physical sciences are covered

systematically by NSRDS data centers, and data bases with broad utility are

prepared and disseminated. These centers can also take on special compilations of data addressing specific applications. The existence of an ongoing

program permits the collection of data for these special compilations to be

carried out in an efficient and timely manner.

This Report on the Heating Values of Natural Gas and Its Components was

prepared with the assistance of the Chemical Thermodynamics Data Canter of

the National Bureau of Standards. We hope that it will provide a useful

source of reference data and computational methods for all those concerned

with the heating value of natural gas.

David R. tide, Jr. Chief

Office of Standard Reference Data
Dr. George Thompson Armstrong, the senior author, died on March 9,

1982. At that time an almost complete, well written draft was at hand and

had been reviewed by several of his colleagues, including myself. It has

been my privilege to help prepare the final version. The original organization and wording have been retained wherever possible. Several sections

have been completed, based on indications in the text. New atomic masses

and heat capacities have been selected and a discussion of the enthalpy of

combustion of methane (in Appendix 7) has been written. All of these were

influenced by notes and letters, but, at times, go beyond them. I hope

that the spirit and quality of the original has been maintained.

There is one section missing: acknowledgements. Neither Mr. Jobe

nor I can reconstruct a record of the many persons who were consulted,

who advised or who sent in material. We thank all who have helped and hope

that this report will both be useful to them and meet with their approval.

David Garvin

Chemical Thermodynamics Data Center
Abstract

This document gives the basic data needed, recommended procedures

and illustrative calculations for computing heating values of natural

gas mixtures from the composition of the mixtures and the properties of

the components at commonly used reference conditions for gas measurement.

Much of the data is given in the form of tables, and sufficient information is given for calculating properties at conditions other than those

for which tables are given. Symbols and terms used are defined, units

of measurement are defined and conversion factors and physical constants

are given.

The standard enthalpies of combustion and heat capacities of the

pure hydrocarbon gases to are selected from prior initial evaluations

of experimental measurements. The enthalpies of combustion of the ideal

gases at the reference temperatures 273.15 K (0 °C), 288.15 K (15 °C),

288.71 K (60 °F), and 298.15 K (25 °C) on a molar basis, and a volumetric

basis are given. Tables are for the dry gases, and information is given

to calculate the enthalpy of combustion of the ideal water-saturated gas

on a volumetric basis. The calculation of enthalpies of combustion of

ideal gas mixtures on a molar, mass, or volumetric basis is described.

Second virial coefficients as functions of temperature for the pure

substances and for binary interactions with methane as one component are

presented as selected from recent compilations based on experimental

measurements.

Tables are given for molar volumes, ethalpic effects (H-H°), and

the heating values of the dry real-gas hydrocarbons on a molar basis, a

mass basis, and a volumetric basis at two reference conditions, 288.15 K
(15 °C), 101 325 Pa; and 288.71 (60 °F), 101 560 Pa (14.73 psia). The

procedure used in calculating these tables is described and information

is provided for making similar calculations for other reference conditions.

An analysis of the uncertainties of the data is presented, together with

procedures for calculating the propagation of errors and the effects of

these errors on calculated heating values. Supporting data, sources of

the data, and discussions of the relationships involved are presented in

a series of appendixes.

KEYWORDS: Calorific value, enthalpy of combustion; fuel gas mixtures;

heat capacities; heating value; hydrocarbons; liquefied natural gas;

natural gas; propagation of errors; reference conditions.

i V
Table of Contents

Page number

Preface i

Abstract i i i

1. Introduction 1

2. Symbols 4

3. Definitions of terms 5

4. Units of measurement 6

5. Physical constants 7

6. Reference conditions of measurement 10

7. Thermodynamic data for auxiliary substances 12

8. Thermodynamic quantities for hydrocarbons - ideal gas,

molar basis 15

9. Standard- state volumetric properties of hydrocarbons 18

10. Properties of the real gases 24

11. Volumetric enthalpy of combustion of the real gas 32

12. Calculating the heating values of gas mixtures 37

13. Uncertainties 48

General list of references 55

Figures 1 and 2 58,59

Appendix 1. Proposed composition limits 60

Appendix 2. Symbols 61

Appendix 3. Definitions of terms used in this document 64

Appendix 4. Units of measurement and conversion factors 72

Appendix 5. Discussion of physical constants 74

Appendix 6. Discussion of reference conditions 92

Appendix 7. Thermodynamic data: their sources and

\/

uncertai nti es 96
Appendix 8. Relationships between thermodynamic

in Tables 3 and 4

quantities

128

Appendix 9. Non- ideality effects and the virial

of state

equation

133

Appendix 10. Propagation of uncertainties 147

VI
List of Tables

:

Table 1.

a.

b.

Table 2.

Table 3.

a,

b.

Table 4..

a.

b.

Table 5.

Table 6.

Page No.

Physical constants 8

Relative atomic and molecular masses used in this

document 8

Other physical constants used in this document 9

Reference conditions of measurement used in this document 11

Thermodynamic data for selected (auxiliary) substances: 13

Standard heat capacity at r = 298.15 K, [H°(298.15 K) -

for = 288.15 K, 288.71 K and 273.15 K 13

Standard enthalpy of formation at r = 298.15 K, 288.15 K,

288.71 K and 273.15 K 14

Thermodynamic data for gaseous hydrocarbons (ideal gas,

molar basis) 16

Heat capacity and enthalpy difference between 298.15 K and

288.15 K, 288.71 K and 273.15 K 16

Enthalpy of combustion (in kJ mol ^) at 298.15 K, 288.15 K,

288.71 K and 273.15 K 17

Molar volume and density at various reference

conditions 21

Combustion data for selected gaseous hydrocarbons

(ideal gas, volumetric basis) in SI units at ISO/ANSI/

ASTM reference conditions of temperature and pressure,

p = 101.325 kPa; T = 288.15 K 22
Table 7. Combustion data for selected gaseous hydrocarbons (ideal

gas, volumetric basis) in SI and U.S. Customary units at

ANSI/ASTM/API reference conditions of temperature and

pressure p = 101.560 kPa; T - 288.71 K 23

Table 8. Virial coefficients for pure substances and mixtures 26

a. Second virial coefficients, B{T)

,

for pure substances 26

b. Second virial coefficients, , for binary

mixtures 28

Table 8. Enthalpy of combustion of real-gas hydrocarbons on

molar, mass and volume bases; in SI units and at ISO/

ANSI/ASTM metric reference conditions p = 101.325 kPa;

T = 288.15 K 30

Table 10. Enthalpy of combustion of the real-gas hydrocarbons on

molar, mass and volume bases; in U.S. Customary units

and at ANSI/ASTM/API reference conditions p = 101.560 kPa

(14.73 Psi); T = 288.71 K (60 °F) 31

Table 11. Enthalpy correction for the real-gas and molar volume,

V , at ISO/ANSI/ASTM metric reference conditions p

=

m 101.325 kPa; T = 288.15 K 35

Table 12. Enthalpy correction for the real-gas and molar volume,

V , at ANSI/ASTM/API U.S. Customary reference conditions

m p = 101.560 kPa; T = 288.71 K 36

Table 13. Sample calculation of enthalpy of combustion of a

gaseous mixture assuming ideal gas behavior 40

Table 14. Sample calculation of enthalpy of combustion of a realgas mixture assuming no interactions between different

vi i i

substances 43
Table 15. Estimated uncertainties in thermodynamic properties 51

Table Al, Proposed composition limits for pipeline quality natural

gas 60

Table A2. Thermodynamic symbols used in this document 61

Table A4. Measurement quantities defined exactly in terms of

SI units 72

Table A5. Atomic masses 76

a. Recent sets of relative atomic masses of the chemical

elements and their period of recommendation by the

International Union of Pure and Applied Chemistry

(lUPAC) 76

b. Relative molecular masses of some important molecules as

formally calculated from different sets of relative

atomic masses (rounded to four decimal places) 77

c. Assignment of uncertainties in relative atomic and

molecular masses for the 1981 set recommended by the

lUPAC Commission on Atomic Weights and Isotopic

Abundances 78

Table A7a. Heat capacity equations for auxiliary substances and

hydrocarbons for the range 268 to 308 K for the equation

Cp/i? = a + 2) (2* - 273.15 K) + c{T - 273.15 K)^ 102

Table A7b. Enthalpy of combustion of selected hydrocarbons and

sources of the data 103

Table A7c. Heating value of methane: sources of data 107

Table A7d. Data points and statistics for studies of methane by

Rossini and by Pittam and Pilcher 108
Table A7e. Enthalpy of combustion of methane as an ideal gas and

relative atomic mass scales given in compilations of

thermodynamic data 109

Table A7f. Vapor pressure of water 110

Table A7g. Estimates of uncertainties s^ in enthalpy of combustion

or s^ in enthalpy of formation by various authors 112

Table A7h. Estimated total uncertainties-first test 113

Table A7i. Estimated total uncertainties-second test 114

Table A7j. Estimated uncertainties in of hydrocarbons from

Scott's correlation of thermodynamic functions 115

Table A7k. Comoarative calculations of C° 116

P Table A9a. Constants for quadratic equations for the second virial

coefficients as functions of T in the range 273 to 300 K 144

Table A9b, Constants for quadratic equations for the second virial

coefficients of interaction, 5(l,i), as functions of

temperature for the range 273 to 300 K 145

Table A9c. Enthalpy differences H-H° between the real and ideal

, gases for pure substances p

= 101.325 kPa at various

reference temperatures 146

List of figures

Figure 1. Interrelationships among enthalpies of formation and

combustion at two temperatures 58

Figure 2. Relationships among enthalpies of combustion for the

real and ideal gas on molar, mass and volume bases 59

X
1

»*

HEATING VALUES OF NATURAL GAS AND ITS COMPONENTS

1. Introduction

This document provides the basic information for calculating heating

values of natural gas mixtures from a knowledge of the gas composition.

While it was prepared specifically to meet the needs of the liquefied

natural gas industry, the information is equally well applicable to

native or processed natural gas mixtures that have not been liquefied.

The substances treated are the saturated hydrocarbons to Cg, a

few of the more commonly encountered cyclic and unsaturated hydrocarbons

,

and some non-hydrocarbon gases frequently found in natural gas. A

broader range of materials, but for fewer reference conditions, is given

in an earlier study [1].^

The approximate limits of composition and other properties of

gaseous fuel mixtures to which this document should apply are given in

Appendix 1.

The presentation is in handbook style, with sufficient detail to

allow the use of the information in a variety of ways to meet various

requirements. The information is sufficient to allow calculations at.

various base conditions of measurement in use in the natural gas industry,

to allow calculations of greater or lesser accuracy, and to allow the

uncertainty of the results to be estimated.

The data and procedures given should not be construed as providing

a definitive substitute for laboratory measurement of heating value,

which can be reliably performed using well known standard methods of

test.

^Numbers in brackets in the text refer to references in the general list at

the end of this document. Appendices have their own reference lists.
2

In many cases, however, an estimate of the calorific value is

adequate. These include engineering design calculations for power

plants and power plant components, in establishing the adequacy of fuels

from certain sources or processes for selected purposes and in estimating

processing requirements or mixing proportions needed to obtain gas

mixtures of specified performance. Moreover, the estimates are pertinent

and usable in custody transfer operations to the extent agreeable to the

parties involved. Indeed, the study made here of the uncertainties in

the estimates suggests that the calculation of heating values may be

used in many routine conditions.

The organization of this document is described here with the aid of

two figures. First there are necessary preliminaries: definition of

reference conditions (Table 2), molecular weights and conversion factors

for energy units (Table 1). Then the basic thermodynamic data are

given. The starting point is a set of carefully evaluated enthalpies of

formation for the ideal gases at 298.15 K (25 °C) from which best values

for the enthalpies of combustion of selected hydrocarbons are derived

and listed in Table 4. Properties of products of combustion and auxiliary

substances are li*sted in Table 3. Three reference temperatures of

interest here are different from that at which the basic data are reported.

The enthalpies of formation and combustion are corrected from 298.15 K

to these temperatures using the scheme outlined in Figure 1 and Appendix 8

and enthalpy differences from Tables 3 and 4.

The rest of the calculations are summarized in Figure 2. Starting

with the molar enthalpy of combustion of the fuel as an ideal gas, point

B, values are derived for the enthalpy of combustion per unit mass, point

E, and per unit volume, dry and water saturated, at points C and D. These

are given, for the ideal gas, in Tables 6 and 7 while the needed molar

volumes are in Table 5. The procedures are in Section 9.
3

Real gas properties are produced by correcting the ideal gas molar

enthalpy of combustion using equation of state data (second virial

coefficients) in Table 8 and procedures described in section 10. This

produces the real gas enthalpy of combustion, point F in Figure 2. From

there the volumetric and mass based quantities are derived, points G, H

and I. The real gas enthalpies of combustion at 15 and 60 °F are

given in Tables 9 and 10, while the molar volumes and enthalpy correction

factors are in Tables 11 and 12. In other tables there is enough information

given to permit calculations for other conditions of temoerature and

pressure.

Procedures for the calculation of the properties of mixtures are

developed in section 12. Both ideal and real gas mixtures are treated,

with the latter being considered two ways: as a mixture of non-interacting

real gases and as a mixture of interacting real gases. All treatments

are based on the virial equation of state.

Uncertainties in the data are discussed in section 13. Those for

the enthalpies of combustion are summarized in Table 15. These are the

limiting factors on the accuracy of the data. They should be used in

all calculations. The numbers in Table 15 indicate that r.he uncertainty

in the enthalpies of combustion of interest here are of the order of

tenths of kilojoules per mole, and, at times, of the order of kilojoules

per mole. This means that in most tables of thermodynamic properties

given here, the right-most digit is not significant. It is provided for

rounding numbers in calculations made by the reader. Differences that are

only in the final digit between the numbers here and in other compilations

should be ignored. They are the result of slight variations in calculation

and rounding procedures. Larger differences usually are due to differing

choices for basic data.
4

2. Symbols

The symbols used in this work are listed in Appendix 2 in Table A2.

They are, so far as practicable, in conformity with the recommendations

of the International Union of Pure and Applied Chemistry (lUPAC) [2,3]

and of the International Organization for Standardization (ISO) [4].
a

5

3. Definitions of terms

The following terms used in this document are defined in Appendix 3

Quantitites and concepts .

atomic weignt net calorific value

<- compression factor (net or lower heating value)

dry gas real gas

enthalpy relative atomic mass

enthalpy of combustion relative molecular mass

enthalpy of formation reference (standard) conditions

equation of state specific heat capacity

gas constant standard state

heat capacity total calorific value

ideal gas (total or gross heating value)

International Practical vapor pressure

Temperature Scale virial coefficient

molar mass volume under standard conditions

molecular weight water saturated gas

Units of measurement,

atmosphere pascal

British thermal unit standard cubic foot (of gas)

joule

newton

torr

The usage of terms is intended to represent best current practice and

to conform so far as possible to recommendations of lUPAC [2,3], ISO [4,5],

and the International Bureau of Weights and Measures (BIPM) [5].
6

4. Units of measurement

The quantities given in this report are based on values measured or

expressed in the International System of Units (SI) [5]. The conversion

factors used in this document for units of energy are:

1055.056 J = 1 BtUj-p

1 MJ m"^ = 26.839 192 BtUj-^ ft“^

(under identical conditions of temperature and pressure). See Appendix 3,

section A3. 2. 2 for a discussion of the various Btu's. Other conversion

factors are found in Table A4 (Appendix 4).
7

5. Physical constants

Relative atomic and molecular masses . (Atomic and molecular weights).

Molecular masses used in this document are listed in Table la. These

molecular masses are based on the 1981 recommendations of the International

Union of Pure and Applied Chemistry Commission on Atomic Weights and

Isotopic Abundancies [6]. Some other sets of relative atomic masses are

discussed in Appendix 5a.

Other physical constants . Table lb lists other physical constants used

in this document. The value for the gas constant is that recommended by

the CODATA Task Group on Fundamental Constants [7]. The acceleration of

gravity and the density of mercury are part of the (historical) measurement

chain for pressure, which is discussed in Appendix 5b.
Relative

atomic

and

molecular

masses

used

in

this

document 8

(T3

•Xto

to

03 P

to3

t/3 X

to

to

03

(V S >

•I- I- -P (0

03 I— f— 3

(UU

CU

CUUc

(O -p

to JD3

tn

03 21 o 1—1 00 c'O <;1- O 1—

u > 00 CT> o CSJ m tn 1—

to r''. 03 • 1“ s- o o o rH 1—1 1—

rH 1—1 rH tH

a -P 03 • « 03 o CSJ to ocj C3 CSJ OCD to 03 1— 3 ro >ct- tn in P's 00 00 03 03 U c s_ C3e: 03 o JZ r— .Q p> o s03 o O p> O

Sa T3 03 > T3 03 JZ c U o o CSJ CSJ 3 c to to 00 00 rH 1—

1—

to rH rH o

03 x:

XXX XXX

XX

X s- +J CSJ ro cn 'a’ tn in to to to

to tp CP CP CP CP CP CP CP CP CP c

03 3 0) tn

03> X (0 to X

to 00

03 03 00 03 cn 00 in

03 s O'. 03 tom CSJ ro in 03 > CT3 CT3 o o o o o o to S- . • • • • to -p 03 c tn 1—1 CSJ ^ to to 00 03

03 o rH ro CO to c tH ^ C\J CSJ E

f— 3 .Q o

03

U S- tQ S03 03 s_ 03 r— U 03 r— o o U 3

s: 1- O

u

a !- 03 a f—

JZ >> o c E o CSJ 03 c CSJ CSJ X X >> U CSJ o XX CSJ CSJ c c o o ty^ t/3 CP CP CP CP 03

03 21 co

i03UO

i. -a

>>coc

00 OS CT3 cn to 'll" OO rH o OS in m to CSJ m

"cr rH rH o o rH rH p>' o rH

os o o o o o o o o o

CT3 csj OCD rH csj 0C3 'lO' 'O' 00

ro rH CSJ rH

m CSJ

s- < oCP

CNJ oCP

O t/> CSJ CSJ CSJ OJ :r az X ^ or

C\i

03E

03 LT3 <

<D

03

>)

i-O

O)

03 P

03UO

s-c

03 PC

03

places

after

addition.

Uncertainties

in

atomic

masses

are

in

the

rightmost

digit.
9

Table lb. Other physical constants used in this document^

gas constant (i?)

standard acceleration

of gravity (g)

standard density of

mercury at r = 273.15 K

8.31441(26) J mol"^ K“^

980.665(0) cm s'^

13.5951(0) g cm’^

^Numbers in parentheses indicate the uncertainties in the last figures of

the number listed. Zero indicates that the number is defined.
10

6. Reference conditions of measurement

Reference conditions of measurement for which this document is

specially intended are listed in Table 2. Reference conditions, sometimes

referred to as "base conditions" are in a state of flux in the gas

industry. Each of the parameters (temperature, pressure, state of gas,

and water content) listed in Table 2 is discussed in Appendix 6. Not

all combinations of temperature, pressure, gas ideality, and water

content discussed there are of probable use in the gas industry. An

attempt has been made to sort out the potentially useful conditions and

five conceivably useful sets of reference conditions are listed in Table

2. This document provides information necessary to convert values of

properties from one to the other. However, the set labelled "ISO/ASTM/ANSI

(metric units)" is considered to be the most useful for the purpose of

this document, and combustion data are provided for these conditions,

with the exception that real gas properties are not specifically listed

for all the substances that are minor components of natural gas. Data

are also given at the slightly different conditions customary in the

USA, the set is labeled "ANSI/ASTM/API (U.S. Customary units)".
11

Table 2. Reference conditions of measurement used in this document

Desi gnation a P State of Gas Water Content

IUPAC(new)[3] 298.15 K 100 kPa ideal dry

STP 273.15 K 101.325 kPa real dry

(0 °C) (760 mmHg)

IUPAC(old)[8] 298.15 K 101.325 kPa ideal dry

(25 °C) (760 mmHg)

IS0/ASTM/ANSI[9]^ 288.15 K 101.325 kPa real dry/sat

(metric units) (15 °C)

ASTM/ANSI/API[9,10] 288.71 K 101.560 kPa real sat

(U.S. customary) (60 °F) (14.73 psia)

^Alternative units for specifying the same temperature should not be

taken to mean a different temperature. That is, 25 °C is identical to

298.15 K; 15 °C is identical to 288.15 K, and 60 °F is identical to

15.555 °C or 288.7055 ... K, in which the fraction is continuing. This

is rounded to 288.71 K. Similarly, different units for expressing the same

pressure should not be construed as different pressures. 101.325 kPa is

identical to 760 Torr and to 760 mmHg. The value 101.560 kPa is a rounded

value corresponding to 14.73 psia. This last value is taken here as exact,

although it may have originated as a rounded value for 30 in Hg. (In contrast

we use 288.706 K for 60 °F, not 288.71 K as appears in most table headings).

See Appendix 6 for discussion of reference conditions and Appendix 5b for

the interrelationships among pressure units.
12

7. Thermodynamic data for auxiliary substances

In this section are given the basic data for products of combustion

and components of air. These are enthalpies of formation, enthalpy

differences and heat capacities, all for the ideal gases. The latter

two types of data are needed to correct the standard enthalpies of

formation at 298.15 K to the various reference conditions of interest in

gas technology.

In Table 3a are listed values of C^(298.15 K) for each of the

auxiliary substances and calculated values of - 7f°(7’2), that is -

{H°{T

2

) ~ for the temperature intervals = 298.15 K; ~

288.15 K; = 288.71 K (60 °F); and = 273.15 K. In Table 3b are

listed selected values of A^°(298.15 K) for each of the auxiliary

substances and values for A^°(288.15 K) , A^°(288.71 K) , and A^°

(273.15 K) derived using the data in Table 3a. The sources of the data

are discussed in Appendix 7. How the data are interrelated and how the

corrections are made are described in Appendix 8.
13

Table 3. Thermodynamic data for selected (auxiliary) substances

Table 3a. Standard heat capacity at r = 298.15 K and [//°(298.15 K) -

for = 288.15 K, 288.71 K and 273.15 K

C°(298.15 K)^ 7f°(298.15 K) - H°{T

J*mol ^*K ^ kJ*mol

^

Substance

(phase)

T - ^2 288.15 K T^2 = 288.71 K

II

CM 273.15 K

Ar (g) 20. 78 0.208 0. 196 0. 520

C(c, graph'ite) 8. 53 .083 - 081 0. 203^^

C0(g) 29. 15 .291 275 0. 728

co^Cg) 37. 12 .369 . 348 0. 914

H2(g) 28. 83 .288 272 0. 717

H20(g) 33. 58 .336 . 317 0. 830^

H20(:^) 75. 29 .753 . 711 1. 888^

H2S(g) * 34. 10 .340 321 0. 349

He(g) 20. 79 .208 . 196 0. 520

N2(g) 29. 13 .291 . 275 0. 728

02^9) 29. 37 .293 . 277 0. 733

S(c, rhombic) 22. 64 .225 . 213

S02(g) 40. 06 .399 • 376 0. 989

Si/i 1 hoit

,

R. C. ; [11], except as noted.

^DeSorbo

,

W. ; and Tyler, W. W. ; [13].

^Osborne

,

N. S. ; Stimson , H. F^ ; Gi nni ngs

,

D. C.; [12].
14

Table 3b. Standard enthalpy of formation, at 2* = 298.15 K, 288.15 K,

288.71 K and 273.15 K

Substance

(phase)

A^°(298.15 K)^

kJ ‘mol

A^°(288.15 K)

kJ*mol ^

A^°(288.71 K)

kJ*mol

^

A^°(273.15 K)

kJ‘mol

^

Ar(g) 0.0 0.0 0.0 0.0

C(c, graphite) 0.0 0.0 0.0 0.0

C0(g) -110.53 ± 0.17 -110.59 -110.59 -110.69

C02(g) -393.51 ± 0.13 -393.50 -393.50 -393.49

H2(g) 0.0 0.0 0.0 0.0

H20(g) -241.814 ± 0.042 -241.715 -241.721 -241.561

H20(£) -285.830 ± 0.042 -286.148 -286.131 -286.634

H2S(g) -20.63 ± 1.00 -20.46 -20.47 —

He(g) 0.0 0.0 0.0 0.0

N2(g) 0.0 0.0 0.0 0.0

02^9) 0.0 0.0 0.0 0.0

S(c, rhombic) 0.0 0.0 0.0 0.0

SO^Cg) -296.81 ± 0.21 -296.69 -296.70 - —

^Values at 238.15 .K were taken from CODATA Bulletin No. 28 and CODATA Special Report

8 [14].
15

8. Thermodynamic quantities for hydrocarbons - ideal gas, molar basis

In this section are given values for the thermodynamic properties

of selected hydrocarbons. The basic data are for the ideal gases and on

a molar basis. Those are enthalpies of combustion, heat capacities and

enthalpy differences. Combustion data are given at all three reference

conditions likely to be used in gas technology.

The ideal -gas values of thermodynamic quantities for the hydrocarbons

on a molar basis are listed in Tables ^a and 4b. The sources of the

data and derivation of the table are discussed in Appendix 7. The

values of C°(298.15 K), [if°(298.15 K) - 7/°(288.15 K)], [//°(298.15 K) -

^f°(288.71 K)], and [if°(298.15 K) - if°(273.15 K)], are given for each

hydrocarbon in Table 4a. The values of -A^i/°(298. 15 K) , -A^//°(288. 15 K)

,

-A^/f°(288. 71 K) , and -A^7f°(273. 15 K) are given in Table 4b. In Appendix

8 is found a discussion of the relationships among quantities in Tables 3a,

3b, 4a and 4b and some illustrative calculations are given.
16

Table 4. Thermodynamic data for gaseous hydrocarbons (ideal gas, molar basis)

Table 4a. Heat capacity and enthalpy

and 288.15 K, 288.71 K

di fference

and 273.15

between

K

298.15 K

Cp(298.15 K) 7/°(298.15 K) -

J

Substance

(phase)

•mol ^*K

^

= 288.15

kJ*mol

^

2

’ 2

/K = 288 .71 ?’2/K= 273.15

methane 35.71 0.355 0.336 0.882

ethane 52.48 0.519 0.490 1.275

propane 73.59 0.726 0.686 1.779

n-butane 98.44 0.972 0.918 2.384

2-methylpropane 96.59 0.953 0.900 2.332

(iso-butane)

n-pentane 119.96 1.185 1.120 2.910

2-methylbutane 118.94 1. 174 1.109 2.880

(i so-pentane)

2,2-dimethyl- 120.96 1.194 1.128 2.923

propane

(neopentane)

n-hexane 142.62 1.409 1.331 3.459

2-methyl- 142.35 1.406 1.328 3.458

pentane

3-methyl - 140.20 1.384 1.307 3.391

pentane

2,2-dimethyl- 141.55 1.396 1.319 3.419

butane

2,3-dimethyl- 139.56 1.378 1.302 3.380

butane

cyclopropane 55.59 0.546 0.516 1.328

cyclobutane 70.86 0.696 0.658 1.693

cyclopentane 83.11 0.814 0.769 1.971

cyclohexane 105.25 1.033 0.976 2.509

acetyl ene 44.06 0.437 0.413 1.077

ethyl ene 43.55 0.423 0.400 1.042

propene 64.41 0.636 0.601 1.561

benzene 85.58 0.841 0.795 2.050
17

Table 4b. Enthalpy of combustion (in kJ mol at 298.15 K, 288.15 K,

288.71 K and 273.15 K

Substance

T = 298.15 K

-A H°/kJ mol

^

cT = 288.15 K T = 288.71 K T = 273.

methane 890.31 891.24 891.19 892.65

ethane 1559.84 1561.29 1561.21 1563.49

propane 2219.90 2221.83 2221.12 2224.75

butane ZS71.25 2879.61 2879.48 2883.67

2-methylpropane 2368.72 2871.10 2870.97 2874.72

n-pentane 3535.77 3538.60 3538.44 3542.89

2-methyl butane 3528.87 3531.72 3531.55 3563.02

2,2-dimethyl- 3514.60 3517.42 3517.26 3521.71

propane

n- hexane 4194.75 4198.04 4197.85 4203.03

2-methyl- 4137.64 4190.94 4190.75 4195.93

pentane

3-methyl- 4190.32 4193.64 4193.45 4198.67

pentane

2 ,2-dimethyl

butane

4176.34 4179.65 4179.45 4184.66

2,3-dimethvl- 4184. 17 4187.49 4187.30 4192.53

butane

cyclopropane 2091.37 2092.87 2092.78 2095.15

cyclobutane 2745.16 2747.19 2747.08 2750.28

cyclopentane 3319.59 3322.19 3322.04 3326.14

cyclohexane 3952.96 3956.02 3955.84 3860.67

ethyne 1299.59 1299.91 1299.86 1300.40

ethene 1410.97 1411.90 1411.85 1413.33

propene 2058.44 2059.85 2059.77' 2061.99

benzene 3301.5KAPI) 3302.94 3302.82 3305.11
18

9. Standard- state volumetric properties of hydrocarbons

The enthalpy of combustion of unit volume of gaseous hydrocarbon is

strongly dependent on the temperature and pressure of the gas. In Table

5 are presented conversion factors (V = molar volume and p = amount of

substance (molar density) used to convert standard molar enthalpies of

combustion to standard enthalpy of combustion per unit volume of the

ideal gas for all the various reference conditions listed in Table 2.

The relationships between these quantities are discussed in Appendix 8.

The values of standard volumetric enthalpy of combustion in SI

units for the ISO/ANSI/ASTM metric reference conditions are given in

Table 6. The relationship by which these values were calculated is:

AH°(T,p)/kJ m'^ = [A^/f°(n/kJ mor^]/[V^^(r)/m^ mol'^], (9.1)

or

A^i/°(r,p)/kJ m

^ = [A^/f°(r)/kJ mol • [p^(7’)/mol m ^], (9.2)

with

v''^^ = RT/p ’ (9.3)

Either equation may be used. The use of these equations to calculate a

standard volumetric enthalpy of combustion is illustrated in the next

paragraph.

To convert the standard molar enthalpy of combustion of methane at

288.15 K to the standard volumetric enthalpy of combustion at T = 288.15

K and p = 101.325 kPa, dry basis:

A^H°(CH^, 288.15 K, 101.325 kPa)

= (-891.24 kJ mor^)/(0. 023645 m\ol’^)

= -37693 kJ m’^

-3 = -37.693 MJ m
19

This reproduces the value given for methane in Table 6. Equation 9.3 has

been used to calculate for this example and for all later conversions.

m The values in Table 5 have been rounded so that the uncertainty is in the

last decimal place shown.

Table 7 gives the standard volumetric enthalpy of combustion for the

U.S. Customary reference conditions: 7’ = 60 °F (288.71 K) and p = 14.73

-3

psia (101.560 kPa). The values are given in U.S. Customary units (Btu ft ),

but also are given in SI units to permit ready comparison with data in

Table 6. It must be emphasized that both the temperature and pressure are

different for the metric base conditions so that the conversion of values

-3

from Table 6 to Table 7 involves other factors than the ratio (Btu ft /MJ

m ^): 1 MJ m

^ = 26.8391924 Btu ft The values in Table 7 are calculated

directly from the values in Table 4a by use of the appropriate conversion

factors, equation 9.3, as follows:

A^H°(C^H^, g, 288.71 K, 101.560 kPa)/MJ m’^ =

[A^H°(C^Hj^, g, 288.71 K)/MJ mol"^]

The process can be illustrated for methane. From Table 4b, A^/f°(CH^, g,

288.71 K) = -891.19 kJ mol'^ = -0.89119 MJ mol’^. From Table 5 for the dry

gas = 0.023635 m^ mol ^ at 288.71 K and a pressure of 101.560 kPa.

A^H°(CH^, g, 288.71 K, 101.560 kPa, dry)

= -(0.89119 MJ mol“^)/(0. 023635 m^ mol’^)

= -37.706 MJ m"^

-3

To convert to units of Btu ft multiply this by the conversion factor

1 MJ m

^ = 26.8391924 Btu ft The product gives:

A^H°(CH^, g, 288.71 K, 101.560 kPa, dry)

= -(37.706 MJ m“^) x (26.8392 Btu ft"^/MJ m‘^)

= -1012.00 Btu ft‘^.

reproducing two values in the table.
20

To find the standard volumetric enthalpy of combustion of the

3 -1

water-saturated gas, use the conversion factor, 0.024054 m mol , that

is appropriate for the water-saturated gas at 288.71 K and 101.560 kPa.

Similar calculations can be made for the other gases (or for mixtures of

ideal gases) at any of the reference temperatures.

The calculation of the factors found in Table 5 is discussed in

Appendix 8. The vapor pressure of water used for calculating the factors

for the water-saturated gas, taken from Wexler's correlation [16], is

given in Table A7f in Appendix 7.
DRY

GAS

WATERSATURATED

GAS

21 (

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fO OO tn

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05 o rH tn |H« pH 05 CM 00 tn to tn to

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to

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s-

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s-3

10

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<U sz

-p

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O'

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ID

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if) to

05

S-o.

ro -PO and

p(H

20)

is

the

vapor

pressure

of

water

[16].

Only

p(CH^)

is

used

in

the

calculation

of.

Values

are

rounded.

The

equation =

RT/p

was

used

to

convert

data

in

Table 4

to

those

in

Tables 6

and

7.

60

°F.

288.706 K

used

in

calculations.
22

Table 6. Combustion data for selected gaseous hydrocarbons (ideal

gas, volumetric basis) in SI units at ISO/ANSI/ASTM reference

conditions of temperature and pressure

p = 101.325 kPa; T = 288.15

-A m

c Substance Formula Dry Basis Wet Basis (sat)^

methane CH4 37.693 37.059

ethane 66.031 64.920

propane 93.967 92.386

n-butane ^4^10 121.787 119.737

2-methylpropane Vxo 121.426 119.383

n-pentane '= 5^12 149.657 147.139

2-methyl butane S^12 149.366 146.853

2, 2- dimethyl propane ^5^12 148.762 146.258

n-hexane C6«14 177.547 174.559

2-methyl pentane C

6

H

14

172.247 174.264

3-methylpentane ^6^14 177.361 174.376

2 ,2- dimethyl butane '^6^14 176.769 173.794

2, 3-di methyl butane ^6^14 177.101 174.120

cyclopropane 88.513 87.024

cyclobutane '^4«8 116.186 114.231

cyclopentane 150.505 138.140

cyclohexane ^6^12 167.311 164.495

ethyne 54.977 54.052

ethene C2H4 59.713 58.708

propene C

3

H

6

87.117 85.651

benzene ‘^6^6 139.691 137.340

^pCH^O) = 1.70532 kPa

I
23

Table 7. Combustion data for selected gaseous hydrocarbons (ideal

gas, volumetric basis) in SI and U.S. Customary units at

ANSI/ASTM/API reference conditions of temperature and pressure

p

= 101.560 kPa; 2* = 288.71

Substance

1

-A H°/MJ m

c

Ory Basis

-A H°/Qtu ft'^

c

Water-Saturated Basis‘S

-A x¥°/MJ m'^ -A H°/Qtu ft'

c c

methane 37.706 1011.99 37.050 994.38

ethane 66.054 1772.83 64.904 1741.98

propane 94.00 2522.87 92.364 2478.97

n-butane 121.83 3269.78 119.71 3212.89

2-methylpropane 121.47 3260.12 119.36 3203.40

n-pentane 149.71 4018.06 147.10 3948.15

2-methyl butane 149.42 4010.24 146.82 3940.47

2 ,2- dimethyl propane 148.81 3994.01 146.22 3924.52

n-hexane 177.61 4766.85 174.52 4683.92

2-methyl pentane 177.31 4758.79 174.22 4675.99

3-methyl pentane 177.42 4761.85 174.33 4679.01

2, 2- dimethyl butane 176.83 4745.96 173.75 4663.38

2, 3- dimethyl butane 177.16 4754.87 174.08 4672. 14

cyclopropane 88.544 2376.45 87.003 2335. 10

cyclobutane 116.23 3119.44 114.20 3065.16

cyclopentane 140.55 3772.33 138.11 3706.70

cyclohexane 167.37 4492.04 164.46 4413.88

ethyne 54.997 1476.08 54.041 1450.40

ethene 59.735 1603.23 58.695 1575.34

propene 87.147 2338.96 85.631 2298.27

benzene 139.74 3750.50 137.31 3665.25

Molar volumes at 60 °F (288.706 K) of 0.0236355 and 0.0240540 m^ mol' have been used

to obtain A H°/MJ mol

c ^P(H20) = 1.76734 kPa.
24

10. Properties of the real gases

The effects of non-ideality on the properties of the real gases are

based on the equation of state of the gases as represented by the virial

coefficients. The second virial coefficients, B{T)

are given for the pure

substances in Table 8a at the four temperatures of interest. The derivation

of this table and the relationships to the enthalpy and the volume of the

real gas are discussed in Appendix 9. In addition, a second degree equation

is given there for the virial coefficient of each gas. Table A9a. Table A9c

in the Appendix gives the enthalpy increment, H-H°on a molar basis caused

by the non- ideality of the gas.

Table 9 gives the real -gas enthalpy of combustion for the hydrocarbons

in SI units for the ISO/ANSI/ASTM metric reference conditions of measurement

(T ~ 288.15 K, p = 101.325 kPa) on molar, mass and volume bases. Table 10

gives the real -gas enthalpy of combustion in U.S. Customary units for the

ANSI/AST.M/API reference conditions for U.S. Customary units, T = 288.71 K

(60°F), and p = 101.560 kPa (14.73 psia), also molar, mass and volume bases.

The corrections themselves are included in Tables 11 and 12.

Table 9 is derived directly from Table 4a and the virial coefficients

using the cycle (all processes at the same temperature and pressure):

real: C^Hj^(g) + (a + b/4) 02 (g) aC02(g) + (b/2) H20(g)

I'l 4-2 t3 t4

ideal: C^H^(g) + (a + b/4) 02 (g) ^ aC02(g) + (b/2) H20(g)

A^/f(real gas) = - [//-//°](C^Hj^,g) - (a+b/4)[f/-/f°](02,g)

+ + a[H-/f°](C02,g)

+ (b/2)[Zf-/2°](H20,g)

The calculation is made for methane at 288.15 K and 101.325 kPa as an

example. For methane a = 1, b = 4; = -17.0 J mol ^ (Table A9c); for
25

oxygen, - -8.72 J mol for CO2, H-H° - -42.7 J mol ^ and for H20(l),

is taken to be zero. A g, 101.325 kPa, 288.15 K) is -891.24

kJ mol ^ (Table 4b). From these values;

g, 101.325 kPa, 288.15 K)/kJ mol“^ =

-891.24 - 0.0427

-(-0.0170) - 2 (-0.00872) =

-891.2483

which rounds to -891.25 kJ mol The adjustment is barely significant in

this case.

By a similar procedure, as indicated in Appendixes 8 and 9, A^/f - A^7f°

can be calculated for any hydrocarbon and any base conditions of temperature

and pressure for which data are given, using Table 4a or 4b and Table A9c of

the Appendix.
Table

8a.

Second

virial

coefficients,

B{T)

for

pure

substances

26

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Table

8b.

Second

virial

coefficients,

for

binary

mixtures

28

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Q. <D CO ^ E 1— O + •' p On-ct >>

iH 3 -C CD + JZ

. \ Q. 0! C^ P r- r—

1

-p r— C •> r— m d) O dJ CM CU CO CO cn On CM E CM C CD E

-M CM jz cn • r—

CO ~ •1— 0) 4-> r—

1

c «• •P - O CO X cn •a C\J r— <V <u cn (u cn 1 iH d> CM 1 CO

XJ E a. - E •> CM » xz * CM ** fO 1 CM 1 CM 1 CM -CM 1 CM •> CsJ (— CM C 1—1 CM 1—

1

CM 1—1 C 1—

1

CM

<T>X

ac

cuQ.

Q. <CC

C

(U>

CTD

S-

<T3

to

0)U

c

(U

t.

O) 4-

<U

i- T3C

(T3

to

Ol +Jo

c

+Jo

o
30

Table 9. Enthalpy of combustion of the real-gas hydrocarbons on molar,

mass, and volume bases; in SI units and at ISO/ANSI/ASTM

metric reference condition

P = 101.325 kPa; T = 288.15 K

Substance

-A H

c -A Hc -A H

c kJ mo 1 ^ MJ kg"^ MJ m'^

methane 891.25 55.5537 37.7688

ethane 1561.3 51.922 66.597

propane 2221.8 50.384 95.675

n-butane 2879.4 49.540 126. 197

2-methyl propane 2870.9 49.394 125.468

n-pentane 3538.1 49.038 159.726

2-methyl butane 3531.5 48.946 157.583

2,2-dimethlypropane 3517.1 48.748 155.848

n-hexane — — —

2-methylpentane 4190.4 48.626 193.840

3-methyl pentane 4193.1 48.657 193.475

2 , 3-di methyl butane 4187.1 48.588 191.375

ethene 1411.9 50.328 60.0977

propene 2059.8 48.949 88.4289

benzene 3302.5 42.278 150.785
31

Table 10. Enthalpy of combustion of the real-gas hydrocarbons on

molar, mass, and volume bases; in U.S. Customary units and

at ANSI/ASTM/API reference conditions

p = 101.560 kPa; T = 288.71 K

(p = 14.73 psia; T = 60 °F)

Substance

-A H

c -A H

C V -A H

C V

BtUjjmol ^ BtUjjlb ^ MJ m'^ Btu ft

^

methane 844.69 23882.4 37.7804 1014.00

ethane 1479.7 22321 66.6159 1787.92

propane 2105.7 21660 95.6987 2568.48

n-butane 2729.0 21298 126.218 3387.59

2-methyl propane 2721.0 21235 125.488 3367.99

n-pentane 3353.3 21082 159.714 4286.59

2-methyl butane 3347.0 21042 157.598 4229.81

2 ,2-dimethlypropane 3333.4 20957 155.858 4183.11

n-hexane — — — —

2-methyl pentane 3971.6 20905 193.828 5202.19

3-methyl pentane 3974.2 20918 193.458 5192.28

2, 3- dimethyl butane 3968.5 20888 191.378 5136.44

ethylene 1338.2 21637 60.1167 1613.49

propene 1925.3 21043 87.2863 2342.62

benzene 3130.1 18176 150.776 4046.72
32

11. Volumetric enthalpy of combustion of the real gas

The molar volume of the real gas and the volumetric enthalpy of

combustion at ISO/ANSI/ASTM metric reference conditions (p = 101.325 kPa; T

288.15 K) are given in Table 11. An estimate of the fractional uncertainty

of the molar volume is also given. The uncertainty of the molar volume

determines the uncertainty in the volumetric enthalpy of combustion.

The molar volume of the real gas and the volumetric enthalpy of

combustion at ANSI/ASTM/API U.S. Customary reference conditions are given

in Table 12. An estimate of the uncertainty of the molar volume is also

given. This factor also gives the approximate contribution of the

uncertainty in the volumetric enthalpy of combustion. The molar volume

is given both in cubic metres and cubic feet, and the enthalpy of combustion is given in both megajoules per cubic metre and in British thermal

units per cubic foot.

The calculation of the real gas molar volume is based on the equation:

pV /RT = 1 + B(T)/V (11.1)

m m for which values of B(T) are taken from Table 8a. This calculation will

be illustrated for methane for the conditions p = 101.325 kPa; T = 299.15

K. From Table 8a, S(288.15 K) = -(47.0 ± 1) cm^ mol ^ for methane. The

exact quadratic solutiorv of this equation is:

^m'^^m^ = I

+ ^

(1 + , (11.2)

where the ideal gas volume is RT/P,

= (8.31441 X 288.15/101325)

= 0.02364468 m^ mol"^,

or may be obtained from Table 5. To make commensurate with B, ^ m m 3-1

is converted to 23644.7 cm mol . Substitution of these values gives:
33

3 -111 1 /?

Vjj^/23644.7 cm'^ mol | ^

(1 - 4 x 47.0/23644.7)'^'^^

= ^

+ I

(0.992049)^^^

= 0.9980083

V = 23644.7 cm^ mol'^ x 0.998008 = 23597.6 cm^ moT^

m y = lO"^ cm’^ X 23597.6 cm^ mol'^,

m = 0.023598 m^ moT^,

in conformity with the value in Table 11.

The volumetric enthalpy of combustion of the real gas is obtained

from the molar enthalpy of combustion of the real gas at the same reference conditions (Tables 9 and 10) using the following relationship;

A i//MJ m'^ = 0.001 (A H/kJ mor^)/(V /m^ mol”^) (11.3)

c c m This may be illustrated for methane for which A 7f/kJ mol ^ - -891.248

c (from Tables 4 and 9) and niol ^ = 0.0235976 (from Table 11). The

value of;

A /f/MJ m‘^ = -891.248 / 0.0235976,

c = -37768.6 kJ m’^,

= -37.769 MJ m’^.

in conformity with Table 9.

-3

The volumetric enthalpy of combustion in Btu ft in Table 10 is

"3

obtained from the volumetric enthalpy of combustion in MJ m in Table

12 using the units Btu^j and foot defined exactly in Table A4 in terms

of the joule and the metre.

The molar volumes have an ascribed uncertainty because of the

uncertainty in the virial coefficients. This uncertainty is significant,

particularly in the cases of the higher hydrocarbons. The uncertainty

in the molar volume may be calculated by differentiating equation (11.2)

to obtain the relative error;
34

6V = (65/y^^)/(l + mm m m

3 and substituting 6B from Table 8a (last column). Taking the value 1 cm

*1 "6 3 "1

mol (= 10 m mol ) for methane from Table 8a, we calculate

^^m'^^m^ = (1/23645)7(1 + 4- (-47)/23645)°‘

= 3.79 X lO‘Vo.992

= 3.8 X lO’^
It 35

Table 11. Enthalpy correction to the real gas and molar volume,

at ISO/ANSI/ASTM metric reference conditions

p = 101.325 kPa; T = 288.15 K

= 0.023 644 7

A /f - A 7f' c c m 6V /y ' m m Substance

J mol“^ 3 ,-l

m mo 1

methane -8.3 0.0235976 4 X 10‘^

ethane 7.0 0.02.34439 8 X lO"^

propane 54.5 0.0232222 8.6 X lO"^

n-butane 154.9 0.0228167 8.6 X lO"^

2-methylpropane 191.9 0.0228815 1.8 X lO"^

n-pentane 482.3 0.0221511 2.7 X lO'^

2-methyl butane 260.1 0.0224102 1.8 X lO'^

2 ,2- dimethyl propane 284.3 0.0225677 1.5 X lO'^

n-hexane — — —

2-methyl pentane 473.6 0.0216181 9.2 X lO"^

3-methyl pentane 482.6 0.0216727 —

2, 3- dimethyl butane 363.6 0.0218792 —

ethene -12.0 0.0234936 4 X lO"^

propene 32.1 0.0232935 —

benzene 428.0 0.0219024 1.4 X 10“^

^Relative error in the real -gas molar volume.
36

Table 12. Enthalpy correction to the real gas and molar volume, V^, at

ANSI/ASTM/API U.S. Customary reference conditions

P = 101.560 kPa; T = 288.71 K

= 0.023 635 4 m^ mol'^; 1 ft^ = 0.028 316 847 m^)

Substance

A - A V

c c m

1 .-1 3 ,-l

J mol m mol

y

m 6V/V

m m

methane -8.0 0.023 5889 0.833 034 4 X lO"^

ethane 7.4 0.023 4359 0.827 631 8 X lO'^

propane 55.2 0.023 2152 0.819 837 8.6 X 10‘^

n-butane 181.7 0.022 8121 0.805 602 8.6 X 10‘^

2-methyl propane 186.7 0.022 8770 0.807 894 1.8 X lO’^

n-pentane 476.3 0.022 1519 0.782 287 2.7 X lO"^

2-methlybutane 261.3 0.022 4069 0.791 292 1.8 X loT^

2,2-dimethylpropane 285.3 0.022 5652 0.796 883 1.5 X lO"^

n- hexane — — —

2-methyl pentane 476.9 0.021 6185 0.763 450 9.2 X lO'^

3-methyl pentane 480.9 0.021 6737 0.765 399 —

2,3-dimethylbutane 361.9 0.021 8778 0.772 607 —

ethene -11.7 0.023 4854 0.829 379 4 X 10“^

propene 33.0 0.023 5982 0.833 362 —

benzene 425.5 0.021 9026 0.773 483 1.4 X 10‘^
37

12. Calculating the heating values of gas mixtures

The heating value of any fuel gas mixture can be calculated from

the molecular composition of the gas and information about the heating

values and volumetric behavior of the individual components. It is

impractical to calculate heating values for all probable compositions

and so the information presented here is limited to procedures for

maicing the calculations.

Composition . It is essential that the composition of the gas be known

in order to use these procedures. The composition values must be identifiable as mole fraction or mass fraction of each component and these

fractions must account for all of the gas.

The most useful description of the composition is the mole fraction,

X, and most of the procedures described here are based on this parameter.

The mole fraction of component i is x(i), where i has the values 1 to id,

and n is the total number of components. It is necessary that:

x(l) + x(2) + . . . + x(n) = 1. (12.1)

Methods of analysis based upon cal ibration with mixtures of known

composition give results in the same terms as the calibration sample.

Therefore the meaning of the composition of the calibration sample should

be unambiguous if the results of analyses are to be used for the calculation

of heating value. Volumetric analyses of gases, give volume fractions of

the components that approximate the mole fractions. It should be realized

however that volumes of gases are not precisely additive when the gases are

mixed, and so volume fraction is somewhat ambiguous. Masses are additive

and amounts of substance (moles) are additive and so compositions of gases

stated as mass fraction or mole fraction are unambiguous.
38

Three procedures for calculating heating value based on information

in this document will be described:

(1) The heating value of an ideal gas mixture.

(2) The heating value of a real gas mixture, neglecting molecular

interactions between different substances.

(3) The heating values of real gas mixtures including molecular interactions between different substances.
39

12.1 Heating value of an ideal gas mixture

The composition of the gas mixture in terms of the mole fractions

x^i) of the individual substances present is presumed to be given, and

the sum of the x(i) is unity (equation 12.1).

The enthalpy of combustion of the ideal gas mixture is calculated

by adding the molar enthalpies of combustion of the individual components

weighted according to their mole fractions using the equation:

A //“(mixture) = x(l)A H°(l) + x(2)A H° (Z) + ... + x(n)A H°(n)

where is the enthalpy of combustion per mole, of the ideal gas.

The enthalpies of combustion are taken from Table 4b. By summation

of terms as illustrated in Table 13 for a typical gas, the ideal -gas

heating value is calculated.

For the example, the value found by this procedure is A^//° (mixture) =

-922.01 kJ mol Using the appropriate molar volume from Table 5, this

can be converted as shown in Table 13 to A H° = -38.994 MJ m

^ or = -1046.6

c V -3

Btij ft , where is the enthalpy of combustion of unit volume of the

gas mixture (dry).

For ideal gases the volume fractions are proportional to the mole

fractions, and so an alternative approach may be used. Heating value

per unit volume A H° can be taken from Table 6, and the same form of

equation can be used.

A^H° (mixture) = x(l)A^/i°(l) + x(2)A^H°(2) + ... +x(7})A^/f°(ri) (12.3)

As shown in Table 13 these data lead to the same result.

For U.S. Customary reference conditions. Table 4b and Table 5, or

Table 7, may be used in the same way using either alternative approach.
40

Table 13. Sample calculation of enthalpy of combustion of

a gaseous mixture assuming ideal gas behavior

(p = 101.325 kPa; T = 288.15 K)

(V = 0.023 6447 m^ mol’^)

m

Component

(i) M( 1 )^ x(i

-A //°(i) - cm -A //°(i)

c x(i)A 7/°(i)

kJ mol

^ kJ mol^ MJ m"^ MJ m“^

CH^ 16.043 0.9000 891.24 802.116 37.693 33.9237

C

2 H 4

28.054 0.0037 1411.90 5.224 59.713 0.2209

30.070 0.0550 1561.29 85.871 66.032 3.6318

N

2

28.0134 0.0313 0 . 0 ^ 0.0 0.0 0.0

n-C4Hio 58.123 0.0100 2879.61 28.796 121.787 1.2179

Sums 1.0000 -A H° = c m

= 922.007 -A H°

C V = 38.9943

-A H° (mixture) = cm 922.007 kJ mol‘^

0.023 6447 m^ mol = 38 994. 2 kJ m"^

-A H° (mixture) =

C V

^ ^ 38 994.2 kJ m*^ • 26.839 192 (Btu/ft^)/(MJ m

- 3

)

= 1046.6 Btu ft

^ M(i) is molecular weight and x(i) is mole fraction.

^Molecular nitrogen is the defined combustion product. Corrections are

made for nitrogen oxides in the calorimetric experiments.
41

12.2 Heating value of a real gas mixture, neglecting

interactions between different substances

The composition of the gas mixture in terms of the mole fractions

x(i) of the individual substances present or in terms of the mass fractions

p/(i) of the individual substances present is presumed to be known and

Ix(i) is unity (equation 12.1) or Iiv(i) is unity (equation 12.4).

w{l) + w{2) + ... + v(z2) = 1 (12.4)

The enthalpy of combustion of the real gas mixture on a molar basis,

A^H^(mixture) is calculated by the equation:

A^/f|^(mixture) = x(l)A^if^(l) + x(2)A^/f^(2) + ... + (12.5)

Similarly the enthalpy of combustion of the real gas mixture on a mass

basis, AH (mixture) is calculated by the equation:

c w

A^H^(mixture) = w(l)A^H^(l) + + ... + (12.6)

Values for AH (i) or A H (i) are obtained from Table 9 for ISO/ANSI/ASTM

c m^ ^ c w^

metric reference conditions and from Table 10 for ANSI/ASTM/API U.S. Customary

reference conditions.

The enthalpy of combustion of the real gas mixture on a volumetric

basis, -A H (mixture), is obtained from -A H (mixture) using the relationship

5 c V ’ c m

^

-A H (mixture) = -A H (mixture)/V’„(mixture) . (12.7)

c V cm m The value for -A^H^(mixture) to use in equation (12-7) is obtained from

equation (12-5). The value of V^(mixture) is obtained from equation

( 12 - 8 ).

V^(mixture) = x(l)V’^(l) + x(2)V^(2) + ... + x(n)V|^(i2 ) . (12.8)
42

This equation assumes that volumes of real gases are additive in mixtures.

These calculations for a simple mixture are illustrated in Table 14.
43

Table 14. Sample calculation of enthalpy of combustion of a real-gas

mixture assuming no interactions between different substances

(p = 101.325 kPa; T = 288.15 K)

Component

(i) M(i) x(i) V') -A /f (i) cm -x(i)AcH^d) x(i)y^(i)

^

3 ,-l dm mol kJ mol

^

kJ mo 1 ^ ^ 3 ,-l dm mol

(Table 11) (Table 9)

CH4 16.043 0.9000 23.5976 891.25 802.125 21.2378

C2H4 28.054 0.0037 23.4936 1411.9 5.224 0.0869

30.070 0.0550 23.4439 1561.3 85.872 1.2394

'2 28.0134 0.0313 23.6376 0.0^ 0.0 0.7389

^'VlO 58.123 0.0100 22.8167 2879.4 28.794 0.2282

Sums 1.0000 -A H

c m = 922.015 V

m = 23.5821

-A n (mixture) = -A i/ (mixture)/y„(mixture) cv cm m = 0.922 015/0.023 5821

= 39.098 MJ m'^

-3

This is greater than -A^//°(mixture) (38.994 MJ m from Table 13) on

the assumption of ideal-gas behavior by a factor of 1.0027, or different by 2.7 parts in 1000.

^See note b, Table 13.
44

12.3 Heating value of a real gas mixture including

molecular interactions between different substances

The composition of the gas mixture in terms of the mole fractions

x(i) of the individal substances present, or in terms of the mass fractions

w(i) of the individual substances present, is presumed to be known, and

Ix(i) is unity (equation 12.1) or lw{i) is unity (equation 12.4).

The calculation of the heating value involves five steps: (1) the

calculation of the ideal gas molar heating value of the mixture using

equation (12.2); (2) the calculation of the virial coefficient of the

gas mixture and its temperature derivative using equations taken from

Appendix 9; (3) calculation of the molar volume of the real gas mixture

based upon the virial coefficient of the gas mixture from Step 2; (4)

adjustment of the molar heating value for the term H - H° based on the

virial coefficient of the gas mixture and its derivative; (5) calculation

of the heating value on a mass basis and the volumetric heating value of

the real gas mixture. The molar heating value A^i/^( mixture) can be

converted to the heating value on a mass basis A H (mixture) by use of

G W

the mass fractions of the substances in the gas. From the molar volume

calculated in step 3, the molar enthalpy of combustion can be converted

to the volumetric enthalpy of combustion, A^ff^(mixture).

This calculation procedure is considered to be the most accurate of

those presented. It is applicable to both dry and humid gases, including

the water-saturated gas.

Step 1 . Calculation of heating value of the ideal gas mixture. This

step is carried out as described in Section 12.1 and illustrated in

Table 13.
45

Step 2. Calculation of the second virial coefficient S(mixture) of the

real gas mixture, and its derivative dS/dr. As described in Appendix 9,

the second virial coefficient S(mixture) is a function of T and of the

virial coefficients of the components and of the virial coefficients of

the binary mixtures of the components. From Appendix 9 we use the

simplified equation (12.9) for which data are given in Tables 8a and 8b.

5(mixture) = [x(l)]^5(l) + [x(2)]^S(2) -i-. . . + [x(ii)

+ 2x(l)x(2)5(l,2) + 2x(l)x(3)5(l,3) + ... 2x(l)x(z2)5(l

,

21 ) (12.9)

Component 1 is taken to be methane in every case (See Appendix 8). This

equation is applicable only to natural gas containing methane as the

major component. For other gases See Appendix 9.

Using values of x(i) from Step 1, values of S(i) from Table 8a and

values of 5(1, i) from Table 8b, calculate 5(mixture) using equation

(12.9). The values of 5(i) and 5(1, i) must be those for the temperature

at which the heating value is desired. For calculations at temperatures

other than those listed in Table 2, see the procedures described in

Section 10 and Appendix 9. Calculate c?5(mixture)/dr using equation

( 12- 10 )

d5(mixture)/dr = [x(l)]^[d5(l)/dr] + [x(2)]^[d5(2)/d2’] (12.10)

+ ... + [x(22)]^[d5(n)/dr]

+ 2x(l)x(2)[d5(l,2)/dr] + 2x(l)x(3)[d5(l,3)/d7’]

+ ... + 2x(l)x(22)[d5(l,22)/d7’].

For equation (12.10) obtain values of d5(i)/d7’ and d5(l,i)/dr at the

desired temperature from Table A9b in the appendix. The calculation of
46

dS(mixture)/d7’ may be omitted if step 4, calculation of the value H - H°

is omitted.

Step 3 . Calculate V^Cmixture) for the gas mixture using equation (12.11).

V^(mixture)/V^(ideal ) ~ ^ J 4B(mixture)/V^^]^^^ (12.11)

where may be taken from Table 5 or may be calculated as = RT/p,

and S(mixture) is obtained from equation (12.9).

Step 4 . Correct to the real gas by calculating H - H° for the gas mixture

using equation (12.12)

H - H° = (RT/V^)lB - TidB/dT)] (12.12)

For this equation use ^(mixture) and dB(mixture)/d2’ from step 2, V’^(mixture)

from step 3 and T, the selected temperature for which the heating value

is required. Calculate A if - A if° as described in Section 10. In order

to do this, calculate the equivalent molecular formula of the gas mixture

C^Hb to be used in the equation:

C^H^ + (a + b/4)02(g) = a C 02 (g) + (b/2 )H 20 (l).

Step 4 and the required calculation of dS(mixture)/dr can be omitted

with errors not exceeding an estimated 50 J mol

Step 5 . Calculate A^if^(mixture) , and A^if^(mixture) using equations

(12.13) and (12.14)
47

A H (mixture) = A /f°(mixture) + (A i7 cm cm cm

Step 1

A //°)(mixture)

c m^ ^

Step 4

(12.13)

A H (mixture) = A (mixture)/y (mixture) CM ^ c m^ ^ ^

Step 5 Step 3

(12.14)

(or alternatively obtain A^/f°(mixture)/y^^(mixture) as described in Step

4 ).
48

13. Uncertainties

The following sources of uncertainty are considered in this document:

(1) Uncertainty in the standard molar enthalpy of combustion of the

pure substance at reference temperature T = 298.15 K.

(2) Uncertainty introduced by conversion of data at the reference

temperature 298.15 K to r = 288.15 K, 288.71 K, and 273.15 K, for

the pure substances.

(3) Uncertainty introduced by the conversion of the standard molar

enthalpy of combustion to standard specific enthalpy of combustion

for the pure substances (mass basis).

(4) Uncertainty introduced by conversion of the standard molar enthalpy

of combustion to the standard volumetric enthalpy of combustion of

the pure substances.

(5) Uncertainty in the relation of the molar or specific enthalpy of

combustion of the real gas to that of the ideal gas for the pure

substances.

(6) Uncertainty in the relation of the volumetric enthalpy of combustion

of the real gas to that of the ideal gas for the pure substances.

(7) Uncertainty in the enthalpic effects of mixing of components of a

gas mixture.

(8) Uncertainty in the volumetric effects of mixing of components of a

gas mixture.

The inaccuracy in the enthalpy of combustion of a given quantity of

gas depends upon errors in actual measurements of mass or volume, temperature, pressure, and composition of the gas under consideration. The

detailed consideration of these errors is considered to be beyond the

scope of this document.
49

The uncertainties in the source data are included in various tables:

(a) Uncertainties of enthalpies of formation of auxiliary substances,

Table 3b; (b) Uncertainties in enthalpies of combustion, heat capacities

and enthalpy increments, Table 15; (c) Uncertainties in virial coefficients.

Tables 8a and 3b; (d) Uncertainties in physical constants Table 1 and

later in this section. In addition some discussions of these uncertainties

are given in Appendixes 7 and 10.

Propagation of uncertainties .

The propagation of uncertainties in the calculation of the quantities in

this document is carried out using the equation recommended by the lUPAC

Commission on Thermodynamics [17].

For the function:

F = nx,Y,Z) (13.1)

that is a function of three independent variables, the estimated uncertainty s„ in F is given by:

r (s^)^ = idF/dx)^{s{x)]^ + OF/3y)^{s(y)}^ +

(dF/dZ)^{s(Z)}^ . (13.2)

The application of this propagation-of-error equation is discussed in

Appendix 10 for particular classes of functions important in this document.

The variables are considered to be independent in each step of

calculation except in the case of fuel mixtures for which there is a

constrai nt:

lx. = 1 . (13.3)

The procedure for reducing this set of variables to an independent set

is discussed in Appendix 10 in connection with the propagation of errors

for virial coefficients of mixtures.
50

The calculation of desired quantities from observed data follows a

logical sequence in this document. The flow charts of the calculations

showing many of the equations used are given in figures 1 and 2 which

reflect the order used in the presentation of the material in the main

text to this point. The calculation of uncertainties and their propagation

is carried out stepwise, as listed above, the uncertainty of each quantity

incorporating the uncertainties in the variables involved, for which the

uncertainties would have been calculated in a previous step.
51

k

to

0)

sO)

tu -pc

-pc

fO p

s_

cuu

c

3

D

01 -P

fO

to LU

03

ID

rH o o OS o C7J OO OO OO Lf) IX) tn OS o o 00 o o o tn tH

1 1 o o

O rH rH

CO rH rH o rH o rH rH rH rH »H rH rH rH o rH rH rH rH rH rH tH o rH

a; CSJ o o o o o o o o o o o o o o o o o o o o o o 0 X X

1 II o o o o o o o o o o o o o o o o o o o o o o O OO lD

c\j

tn o

E

to

C\J

b=

03C

>>

-Co

C\J

LD

00

CT>

C\Ja. o

o

E

o

a:

o

oE

lD

^<4 00

on

csj o

a:<

VW

CO

u

I

Lf5

I I Cl (Ti ^ Lf) cf OO OO LD Lf) tn to to to OO tn to o o o

o C\i rH o o o o C3 o o o o o o o o o o o o o o o o rH o rH

5- • o o o o o o o o o o o o o o o o o o o o o o

Q. SJ 00 X « X X

00 o o o o o o o o o o o o o o o o o o o o o o tn CNj

tn CSJCSJ«:j-CSJ«^-OOOOOOOOOtDir)'^CSJC\JOC\JC\JC\10'^C\JO<M «^^ro^ootf3Lf)LnLnLnLf)tjOtDoo'^':fLn>^':3‘<^‘tDO^oo

ooooooooooooooooooooooooo

OLDtooororo'^LntDLnoocMoooO'—itoooocvjto

r^tD«!d-LnLnt-trHr-!rHCSJCSlC\JCNJC\lt—

CSJ to rHCSJ-;l-c^tDtDOOOLf)LntDC\J^Or-JOOC^C\jr^«:t OO«:3"^’^'!d'tf)tDOO0^<TlinLf)f^tDtD'^tDLn

00

CSJ o

OrHt-liHTHOOOOtHrHOOOOOOOOOO o o

0)c 0) cu

03 c c

Q. OJ OJ OI O 0) OJ p p

c OJ i- c c 3 3 OJ ,-~s 03 c Cl 03 03 P P p p

Q. 03 r— p P r— 0) OJ O p >) c c >> >» C 0) c 0) X 3

s- 3 x= 0) OJ P P 03 C 03 c o CT

01 Q. OJ X3 p a. Cl P P P 03 P 03 •r— u OJ <— c >, 0) OJ 1— 1— OJ CU O P c X a 1— c OJ 0) c >» 03 E c >1 > E E S- 3 OJ OJ 0) OJ s—

/

(0 c 0) C 03 x: .p £ •r^ 03 .c. x: •f— PP pp cu OJ c c c c c

-p 03 c 03 -P P c P “D X p p O XJ O O o o c c OJ 0) OJ o S- o

to JZ 03 Q. 3 0) 0) <u 1 cu 0) 0) 1 1 ^ r— 1— >i 0) p N 0)P OJ p

X3 p o xa E Q. E CSJ x: E E CSJ 00 u u u u p p o c >> S- p

3 0) -p i~ 1 1 1 1 1 1 1 •• •> >1 >> >> >,p p i~ 0) X OJ OJ OJ to E 0) a. c CSJ c CSJ CSJ c CSJ OO CSJ CSJ u u u u 0) OJ pp o u 3 u
52

In order to place the overall uncertainties of the values at various

stages listed above in proper perspective, the sources of uncertainty

are first discussed for specific factors:

(1) Relative atomic masses (atomic weights)

On the basis that the uncertainties in the atomic masses of

the elements have the values given in Table A3d the uncertainties in the molecular weights of the compounds of interest are

as follows, where s is the uncertainty in atomic weight units

and

s/M is the fractional uncertainty.

Substance s 10^ s/M

atomic mass unit

CH^ 6.2 0.4

CH 3.8 0.3

CO

2

9 0.2

H^O 4.6 0.3

The values for CH^ and CH represent the extremes of C:H ratios encountered in this document, and relative uncertainties for other hydrocarbons

will fall in between. The relative uncertainty introduced by those of

-5

the atomic masses is 3 or 4 x 10 . The calculation of these values by

simple addition of uncertainties, and an alternative calculation based

on quadratic addition of uncertainties are shown in Table A5d, along

with some comments about the sources of the estimates.

(2) Uncertainty in the gas constant R

The uncertainty in R is 0.00026 in 8.31441 J mol ^ K

^ or a

relative uncertainty of 31 x 10
53

(3) Uncertainty in the temperature scale

The practical temperature scale is exactly equal to the thermodynamic temperature scale at r = 273.15 K. At T near 294 K,

the international practical temperature scale differs from the

thermodynamic temperature scale by amounts estimated by Guildn&r

and Edsinger [18] at about 0.0025 to 0.0030 K and at T near 315 K

by about 0.0065 to 0.0082 K, with deviations decreasing to very

small values as T approaches 273.15 K. From these we can estimate

that at r = 298.15 K the error will be about 0.004 K, which cor-

-5

responds to a fractional error in temperature of 1.3 x 10 . This

is the maximum error that would be expected in the temperature

scale in the range 273.15 K to 298.15 K. In addition, by rounding 288.7055 K (60 °F) to 288.71 K an additional error of 0.0045 K

can be introduced if this approximation is made, and additional

-5

relative error of 1.6 x 10 . (Usually, we have avoided this by

using 288.706 K).

(4) Relative uncertainties in the ideal gas molar volume

We use equation (25) Appendix 10 to calculate the relative

id id uncertainties . For defined reference conditions p l/(r,p) T,p

is defined constant and so S = 0; r can be considered as defined

P

constant or as defined by IPTS-68 which differs from T (thermodynamic) as described in paragraph 3. Thus s can be taken as zero

-5 or s/T can be taken as 0 at r = 273.15 K, and 1.3 x 10 at other

temperatures used in this document. For we take the uncertainty

stated above for i?, 26 x 10 ^ J mol ^ K or s^/R = 31 x 10

Using these data, and assuming the additive worst case of calculations using T = 288.70555 rounded to 288.71 and an error of 1. 3 x

10 in the deviation of T at 60 °F, we obtain the relative errors
54

in molar volume tabulated in the last column below,

valid at all standard pressures at the temperatures

i ndicated.

T’/K lO^s^/R lO^s^/T s /p ^V{T,py^p,T

273.15 31 0 0 31 X lO'^

288.15 31 8 0 32 X lO"^

288.71 31 29 0 42 X 10“^

-6

298.15 31 13 0 34 X 10

°

For the equations to be used for calculation of uncertainties

of types 2 through 8, listed at the beginning of this section

see the section on propagation of uncertainties in Appendix 10.
55

General list of references

[1] (a) Armstrong, G. T. ; Domalski, E. S. ; Minor, J. L. Jr.; "Standard

combustion data for the fuel gas industry," 1972 Operating Sect ion

Proceedings, American Gas Association, Arlington, VA. , pp. D-74 to

D-87. (b) Armstrong, G.T. ; "Calculation of the heating value of a

sample of high purity methane for use as a reference material'; Nat.

Bur. Stand. (U.S.) Technical Note 299, U.S. Government Printing Office,

Washington, D.C.

[2] McGlashan, M. L. ; Paul, M. A.; Whiffen, D. H. ; "Manual of symbols

and terminology for physicochemical quantities and units", Pergamon

Press; Oxford (1979). See also: Pure Appl . Chem. (1979), 1-41.

[3] lUPAC Commission on Thermodynamics; "Manual of symbols and terminology

for physicochemical quantities and units. Appendix IV; Notation

for states and processes, significance of the word standard in

chemical thermodynamics and remarks on commonly tabulated forms of

thermodynamic functions."; Pure and Applied Chemistry, (19S2) in

press. Provisional version in Pure and Applied Chemistry, (1979)

bl 393-400.

[4] International Organization for Standardization. ; "Quantities and

units of physical chemistry and molecular physics," International

Standard ISO 31/VIII First edition 1973-04-01.

[5] (a) International Organization for Standards; "SI units and recommendations for the use of their multiples and of certain other units";

ISO/1000-1973. (b) Page, C. H. ; Vigoureux, P. editors; "The International system of units (SI)"; NBS Special Publication 330, (1974).

(c) American Society for Testing and Materials (ASTM). "Standard

for metric practice"; ANSI/ASTM E 380-76; IEEE Std 268-1976.
56

[6] lUPAC Commission on Atomic Weights and Isotopic Abundances; "Atomic

weights of the elements 1981"; Pure and Applied Chemistry (1982) in

press.

[7] CODATA Task Group on Fundamental Constants; "Recommended consistent

values of the fundamental constants, 1973"; CODATA Bulletin 11,

December 1973. See also Cohen, E.R.; Taylor, B.N.; J. Phys. Chem.

Ref. Data (1973), 2, 663.

[8] lUPAC Commission on Thermodynamics and Thermochemistry; "A guide to

procedures for the publication of thermodynamic data."; Pure and

Applied Chemistry (1972), 399-408.

[9] (a) International Organization for Standardization; "Petroleum liquids

and gases - Measurement - Standard reference conditions"; ISO 5024-

1976 (E). (b) ASTM Committee D-3 on Gaseous Fuels; "Standard methods

for volumetric measurement of gaseous fuel samples"; ANSI/ASTM D 1071-

78a, Am. Soc. Testing and Materials, Philadelphia 1980. In Annual Book

of ASTM Standard's, (1980), part 26.

[10] ANSI Committee Z 132, "Base conditions for the volumetric measurement

of natural gas"; American National Standard Z 132.1 - 1969; API

Standard 2562, First Edition.; "Base conditions of pressure and

temperature for the volumetric measurement of natural gas"; American

Petroleum Institute, New York, N.Y. 10020. But see also "Measuring,

Sampling, Testing, and Base Conditions for Natural Gas Fluids"; API

Publ 2529, 3d Edition, (1979), p. 1.

[11] Wilhoit, R. C; Private communication 5 October 1981.

[12] Osborne, H. S. ; Stimson, H. G. ; Ginnings, D. C. ; J. Research Nat.

Bur. Standards (1939), 127.

[13] Desorbo, W. ; Tyler, W.W. ; J. Chem. Phys. (1953), 1660. See also

CODATA Task Group, J. Chem. Thermodynamics (1972), 4, 331-336.
57

[14] (a) CODATA Task Group on Key values for Thermodynamics; "CODATA

recommended key values for thermodynamics 1977"; CODATA Bulletin

No. 28, (April 1978). (b) CODATA Special Report 8, (1980).

[15] Domalski, E. S. ; J. Phys. Chem. Reference Data (1972), 1, 221-277.

[16] Wexler, A.; J. Research Nat. Bur. Standards. A. Phys. Chem. (1971),

75A , 213-230.

[17] Olafsson, G. ; Angus, S. ; Armstrong, G.T.; Kornilov, A.N.; J. Chem.

Thermodynamics (1981), 13, 603.

[18] Guildner, L.A. ; Edsinger, R.E.; J. Research Nat. Bur. Standards (1976),

80A, 703-738.

I
60

Appendix 1. Proposed composition limits

Table A1 gives composition limits suggested by ASTM Committee D-3 [1]

on Gaseous Fuels in a proposed model specification for pipeline quality gas.

Table Al. Proposed composition limits for pipeline quality natural gas

Composition Minimum Value Maximum Value

Major and Minor Components Mole fraction

Methane 0.65 1.00

Ethane -- 0.14

Propane -- 0.05

Butanes -- 0.02

Pentanes and Heavier — 0.005

Nitrogen and other Inert Gases -- 0.18

Carbon Dioxide -- 0.03

Hydrogen -- 0.05

Total Unsaturated Hydrocarbons -- 0.005

Carbon Monoxide -- 0.001

Oxygen - - 0.001 (1000 X 10

Trace Components mg/m^

Hydrogen Sulfide -- 5.7

Mercaptan Sulfur -- 11.5

Total Sulfur -- 22.9

Water Vapor 110.

Other Characteristics Minimum Value Maximum Value

C 3. ^ Heating value under standard conditions

Gross, saturated 35.4 MJ m 42.8 MJ m

^ (950 Btu ftZ ) (1150 Btu ft”'^)

Gross, dry 36.0 MJ m 43.6 MJ m ~ (967 Btu ft'"*) (1170 Btu ft“'^)

Relative density (specific gravity) 0.530 0.790

Liquids: The gas shall be free of water and hydrocarbons in liquid form

at the temperature and pressure at which it is delivered to a purchaser.

Solids: The gas shall be free of solid particulate substances in amounts

deleterious to the materials normally encountered in transportation and utilization of the gas.

Gases: The gas shall be free of other gases that could adversely effect

the transportation or utilization of the gas.

(a)

[ 1 ]

-3 -3

factor 1 MJ m = 26.839 192 Btu ft under identical temperature and

pressure.

Anon; "New standard: D4087. Specification for pipeline quality natural

gas"; ASTM Standardization News (Dec. 1981) 9, [12], p. 38E.
61

Appendix 2. Symbols

Table A2. Thermodynamic symbols used in this document

T thermodynamic temperature (Kelvin scale)

P pressure (absolute)

(g) gas phase

(1) liquid phase

(c) crystal phase

concentration or amount-of-substance density (amount

of substance per unit volume, i.e., moles per unit

volume)

P^'CH^,dry) concentration or amount of substance density of methane

with no water present

p^(CH^,sat) concentration or amount of substance density of methane

when saturated with water vapor

id g)

d m

concentration of saturated water vapor, ideal gas

molar volume of ideal gas in standard state

V

m molar volume of non- ideal gas

r(sat/dry) ratio of concentration of fuel in saturated gas to

concentration of fuel in dry gas

Q heat released per unit quantity of fuel in a specified

process

Q(tota 1 ) total heat of reaction, i.e. with fuel in gaseous

state and water formed as a liquid

or H°(T) standard enthalpy at temperature T

or H{T) enthalpy in a non-standard state at temperature T,

e.g. enthalpy of the real gas
62

g, 298.15 K,

101.325 kPa) enthalpy in a non-standard state of the substance

C H, at r = 298.15 K, p = 101.325 kPa

//(state 2) -

H(state 1) enthalpy difference or increment, written here for

non standard states. Measurable, while //(state) is

not

standard enthalpy of formation from the elements

enthalpy of formation from the elements in a nonstandard state

A^°(T) standard enthalpy of formation at temperature T

Afi/°(C^H^, g,

298.15 K) standard enthalpy of formation of the gaseous substance

A H°

c

C H, at r = 298.15 K a b

standard enthalpy of combustion (ideal gas)

A H

c enthalpy of combustion (real gas)

A H°(T)

c standard enthalpy of combustion at temperature T

A^//(r,p) Enthalpy of combustion at temperature T and pressure p.

B Second virial coefficient in volumetric units

•k B Second virial coefficient in pressure units

k

B{T) or B (D Second virial coefficient, indicating that it is a functior

of temperature

B{i) Second virial coefficient of substance i

S(l,i) Virial coefficient for binary interaction of methane (1)

with another substance (i)

5(mixture) The virial coefficient of a mixture

Note: For methane (1) - mixed with one other component

(i): S(mixture) = [x(l)]^S(l) + 2x(l)x(i )5(l,i ) +

[x(i)]^S(i)
63

x(i) mole fraction of component i

v(i) mass fraction of component i

Mii) relative molecular mass of substance i

V° ideal gas molar volume

m R the gas constant

C° standard heat capacity at constant pressure

P

AC° IC° (products) - IC° (reactants)

P P P

1 compression factor, pV/RT (commonly called compressibility factor)
64

Appendix 3. Definitions of terms as used in this document.

A3.1 Terms relating to quantities and concepts .

A3. 1.1 Atomic weight - the atomic weight is the dimensionless relative

mass of an atom or of a mole of atoms of normal isotopic composition

12 based on a scale in which the isotope C has a relative mass of

12. Preferred name is relative atomic mass, q.v.

A3. 1.2 Compression factor - The compression factor (symbol Z) is the

ratio

Z - pV /RT

m For an ideal gas Z - 1, but for real gases, Z differs from unity by

amounts that are small but significant at reference conditions.

A3. 1.3 Dry gas - Dry gas is a gaseous substance or mixture containing

little or no water vapor.

A3. 1.4 Enthalpy - Enthalpy (symbol H) is a thermodynamic property of a

material, representing the internal energy U plus the pV work

associated with its volume and pressure.

H = U + pV.

Enthalpy is often called heat content. Enthalpy, per se, is not

measurable but enthalpy differences are.

A3. 1.5 Enthalpy of combustion - The enthalpy change occurring when a

combustion reaction occurs with all reactants and products reduced

to the same specified temperature. It is often called heat of

combustion and has a negative value for all fuels. The negative of

the enthalpy of combustion of a fuel is often called calorific

value, heating value, net calorific value (q.v.), or total calorific

value (q.v.). Standard enthalpy of combustion is the enthalpy of
65

combustion calculated for the condition when all substances are in

their thermodynamic standard states (see standard state). For

gaseous substances this state is the ideal gaseous state.

A3. 1.6 Enthalpy of formation - The enthalpy of formation is the enthalpy

change occurring when a substance is formed from the elements; the

elements and the substance being at the same specified temperature.

The standard enthalpy of formation is the enthalpy of formation

when the substance and the elements are in their thermodynamic

standard states. For gaseous elements this state is the ideal

gaseous state.

A3. 1.7 Equation of state - The equation of state of a substance is the

mathematical expression expressing the relationship among the

amount of substance n, the volume it occupies V, the thermodynamic

T and the pressure p. For an ideal gas. V/n = RT/p, where R is

the gas constant. See also virial coefficient.

A3. 1.8 Gas constant - The gas constant (symbol R) is the proportionality

factor relating the thermodynamic temperature T to the pV work

associated with the pressure p and the volume per mole V^{V/n) of

an ideal gas.

A3. 1.9 Heat capacity - The heat capacity at constant pressure, C , of a

substance is an extensive property that is the rate of increase of

enthalpy of the substance with temperature; C = (dH/dT) . If the

P P

amount of substance is one mole, then it is more properly termed

the molar heat capacity, an intensive property.

A3. 1.10 Ideal gas - An ideal gas is a hypothetical gas that conforms

exactly to the ideal gas law:

pV^

= RT

m
66

in which p is the pressure, V is the molar volume, T is the thermom dynamic temperature, and i? is the gas constant.

A3. 1.11 International Practical Temperature Scale - The International

Practical Temperature Scale (IPTS) is the closest feasible approximation to the thermodynamic temperature scale as realized by a series

of defined fixed temperatures and interpolation formulas and agreed

upon at intervals by the International Bureau of Weights and Measures

(BIPM) with advice from the Consultative Committee on Thermometry

I

) (CCT). The scale is changed slightly from time to time, and the

date of adoption of a new set of values is specified; e.g., IPTS-68

is the International Practical Temperature Scale of 1968. For very :i

precise work the temperature T may also be identified by the year ]

of the IPTS; thus !Tgg is a temperature measured on the scale IPTS-68.

A3. 1.12 Molar mass - The molar mass of a substance is the mass of a

mole of the substance of specified formula, based on a scale on

12 which one mole of the isotope C has a mass of 12 grams (exactly).

A3. 1.13 Molecular weight - A dimensionless relative mass of a molecule

(or of a mole of molecules in which all the atoms have their normal

12 isotopic compositions) based on a scale in which the isotope C

has a relative mass of 12. Preferred name is relative molar mass,

q.v.

A3. 1.14 Net calorific value - The net calorific value (net, lower or

inferior heating value) is the heat evolved by the complete combustion,

at constant pressure, of a unit amount of gas with air, when the gas, air,

and products of combustion all are at a specified reference temperature

and pressure, and all the water formed by the combustion reaction remains in the the vapor state. See notes under total calorific value.
67

A3. 1.15 Real gas - "Real gas" is a descriptive term identifying actual

gas behavior as contrasted with the hypothetical ideal gas (q.v.)

The actual volume occupied by a given amount of a real gas at a

given temperature deviates from that calculated by the ideal gas

law by amounts that are small but significant at reference conditions.

A3. 1.16 Relative atomic mass - The relative atomic mass is the dimensionless

nc relative mass of an atom (or of a mole of atoms of normal isotopic

12 composition) based on a scale in which the isotope C has a relative

e atomic mass of 12. This term is preferred to "atomic weight", but

is identical in meaning.

A3. 1.17 Relative molecular mass - The relative molecular mass, formerly

called "molecular weight", is the ratio of the average mass per

formula unit of a substance to 1/12 of the mass of an atom of

nuclide

A3. 1.18 Reference (standard) conditions - Reference (standard) conditions

are conditions of temperature, pressure, and degree of saturation

with water to which measurements of gas volumes or proparties are

reduced to allow comparability among measurements made under possibly

different conditions. See table 2.

A3. 1.19 Specific heat capacity - The specific heat capacity of a material

is the heat capacity (q.v.) per unit mass of material. The term "specific

heat" is ambiguous and incomplete and is now deprecated.

A3. 1.20 Standard state - The standard thermodynamic properties for a gaseous

substance, whether pure or in a gaseous mixture, apply to the pure substance at the standard state pressure and in a hypothetical state in

which it exhibits ideal gas behavior. These conditions of temperature,

pressure and ideal behavior define the standard state. When a property

of a gas refers to the gas in its standard state it is designated with
68

a superscript i.e. H° is the enthalpy of a substance in the standard

state.

A3. 1.21 Total calorific value - The total calorific value (total, superior

or gross heating value) of a gas is the heat evolved by the complete

combustion, at a constant pressure, of a unit amount of gas with air,

when the gas, air and products of combustion all are at a specified refen

temperature and pressure, and all the water formed by the combustion

reaction is condensed to the liquid state.

Note 1. The heat evolved is the negative of the enthalpy change

for the specified process. The heat may be measured in

joules or British thermal units.

Note 2. The amount of a gas may be measured as an amount of

substance (moles), as a mass (kilograms or pounds), or as

a volume measured under standard reference conditions

(cubic meters or cubic feet).

Note 3. See table 2 for reference conditions that are in use or

have been used.

Note 4. The gross heating value of a gas is greater than the

gross heating value of the liquid of the same composition

by the enthalpy of vaporization of the liquid.

Note 5. The gross heating value of a gas is greater than its net

heating value by the enthalpy of vaporization of the water

formed in the combustion.

A3. 1.22 Vapor pressure - The vapor pressure of a substance is the

pressure in the vapor of a substance when equilibrium exists between

the vapor and a condensed phase of the substance.

A3. 1.23 Virial coefficient - The virial coefficients are the constants

S, C, Z) or 5*, C*, Z)*, in the virial equation of state which describes

the behavior of a real gas and which may take the form:
69

pVjRT =1 + B{T)/V + C(7’)/y^ + D{T)/V^ +

or the form:

pV/RT = 1 + {T)-p + C^{T)-p^ + ...

* The virial coefficients are functions of temperature T. B{T) or B (T) is

* the second virial coefficient; C(T) or C (T) is the third virial coefficient,

and so on. In this document no terms higher than B are used. See

Section 10 and Appendix 9.

A3. 1.24 Volume under standard conditions - The volume under standard

conditions, symbol ''s the volume occupied by a measured or

specified amount of gas when under standard conditions of temperature,

pressure and degree of saturation with water, (cf. standard cubic

foot). This is a volume and as such has the unit cup-ic metre or

cubic foot.

A3. 1.25 Water-saturated gas - A gaseous mixture in which water vapor is

present at the saturation vapor pressure for the temperature and

other conditions of measurement.

A3. 2 Terms that are units of measurement .

A3. 2.1 Atmosphere - The atmosphere (symbol atm) is an obsolescent unit

of pressure, equivalent in various usages to 760 mmHg, 30 inHg, and

14.7 psia (more or less). It has been most recently defined as

101.325 kPa (exactly).

A3. 2. 2 British thermal unit (symbol Btu) - The British thermal unit is

a customary (U.S. or British) umt for the measurement of amounts

of energy, particularly when observed as heat. It was initially

defined in terms of the specific heat capacity of water as the

amount of heat needed to raise the temperature of one pound (mass)

of water one degree Fahrenheit, but is now by consensus usually

defined in terms of the joule. The International Tables British
70

thermal unit (symbol Btu^-j.) is used in this document and is defined

by the relationship 1.8 Btu lb ^ = 4,1868 J g This gives it the

exact value, 1 BtUj-r = 1055.05585262 J, which may be rounded to

1055.056 J. Gas industry measurements until a few years ago were

often made using a British thermal unit defined as the heat needed

to raise the temperature of one pound of water from 58.5 to 59.5 °F

(the Btu^gp). The value depends upon the value used for the specific

heat capacity of water, but differs from the BtUjy by about 0.252

J, a significant amount, See ref. [1] in the General List. This

unit is now deprecated. For each defined Btu there is a corresponding

calorie. They are related by 1 Btu/lb*°F = 1 cal/g‘°C,

A3. 2. 3 Joule - The joule is the work done when the point of application

of a force of one newton is displaced a distance of one metre in

the direction of the force. It is also the value of the pV term of

3 a gas occupying a volume V = 1 m at a pressure p = 1 Pa.

A3. 2. 4 Newton - The newton is that force which, when applied to a body

having a mass of one kilogram gives it an acceleration of one metre

per second per second.

A3. 2. 5 Pascal - the pascal is the pressure produced by a force of one

newton on an area of one square metre.

A3. 2, 6 Standard cubic foot of gas - The standard cubic foot (symbol

scf) of a gas is the quantity of any gas that, at standard temperature

and under standard pressure, will fill a space of one cubic foot

when in equilibrium with water vapor. See Section 6 for standard

conditions. Note that this is a unit of amount of substance. Cf.

"vol ume under standard conditions"

which is a preferred term for

describing volumetric measurements of gas.
71

A3. 2. 7 Torr - The torr is an obsolescent unit of pressure defined as

1/760 atm. By this definition it is essentially equal to 1 mmHg,

but the dependence on the value for the density of mercury has been

removed.
Appendix

4.

Units

of

measurement

and

conversion

factors

72

03 03 05 05 05 •P -P O O m z z CM CO CO Nw/ rH 05 CM E E E CO 03 03 CO CM E CM Q. OCM E CD CO 'a- rH 03 cn CO CM -o CO E 00 05 C5. . • to -p E u

-p c cn CM

u CD "a" tn 03

O • • •r~ 05 E E E cn o CM CO rH cn CO 00 Q. cn c ro Lf) CD CO rH o CO LT5 E f'. JX CD • 3 ro 'O' 00 > cn sa- 00 CO rH o CD 00 'a- rH CO LT5 CD CM

l-H CM o rH CM cn lD 05 CO 00 CM CD o o CM o CM cn

CO 3 o CO 0T5 o «a' CM 00 CO o o o CO 00 CO

O'

14- LU o o o o o CD CT5

O CD o o o’ o rH CT5 CO o rH o CO

S-OJ

+Ju

(T3 X

0) •a

cuc

0) •a

to

01

4-> c

ra3

cr

OJB

0)

S-3

to

(T3 0) <

Ol

(T3

O

j3E

>>

CO

fO z

00CD

05C

C •<- 05 o

>>

toto

(T3

NO

05 CM csj esj E CO ro CO

(C C-p-O 3 3+J-O

05 •>- <4- T- M- >,

SO < >

ro05

05S-3

1/5 E S- •r

to -P o to

05

i- 05 1— Q.

(0 OOO I

JZuc

-poo OsrO>> o o > >

fO fO

05 -O u c C 3 3 O O D.

Uc

05

s-

(T3 3

O"

to POo

T3

Sro

Uc oo

05 05

s- s- (0 fO 3 3

O" 0“

to to

uu 3 3 U U

•O

S03>>

JO3U

05

S- P o O

O 0 I—

05 C5 S- o X z -P E 05 CD E E 05 z rH Z o cn o 05 E c 'O' tD S- E

03 CD 3 1—

05 \ O' V—/ s- r>- to 03 CO

ID s- 03 3 CM

•r- o S- Ol. O' cn 1— 05 to cn cr a. O ro

•r- o CD S- tn

CD 05 05 u \ CL . i '—' s- cn o

CO o CM 05 • 05 <4- CO U

X

i. rH s- 05 E O o cn Z 03 rH 4- CD c Q. s- CD o to 05 •O o r- o O i- c CO E r— S- 3

fO +J o o (75 05 ro -p Q.
73

-pcoo<

<TJ o 3 z C '—

X • f— CM P

03 IXX 05

C

Q_ • p o p CM o u LD C\J (J5 X X X

p tH ro •d" • CO 05 c , 03 • cn P

05 C5- ro + + XX 03 , 'O' o X >,

03 C\J CJ P o P“ 1/5 > CM O 0 O + CO p O • t— CM o \ X 'O' X u •rTM CM o HJ P Ll_ CO CO X 03 p

cr , , cn 00 CO 00

O

pp o X 03 . Llj CO o \ • . . . * 05 P X

CO o i*; un rH rHp P 'O'pX

p 1—

05

10

o;

0)i- 35 P O P X P QC o X P 03 0 O 0 0 0 X c 03 P \ \ E s X 05 P P En > Q. X E

05 P Prop

05

05

1/5 P

>) 05 P 1/5 X 05 3 3 c C U •r“ 05 •TTM !P P 1/5 P

05 • r— p—

X c E 3 0) 03 03

1/5

X C_) P QC P 3 c O • r- 05 » »

05 P 05 05 05

05 rTM

X P P P

05 05 03 3 3 3 E P O P P P P

03 05 03 03 03 Z £ 05 05 P P p

05 05 05 05 05 05

r*- P P C3. <3. Q. P C35 35 E E E

•f“ 03 05 05 05 05 0) E X '3 3 P P p

05 r—

s-o;cnju03u o

+-> CO

(0co

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(U -pc

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05 JZuoE

i.

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<0E

s05

to

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(0 -p

(Oco

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<0c

1/5p

05 05 0)X 05 . . . 

•1“ 1/5 P 03

X P X P tf— c O 03 O P •r“ 05 05 P r— P P

03 03 P O

O

UU z z

05c

-C

X

+j -r- a» c

1/5 OJ O CL

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05 x: O EE

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03

05p

05 Xp

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05 05

p

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•r- P

05 P CQ I— U

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05 o pCO

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II
74

Appendix 5. Discussion of physical constants

Appendix 5a, Relative atomic and molecular masses

Table A5a gives a listing of six differing sets of relative atomic

masses (atomic weights) for the elements C, H, 0, N that have been

recommended since 1941 plus the set used in the International Critical

Tables. They are coded A to G in order of increasing age. The set

selected for this document is Set A, which was chosen for two reasons:

(1) It is the current recommendation of the lUPAC Commission on Atomic

Weights and Isotopic Abundances [1] and represents the best estimate

based on modern research of the atomic masses. (2) It is only slightly

different than the set that has been recommended for ten years [2]. The

only noticeable difference is the value for hydrogen which, at five

significant figures, rounds to the value given in Set B.

Lines A and A' in Table A5a are both part of the current recommendation, The first line (A) gives values to the decimal place in which

there is some uncertainty. (The uncertainties are given in Table A5c).

Variation in natural terrestrial samples is the principal cause of

uncertainty for the elements of interest here. The five-significant-figure

set coded A' has values that are less accurate, but are more likely to

remain constant for a decade, even with continued research into the

matter.

Table A5b gives for each set of atomic masses the relative molecular

masses of CH^, CH

2

and CH, which represent respectively the maximum H/C

ratio the limiting H/C ratio for homologous series, and the minimum H/C

ratio encountered in the common low molecular mass hydrocarbons. Also

given are the relative molecular masses calculated for H

2

O and CO

2 using

each set.
The uncertainties introduced into the calculations by the variability

with time of the recommended atomic masses are indicated in the footnote

to Table A5b. This also shows the level of error due to atomic mass

changes that can be introduced when measurements made at different times

are compared without adjustment for the changes in molar masses.

An examination of compilations of thermodynamic data for hydrocarbons

has shown that sets B, C, D, F and G have been used. There has been no

clear preference. Nor have the recent recommendations (in the 1970' s)

been used often. Some examples are given in Appendix 7 where sources of

data are presented.

The procedure used in this document to calculate relative molecular

masses is to use Set A and then round the value of the molecular mass to

the decimal place in which the accumulated error appears. This procedure

is displayed in Table A5c. The uncertainties in the atomic masses used

in that table are those recommended by the lUPAC Commission, Because

those uncertainties allow for samples from many sources, the resulting

errors may be considered large for fossil fuels. Nevertheless, the

uncertainty in the relative atomic mass of carbon make it appropriate to

round molecular masses of the hydrocarbons of interest to three decimal

places.
76

Table A5. Atomic masses

Table A5a. Recent sets of relative atomic masses of the chemical

elements and their period of recommendation by the

International Union of Pure and Applied Chemistry (lUPAC)

C H 0 N Year first recommended

A 12.011 1.00794 15.9994 14.0067 1981 [1]

^,(a) 12.011 1.0079 15.999 14.007 1981 [1]

B 12.011 1.0079 15.9994 14.0067 1971 [2]

C 12.011 1.0080 15.9994 14.0067 1969 [3a]

D 12.01115 1.00797 15.9994 14.0067 1961 [3b]

E 12.011 1.0080 14.008 1953

F 12.010 1.0080 16 14.008 1941

G 12.000 1.0077 16 14.008 1923

(c)

(a) The A' line has the five-significant-figures set given by the lUPAC

Commission on Atomic Weights. These are values rounded from Set A.

’ Set A' also includes Ar, 39.948; He, 4.0026; S, 32.06. The certainty of the value for sulfur is limited to four significant figures by

the range of isotopic compositions in normal terrestrial materials.

(b) The relative atomic mass of oxygen in set E and earlier sets is an

integral value by definition.

(c) Prepared by G. P. Baxter for the International Critical Tables. The

first international recommendation (1931) gave C = 12.00 and H =

1.0078. There were several changes during the 1930' s.
77

Table A5b. Relative molecular masses of some important molecules as

formally calculated from different sets of relative atomic

masses (rounded to four decimal places). See Table

1 for recommended values

SET CH4 CH^ CH H^O CO2

A 16.0427 14.0269 13.0189 18.0153 44.0098

B 16.0426 14.0268 13.0189 18.0152 44.0098

C 16.0430 14.0270 13.0190 18.0154 44.0098

D 16.0430 14.0271 13.0191 18.0153 44.0099

E 16.0430 14.0270 13.0190 18. 0160 44.011

F 16.0420 14.0260 13.0180 18.0160 44.010

G 16.0308 14.0154 13.0077 18.0154 44.00

Maximum relative differences in molecular masses from sets A through F.

The letters in parenthesis identify the sets giving the maximum deviation

1 i sted.

CH^, 4.4-10'^(D/F); CH^, 7. 8- 10‘^(D/F) ; CH, 8. 6«10"^(D/F) ; H^O, 6.6-10'^

(E,F/B); CO2, 4.4-10'^(E/A).
78

Table A5c. Assignment of uncertainties in relative atomic and molecular

masses for the 1981 set recommended by the lUPAC Commission

on Atomic Weights and Isotopic Abundances

Assign s (the estimated uncertainty of values in Table 1) as follows;

Element Relative atomic mass Uncertai nty s X 10^

atomic weight unit

Ar 39.948 1.

C 12.011 10.

H 1.00794 0.7

He 4.00260 0.1

0 15.9994 3.

N 14.0067 1.

S 32.06 100.
79

Table A5c. (continued)

Calculation of absolute and relative errors in relative

molecular masses using data in Table A5c.

Molecule Atoms and weights Uncertainties^

Additive Quadrature

s -mS 2 ,.4

10 s 10 s

CH

H^O

c 12.011 10 100

H 4 X 1.00794 4 X: 0.7 8

16.04274 12 (108)-^^" = 10.

Rounded value*^: 16.043

Relative uncertainty: divisor 16.04 0.7 0.6

C 12.011 10 100

H 1.00794 0.7 0.5

13.01894 llT" o

11— II

t-H CM O O

11—

Rounded value: 13.019

Relative uncertainty: divisor 13.02 0.3 0.24

C 12.011 10 100

31. 9988 6 36

c. 44.0098 TT (130)^^^ = 12

Rounded value: 44.010

Relative uncertainty: divisor 44.01 0.2 0.15

H 2 X 1.00794 2 X: 0.7 2

0 15.9994 3 9

18.01528 4.4 =3.3

Rounded value: 18.0153

Relative uncertainty: divisor 18.02 0.2 0.2

^The uncertainties by quadrature (square root of sum of squares) are recommended. See Appendix 10 on propagation of errors.

*^The rounded values also appear in Table I. Other values there were determined by similar procedures.
80

References for Appendix 5a.

[1] lUPAC Commission on Atomic Weights and Isotopic Abundances; "Atomic

I weights of the elements 1981"; Pure and Applied Chemistry (1982) in

j press. '

[2] lUPAC Commission on Atomic Weights; "Atomic weights of the elements |

1971"; Pure and Applied Chem. (1972), 637-649. ^

[3] a. lUPAC Commission on Atomic Weights, "Atomic weights of the elements";!

Pure and Applied Chemistry (1970), 21, 91-108; and J. Am. Chem. Soc.

(1971), 93, 2579. b. Cameron, A. E. ; Wichers, E. J. ; "Report of the

International Commission on Atomic Weights (1961"; J. Am. Chem. Soc.

(1962), 84, 4175.

II
81

Appendix 5b

The density of mercury, the standard acceleration of gravity,

and the significance of the reference pressure 760 mmHg.

The density of mercury serves three slightly different purposes:

(1) to convert barometric and manometric pressure measurements using a

mercury column to a uniform basis in terms of the standard millimetre of

mercury or standard inch of mercury as a unit of pressure; (2) to convert

old data reported in standard millimetres or inches of mercury using

standard barometric tables to a pressure unit consistent with the mechanical

units of pressure, SI units; and (3) to convert accurate pressure measurements made using a mercury column as a sensor directly to pressure in

the SI units. It should be recognized that the actual density of mercury

is a function not only of temperature but also of pressure and isotopic

composition, and furthermore that the experimental determination of the

density as a function of these variables is subject to experimental

uncertainties that change as improved procedures are introduced. We

cannot go into the details of these factors and their treatment over the

years, and shall give only a sketch of the principal points related to

the usage of the millimetre of mercury as a unit of pressure.

The conclusion that has been drawn from this historical survey is

that the standard atmosphere and 760 mmHg are identical today. Except,

perhaps, for measurements of the highest precision, the two units have

been the same for eighty years.

1. The General Conference on Weights and Measures, CGPM, in 1901

-2

(Resolution 3) adopted the value 980.665 cm s for the standard

* (normal) acceleration of gravity [1] . This was confirmed in 1913

[2, p. 16].

A References cited here are listed at the end of Appendix 5b.
82

2. In 1913, Chappuis [3] presented selected values for the density and

coefficient of expansion of mercury and gave P^g(t = 0 °C) = 13.59515

-3

g cm , under normal pressure, using the density of water at 4 °C =

0.999973 g cm He cites earlier work (Chappuis (1907), [4]) for

the coefficient of expansion and cites Marek (1883), [5] (c?^ =

13.5956) and Thiesen and Scheel (1897), [6], = 13.59545) for

the density of mercury at 0 °C relative to that of water at 4 °C.

He took the average of these and chose d^ = 13.59552 from which he

obtained the value listed above for the absolute density of mercury.

The uncertainty in the value given is ± 0.00008 based only on the

range of values from which it was derived.

3. In 1917 Leduc [7] used for standard conditions in determining the

-2

standard density of air: p = 76 cm Hg, g = 980.665 cm s , and

-3

P|^g(at t = 0 °C) = 13.5951 g cm , for which, however, he gave no

citation o.f source.

4. The Smithsonian Meteorological Tables (Fourth Revised Edition,

1918) [8] introduced new tables for converting millimetres of

mercury to millibars using the standard acceleration of gravity

adopted by the CGPM in 1901 (980.665 cm s ^), and the density of

-3

mercury pu (at 0 °C) = 13.5951 g cm , taken from Chappuis [3] and

ng

Leduc [7]. These tables were widely used in the United States for

the correction of manometric and barometric measurements using

mercury as an indicator to pressures.

5. N. E. Dorsey (1926), [9] for the International Critical Tables,

defined the normal atmosphere as the pressure exerted by a vertical

-3

column of liquid 76 cm long, density 13.5951 g cm , acceleration

-2

of gravity being 980.665 cm s . Incidentally, the British atmosphere

is stated to be based on 30 inches (76.2 cm) rather than 76 cm.
83

6. In 1926 Kimball [10], for the International Critical Tables section

on barometry and manometry, gave the standard density of mercury as -3

13.5951 g cm , and the standard acceleration of gravity of 980.665

-2

cm s without reference, but his temperature corrections are referred

to the International Meteorological Tables [11]. (We have not found

those numbers in the IMT, they may have been derived by Kimball).

7. In 1927 the Bureau of Standards, the National Physical Laboratory

and the Physi kal i sche Technische Reichsanstal t submitted a text of

a proposed international temperature scale in which the standard

( normal e) atmospheric pressure is defined as the pressure exerted

by a column 760 mm in height of mercury having an average density

-3

of 13.595 g cm under an acceleration of gravity equal to 980.665

cm s equivalent to 1 013 250 dynes cm ^ [CGPM 1927, ref. 12].

No source is given for the numbers, but they were modified and

reiterated in 1933 [13] with a corrected text in which the density

-3

of mercury is changed to 13.8951 g cm . The number thus conforms

to the value given by the Smithsonian Meteorological Tables. The

numbers are related by the relationship:

1 013 250/(980.665-76) = 13.595098

or by:

12.5951 • 76 • 980.665 = 1 013 250.14

Thus it is apparent that a value for the standard atmospheric

pressure had been chosen to be in close accord with a standard

height (76 cm) of a column of mercury of average density equal to

the best known value at the time, and under the standard acceleration

of gravity. It will be noted that the specified average density is
84

slightly less than the value at normal pressure given by Chappuis

in 1913 [3]. This is consistent with the fact that the average

pressure on the mercury in a barometer is only 1/2 the external

pressure and thus the average density should be less than that

-5 -3

given by Chappuis. The deviation should be about 3 x 10 g cm

which would reduce Chappuis' value to 13.59512 with an uncertainty

of 0.00008. The value rounded to 13.5951 thus contains the last

significant figure.

8. In 1948 the CGPM presented the international temperature scale of

1948 which reiterated the previous standard pressure in all details

[14, pg. 93]. However, a change of emphasis occurred. The standard

-2

pressure [14, pg. 90] is defined as 1 013 250 dynes cm and the

mercury scale is related to it. It adds that except for work of

the highest precision one can accept that the mercury of commerce

-3

has the specified average density 13.5951 g cm in the column of

760 mm.

9. The CGPM, in 1954 considered the definition of the standard atmosphere

in 1948 might be construed as applying only to precision thermometry

[15]. It, therefore, adopted a resolution (Resolution 4) defining

-2 one standard (normal e) atmosphere as 1 013 250 dynes cm or 101

325 newtons per square metre, thus validating definitively the

standard value to be used, and removing any ambiguity about the

calculations given in paragraph 7.

10. In 1957 Cook and Stone [16] reported some measurements of the

density of mercury at 20 °C and reported:

-3

Pl, (20 °C) = 13.545 8924 g cm

ng
85

which reduces at 0 °C to 13.595 0889 g cm Cook (1961, ref. 17)

extended these measurements and reported a summary value including

both sets of P|_|g(at 20 °C, 1 atm) = 13.545 884 g cm ^ which, reduced

to 0 °C, becomes:

P^g(0 °C, 1 atm) = 13.595 080 g cm"^ ,

with a standard deviation of 0.2 x 10 ^ (0.2 ppm). These densities

-3

at t = 0 °C round to 13.5951 g cm . Thus the value used traditionally for barometry and manometry is accurate in the sixth figure.

11. Brombacher, Johnson, and Cross (1960, ref. 18) for the NBS monograph on

mercury barometers and manometers used the value P|^g(0 °C) = 13.5951 g

-3

cm , the standard atmosphere, 1013.250 mbar, and the standard

-2

acceleration of gravity 980.665 cm s

12. In 1960, the eleventh General Conference on Weights and Measures

adopted the International Practical Temperature Scale of 1948

amended 1960 [19, pg. 63]. In the amended text [19, p. 127] the

statemerlts are made that in practice pressures are determined by

means of a column of mercury and that one can assume that the

-3

density at 20 °C of pure mercury is 13 545.87 kg m as an average

in a column of mercury that balances 1 atmosphere. They also

recommended the use of the Potsdam system for determination of

local gravity for realization of the standard atmosphere.

13. In 1964 Bfgg [20] reviewed the density of mercury and used the data

of Cook [17] and the thermal expansion formula of Beattie et al

[21] which is the same as was used by Cook, to calculate a table of

densities of mercury as a function of temperature at P = 1 atm. He

tabulated P|^g(20 °C) = 13.545 88 g cm ^ and P|^g(0 °C) = 13.59508 g

-3 cm which are the rounded values of Cook.
86

14. Bonhoure and Terrien [1968, ref. 22] working at the International

Bureau of Weights and Measures, BIPM, used for the density of their

* mercury column the density at mid height, p(mean) , and used the

formula:

p(mean) = p^/[l - «Hg^^Hg

'

-3

for which p^, the density of mercury at 20 °C = 13.545 892 g cm

obtained from Cook and Stone [13], = 181.1 x 10 ^ K = 39

X 10 m^ N ^ and Pj^ = 101 325 N m The density given at 20 °C

can be derived from the formula and data of Bonhoure and Terrien.

The term -p(p/2 - p^) becomes + 1.975 x 10 and this leads to

P|_jg(20 °C, mean) = 13545.8652 kg m If the more recent work of

Cook [14] is used for p^, p^g(20 °C) = 13545.884 kg cm then

P|^g(20 °C, mean) = 13545.857 kg m

15. In 1968 the 13th CGPM adopted the International Practical Temperature

Scale of 1968 ([23], resolution 8, p. 62, p. 105; full text, pp.

A1-A24). fhe formula for calculating the density of mercury as a

function of temperature and pressure is given as :

p(tgg,mean) = p(20 °C, Pq)/[ 1 + A(tgg - 20 °C) + S(tgg - 20 °C)^]

x[l - x(p/2 - p^)]

where A = 18115 x lO"® °c'^; 5 = 0.8 x lo’® °c"^, x = 4 x lO"^^ n“^

2 “3

m , and p(20 °C, p^) = 13545.87 kg m far pure mercury when p^

=

101 325 N m The term -x(p/2 - p^) has the value + 2.0265 x

10 The value of Pu (20 °C, mean) = 13545.842 kg m ng

The temperature correction to t = 0 °C is:

p(20 °C)/p(0 °C) = 1 - 20 X 18 115 x lO’® + 400 x 0.8 x lO’® =

1 - 0.00362300 + 0.0000032 = 0.9963802

*We use the term p(mean) in all cases where authors have referred to pressures at mid height, at p/2 or average densities.
87

-3

p^g(0 °C, mean) = 13595.053 kg m . The value resulting from a

similar calculation but not using the CGPM (1968) formula v/as found

by Armstrong [24] to be p^g(20 °C, mean) = 13595.052 kg m in good

agreement.

16. Brown and Lane [25] reviewed the density of mercury in 1976. They

cited the formula and constants as given by CGPM in 1968 [23] and

presented a table for Pu (t^-cP ) derived in the same way as that

ng bo 0

of Bigg ^20] except converted to IPTS-1968. They used p

= 101.325

-3

kPa and tabulated Ppjg(*^ Pq) ~ 13595.08 kg m in exact agreement

with the value calculated by Cook [17].

As a result of this survey, it is quite clear that for precise

measurements of pressure using a mercury column (purpose 3 above)

the accurate determination of density by Cook [17] and interpolation

formulas such as are given by Brown and Lane [25] should be used.

However, it is also clear that the standard millimetre of mercury

used in establishing the relationship between the standard atmosphere

specified as 760 mm Hg and the standard atmosphere as 101.325 kPa is

not based on the recent measurements of Cook but is based on much

earlier measurements. The value used for the density of mercury,

-3

p^g(0 °C, average) = 13.5951 g cm , was given official standing by

CGPM In 1927 Cas corrected in 1933) and confirmed in 1948. This

density was derived by Chappuis in 1913 from earlier measurements

and shortened to the present last significant figure (six figures)

by Leduc in 1917. It was adopted by such internationally used

compilations as the International Critical Tables and the Smithsonian
88

Meteorological Tables, and finally placed in the status of a defined

value by CGPM.

The exact process has not been discovered by which the decision was

made that the standard atmosphere should be 101.325 kPa (101 3250

-2

dynes cm ) exactly, rather than a column of height 760 mm of

-3

mercury of density 13.5951 g cm in a standard gravity of 980.665

-2 cm s

It is clear however, on the basis of the CGPM decisions in 1933 and

1948 that the selected pressure was, within the limits of accuracy

of the measurements, identical on the two bases.

We believe the relationships given in paragraph 7, above, constitute

a definition of the unit, mmHg, which is most applicable to pressure

measurements made during the long period from about 1917 to 1957,

that there is no point in refining further an obsolescent unit and

that the accuracy of the quantities measured does not warrant

defining the mean P|^g(0 °^)> to greater than six figures for the

purpose of identifying the pressure reference condition.

The value chosen for this document is therefore:

P|^g(0 °C, mean) = 13595.1 kg m

^

Within the limits of measurement the unit of pressure, mmHg, is

identical to the Torr, i.e.

in IT 101 325 Pa _ TTT .,00 n 1 mmHg = 1 Torr = = 133.322 Pa ,

These correspondences are included in the ASTM standard for Metric

Practice [26]. On this basis a reference pressure of 760 mmHg is

identical to 101.325 kPa.
89

References to Appendix 5b

1. CGPM (1901); Compt. rend, des seances de la troisieme conference

general des poids et mesures, Paris 1901. Paris, Gauthier-Vi 1 lars

(1901); p. 68. (See also Reference 2, page 16).

2. "The international system of units"; NBS Special Publication 330

1977 Edition. Washington, D.C. 20402: U,S, Government Printing

Office (1977).

3. Chappuis, P. ; "Densite et dilatation du mercure" ; Table 40 in H.

Abraham and P. Sacerdote editors, Societe Francaise de Physique

Recueil de Constantes Physiques, Paris: Gauthier-Vi 1 lars , (1913),

p. 139.

4. Chappuis, P. ; "Dilatation du mercure", Trav. Mem. Bur. Int. Poids

Mes. (1907) 13 , 1-31.

5. Marek, W. J. ; Trav. et Mem. Bureau International des Poids et

Mesures (1883), 2, Dl-58, particularly pp. 55-58.

6. Thiesen, M. ; Scheel , K. ; Sell, L. ; "Untersuchungen uber die thermische

ausdehnung von fasten und tropfbar fllissigen korpern". Wissenschaftl iche

Abhandl ungen der Physi kal ische-techni sche Reichs-anstal t , (1895), p.

73-184, (We have not identified a reference to Thiesen and Scheel in

1897. The present citation contains the pertinent data.)

7. Leduc, A.; Traveaux et memoires du Bureau Internationale des Poids

et Measures, (1917), XVI, pp. 1-37 (end of volume).

8. Smithsonian Meteorological Tables; 4th Revised Edition. Washington:

The Smithsonian Institution (1918), pp. iii, xviii.

9. Dorsey, N. E. ; "Selected technical terms"; International Critical

Tables, New York: McGraw-Hill Book Company (1926), Volume 1, p. 24.
90

10. Kimball, H. H. ; "Barometry and manometry"; International Critical

Tables. New York: McGraw-Hill Book Company (1926), Volume 1, pp.

68-70.

11. International meteorological committee; "International meteorological

tables"; Gauthier-Vi liars, Paris, (1890).

12. CGPM (1927); Compt. rend, des seances de la septieme conference

general des poids et mesures, (1927). Paris: Gauthier-Villars

(1927-1928), p. 95.

13. CGPM (1933); Compt. rend, des seances de la huitieme conference

general des poids et mesures (1933). Paris: Gauthier-Villars (1934),

p. 74.

14. CGPM (1948); Compt. rend, des seances de la neuvieme conference

general des poids et mesures (1948). Paris: Gauthier-Villars (1949),

pp. 90.

15. CGPM (1954); Comptes rendus des seances de la dixieme conference

generale des poides et mesures (1954), pg. 80. See also ref. 2.

for the same material.

16. Cook, A. H. ; Stone, N. W. B. ; Phil. Trans. Roy. Soc. (London) (1957), A250

277.

17. Cook A. H. ; Phil. Trans. Roy. Soc. (London) (1961), A254 , 125-154.

18. Brombacher, W. G. ; Johnson, D. P. ; and Cross, J. L. ; "Mercury

barometers and manometers"; Nat. Bur. Standards (1960), Monograph 8.

19. CGPM (1960); Comptes rend, des seances de la onzieme conference

general des poids et mesures (1960), p. 63, p. 127.

20. Bigg, P. H. ; Brit. J. Appl. Phys. (1964), I^, 1111-1113.

21. Beattie, J. A.; Blaisdell, B. E. ; Kaye, J. ; Gerry, H. T. ; and

Johnson, C. A.; Proc. Amen. Acad. Arts Sci. (1941), 71, 371.
91

22. Bonhoure, J. ; Tern'en, J. ; Metrologia (1968), 4, No. 2, pp. 59-68.

23. CGPM (1968); Compt. Rend, des seances de la treizieme conference

general des poids et mesures (1967, 1968), p. 62, p. 105, pp.

A1-A24 (esp. p. A8).

24. Armstrong, G. 7.; "Calculation of the heating value of a sample of

high purity methane for use as a reference material"; Nat. Bur.

Standards (U.S.) Technical Note 299 (U.S. Government Printing

Office, Washington, D.C.), (1966).

25. Brown, I.; Lane, J. E. ; (Herrington, E. F. G. ; editor). Recommended

reference materials for realization of physicochemical properties.

Section; Density. Pure and Applied Chemistry (1976), 45, 1-9.

26. ASTM (1976); "Standard for metric practice"; ASTM desgination

E-380-76 (also American National Standard Z210.1) pp. 29, 35 Philadelphia: American Society for Testing and Materials Annual Book of

ASTM Standards Parts 41 and others, (1976 and later years).
92

Appendix 6. Discussion of reference conditions

The values of pressure, temperature, state of gas, and water content

to be used for defining the reference conditions of measurement of

natural gas are listed in table 2. Various standardizing bodies have

selected different values or conditions of these parameters for "base

conditions" or reference condftfons. The actual values used in the

measurement of the properties of a gas are extremely important in custody

transfer as well as in calcuTations of performance if comparability of

the heating value of the gas is to be realized. This is particularly

true for volumetric measurement of gas because the volume occupied by a

given quantity of the gas is strongly dependent on them. To a lesser

extent, though still significantly, the heating value of a given mass of

gas is dependent on these parameters. The values for these parameters

that have been recommended by various groups are discussed below.

Temperature . For many years in the United States a reference temperature

of 60 °F (15 5/9 °C, 288.7055 ... K, rounded to 288.71 K) has been used

both in the natural gas industry, and in petroleum products industries

as the temperature to which volumetric measurements and measurements of

properties should be referred, whereas European practice has been to use

* 15 °C [9a] . In recent years in an attempt to bring greater conformity

to U.S. and European measurement practices, U.S. practice has been

moving to the adoption of 15 °C (59 °F or 288.15 K) as a reference

temperature. See, for instance [1] or [9bj. ASTM Committee D-3 on

Gaseous Fuels has recently adopted 15 °C as the reference temperature

for metric measurement of natural gas samples [9b] as has the American

Gas Association; and the American Petroleum Institute appears also to be

adopting 15 °C for gas measurements [10].

References cited in this appendix are in the General List.
93

On the other hand, for many years scientifically oriented measurements

universally have often been referred either to 25 °C (298.15 K) or 0 °C

(273.15 K) as a standard temperature [8]. The former temperature is

cited in many compilations of the properties of hydrocarbons. See for

examples Table A7e in Appendix 7. r = 273.15 K is used in Japan as the

reference temperature. Because there is widespread use of each of these

four temperatures information is given in this handbook for converting

measurements from one of these temperatures to another. However, 15 °C

(288.15 K) and 60 °F (288.7055 ... K) are the reference temperatures

selected for presentation of the tables of heating values measured on a

volumetric basis.

Pressure . The situation with respect to reference pressure is much the

same as with respect to temperature. European practice has been to use

101.325 kPa (1 atm). Early gas industry practice in the United States

emphasized a pressure base of 30 inHg (762 Torr, 101.5913 kPa) which converts

to 14.7345+ psia; and was often rounded to 14.73 psia (101.559.77 kPa which

is rounded to 101.560 kPa). However usage of other pressure bases has been

wi despread.

The pressure 14*. 73 psia was adopted by ANSI in 1969 (ANSI Z 132.1 -

1969) [10] apparently without reference to existing standards which

defined the base pressure as 30 inHg, a round number that can be presumed

intended to be exact: ASTM D 1071-55; USA Standard Z 77.5 - 1963 (Later

American National Standard Z 77.5 - 1973) and ASTM D 900-55; USA Standard

Z 68.1 - 1956 (later American National Standard Z 68.1 - 1973). It is

unclear whether 14.73 psia was intended to be equivalent to 30 inHg or

whether it was intended to be exact. Unfortunately the difference is

significant in tabulated values of enthalpies of combustion of gaseous

hydrocarbon compounds on a volumetric basis. The difference is, however.
94

less than other uncertainties in the measurement of total heating value.

Because ASTM has followed the lead of ANSI in the use of this number

(see ASTM D 1071-78a, ref. 9b) we use the value 14.73 pisa as exact in

making the calculations, and not as a shorthand notation for 30 inHg.

In scientific work a slightly different pressure, usually called

the standard atmosphere (1 atm = 101.325 kPa by definition [2,5]) has been

used for many years. This pressure has been recently adopted in ASTM

standards for measurement of gaseous volumes in metric units and by the

American Gas Association and is apparently being adopted by the American

Petroleum Institute [lO]. This pressure has been used also as the

standard state pressure for reporting thermodynamic quantities. However,

recent action by lUPAC Commission on Thermodynamics [3] recommends that

a new pressure, 100.000 kPa (=1 barj, be adopted as the standard pressure

to which the thermodynamic standard state would apply. While widespread

usage of this new reference pressure may not occur until some time in

the future, notice is taken of it in this document.

Information is given to allow calculations to be made for each of

these three reference pressures. However, for consistency with gas

industry measurement practice the tables are given for the reference

pressure of 101.325 and 101.560 kPa.

Gas ideality . The use of real gas properties is universal in practical

measurements in the gas industry. However, the properties of real gas

mixtures are functions of the composition. As a result of the fact that

tabulation of the properties of all real gas mixtures is impossible, and

because the components of natural gas (other than CH^) are generally present

as a small fraction of the total gas it is more useful to emphasize ideal

gas properties for the pure gases and to utilize the virial coefficients

and compositions for calculation of the properties of mixtures on an ad hoc

basis.
95

Water content . Natural gas in production, transmission and distribution

usually has a relatively low water content. Hence, for most purposes

the dry gas is a better reference state than the water-saturated gas.

However, for determination of heating value by the continuous flow

recording combustion calorimeter, the measurement devices require the

gas to be water-saturated before combustion. This is because the wet

test meters used for metering the amount of gas burned are water-filled,

and because the entering gases must be water-saturated in order to

assure that all the water formed is condensed to the liquid state in the

calorimeter in order to determine the gross heating value. Thus for

determination of calorific value, the water-saturated gas is a better

reference state than the dry gas. (In any custody transfer it is necessary to identify which reference state is used. This document provides

factors for converting between dry and water-saturated gas.)

Metric and U.S. Customary reference conditions

Both the temperature and pressure are different in the ANSI/ASTM/API

reference conditions (60 °F, 14.73 Psi; 288.71 K, 101.560 kPa) and in the

ISO/ANSI/ASTM metric reference conditions (288.15 K, 101.325 kPa).

V Conversion of a molar thermodynamic property of an ideal gas from one

reference condition to the other requires correction only for temperature.

For volumetric properties there are both temperature and pressure corrections. For properties of real gases there always are both temperature

and pressure corrections.
96

Appendix 7. Thermodynamic data: their sources and uncertainties

7a. Sources of thermodynamic data

Heat capacity and enthalpy increment

Standard heat capacities at constant pressure and at 298.15 K for

the substances in Tables 3 and 4 are taken from Table A7a which summarizes

an evaluation by Wilhoit [1] . These are selected instead of those of

Rossini et al . [2] because they are based on more recent and more detailed

calculations. Wilhoit has presented his results in the form of polynomials

of degree two that are valid in the temperature range from 268 to 308 K.

The constants of the polynomials, a reference to the method of calculation

of Cp and the value of C° (298.15 K) calculated from the polynomial are

presented in Table A7a.

The polynomials have the dimensionless form:

in which the values of a, h, c are determined by the method of least squares

to fit the data given in the references.

C°^/R = a + b (7’-273.15 K) + c(r-273.15 K)

2

T

2

T

1

References are listed at the end of the appendix.

Tables A7a through A7f are at the end of Appendix 7a.
97

= i?ta(t2 - + (b/l)(t\ - t\)

* (c/3)(t^ - t^)]

where t = T

- 273.15 K, i.e, the Celsius temperature.

The values of (298.15 K) in Tables 3, 4 and A7a differ by less than

1 J mol ^ K

^ from those given by Rossini et al. [2] and by Stull, Westrum

and Sinke [3], except for 2-methyl pentane , 1.84; 3-methylpentane, 2.89;

cyclobutane, 1.36; cyclohexane, 1.02; and benzene, -3.91 J mol In

the last case, the maximum difference the use of the polynomial would

cause in the tabulated enthalpy of combustion is 0.04 kJ mol ^ (in appro-

-1 -5

ximately 3303 kJ mol ); or 1.2 x 10 relative error in the enthalpy of

combustion. This is considered to be well within the uncertainty of the

measured enthalpy of combustion.

Enthalpies of formation

The values of A^°(298.15 K) for the auxiliary substances in Table 3

were taken from the recent recommendations of the CODATA Task Group on Key

Values for Thermodynamics [15]. These do not differ significantly from

those given by Wagman et al., [16], and (except possibly for H

2

S) may be

expected to be used in the next generation of chemical thermodynamic tables

and to be used internationally.

Enthalpy of combustion

The selected values of the standard molar enthalpy of combustion at

298.15 K of each gaseous substance are listed in Table A7b (and are repeated

in Table 4). These are taken from an evaluation by Domalski [17,18b] and

have been previously published by Armstrong, Domalski and Minor [18a]. If

the enthalpy of combustion of the gas depends upon measurements made on

the liquid, for the liquid is also given. For each value, references

are given to the source of the data used in making the selection as well
98

as others examined but not used. The agreement of this selected value with

those in several compilations of critically evaluated data is also shown.

In most cases, the values presented here are consistent with those in the

API tables [2] and their successors [76,77].

Methane is considered here in slightly more detail than the other

hydrocarbons, both because of its importance in natural gas and also as

an illustration of the problems that must be faced when selecting a "best

value.

The sources of original measurements of enthalpy of combustion for

methane are given in Table A7c in chronological order. These eight studies

provide the entire measurement set. (There are fewer for some of the other

hydrocarbons.) Because of the great increase in experimental accuracy and

precision in the early 20th century, only the three measurements reported

after 1930 are of any importance. Of these, only the studies of Rossini

[55] and of Pittam and Pilcher [48] are considered today, because of the

method used and the higher precision of measurement.

Those two sets total to 12 experiments, the values for which, corrected

to 293.15 K and ideal conditions, are listed in Table A7d. The results of

the two studies overlap, their precisions (separately) are similar, and it

is not clear that any of the points can be discarded. If there were no other

considerations, it would be appropriate to pool the data and take an average:

890.65 kJ mol There are, however, several additional points to be considered. i

1. The impurity level of the synthetic methane used by Pittam and Pilchei

* was less than 5 ppm . Rossini also used a sample of synthetic methane with a |

measured 1219 ppm impurity of CO, for which a correction was made. The NBS j

reference sample of methane (83), from a natural gas source and with a 400 ppm

*We are indebted to Prof. H. A. Skinner for information about this sample. j
99

ethane impurity, has about the same enthalpy of combustion as that reported

by Pittam and Pilcher (using Rossini's value for methane).

2. Pittam and Pilcher measured CO

2 production to determine the extent

of reaction, a preferred method, while Rossini measured water production.

Both used flow calorimetry, with about the same precision. Pittam and

Pilcher's value depends, via their calibration, on Rossini's measurement

of the enthalpy of formation of water (adding to the uncertainty) while

Rossini's value for methane does not.

3. Both Rossini and coworkers (39,50,51,52,55,56) and Pilcher and

coworkers (47,48) have measured the enthalpies of the to Cg hydrocarbons.

There is overlap with studies by Good (31,32). There does not seem to be

any systematic difference separating the work in Rossini's laboratory and

that in Pilcher's, but there may be more scatter in the earlier work.

4. Correlations of enthalpies of formation for the alkanes favor the

more uniform values (with respect to the increment for CH

2

) of Pittam and

Pilcher and may favor a value for methane that is between those of the two

studies. The latter may be more of an evaluator's decision than a property

of the correlations: methane occupies a unique position.

A preliminary analysis of these four factors indicates that both

studies should be considered in a reassessment. Neither outranks the

other. An average value of 890.6 ±0.7 kJ mol ^ (2a uncertainty) is an

appropriate choice, considering methane alone.

In our opinion, however, there should be a thorough-going reassessment

of the thermodynamic properties of the lower alkanes as a group befo-'’e any,

probably small, readjustments are made. Somayajulu et al . [98] have suggested

a possible revision. It is probable, however, that an improvement can come

only from new, highly accurate measurements. There also are additional considerations that are related more to gas technology than to the combustion

experiments.
100

5. Recording flow calorimeters that are commonly used in determining

heating values are calibrated against samples of methane for which the

heating values are certified. In the U.S.A. , and possibly elsewhere, these

certifications are now based on analysis of the gas and calculation of the

heating value, using procedures outlined in this document. If the enthalpy

of combustion of methane is revised, a plan must be developed to (a) recertify the standard samples and (b) recalibrate the calorimeters (or

revise the calibration factors now in use). It would be advisable to

develop this plan now and have it agreed upon by the various standardssetting and user groups, '

6. Based on existing measurements a revision of the enthalpy of combustion of methane would be to a value that lies within the uncertainty (2a)

of the present recommendation and the new value would have an equally large

uncertainty. It may then be questioned whether a new value, strongly overlapping with the old one, should be introduced in technical practice.

Because of these last two factors and the desirability of a reassessment of the lower alkanes as a group, we have retained the older value which

is the basis for many existing tables of calorific values. When a revision

is made of zx^//(2S8.15 K) for any of the gases of interest here, the procedures

given in this document and data in the tables will permit straightforward

calculation of new heating values at the various reference conditions.

Compiled data on methane

Several compilers, have reevaluated the original data and have reported

values (primary compilations). Others have adopted and applied these

results (secondary compilations). The more significant compilations of both

types are listed in Table A7e, together with the reported value for methane

(converted to SI units if originally given in other units).

Except for the selection by Pedley and Rylance [45] all of these

values can be traced to Rossini's measurements [55] usually via the API
101

tables [2,76,77]. Slight differences reflect different reanalyses, weighting,

rounding procedures or variant auxiliary data. In the light of the spread

of the data they are immaterial.

In this same table is listed the atomic mass scale used by each

compilation, if it could be identified. If not, the most recent scale

with which molecular masses are consistent is indicated. The atomic

mass scale designations are keyed to Table A5a.

Vapor pressure of water

The values used here are given in Table A7f for each of the reference

temperatures. The correlation equation given in the table may be used for

other temperatures in the same range.
102

Table A7a. Heat capacity equations for auxiliary substances and hydrocarbons for the

range 268 to 308 K for the equation C°/R = a P

+ b{T - 273. 15 K) +

ciT - 273.15 K)-

* [1]

C (298.15 K)

lO^c/K^

P

Compound Formula a lOOij/K

J mol’^ K Reference

methane CH^(g) 4. 1947 0.3539 1.49 35.71 [4]

ethane C2^(g) 5.9569 1.377 1.69 52.48 [5]^

propane C^HgCg) 8.2671 2.286 1.90 73.59 [5]^

n-butane C,Hio(g) 11.109 2.875 1.82 98. 44 [6]

2-methyl propane C4Hio(g) 10.824 3.153 0.82 96.59 [6]

n-pentane Ct^H.,^(g) 13. 587 3.288 2.98 119.96 [7]^" 2-methy1 butane C5H^2^9) 13.412 3.540 1.40 118.94 [7]“

2 , 2-dimethyl - propane 13.584 3.346 0.45 120.96 [7]^

n-hexane CgH^ACg) 16. 134 3.986 3.60 142.62 [7]^^ 2-methyl pentane 16.064 4.172 2.21 142.35 [7]^" 3-methyl pentane 15.768 4.351 1.11 140.20 [7]“’ 2-2-dimethylbutane C6Hi4(g) 15.877 4.579 0.52 141.55 [7]°

2,3-dimethylbutane 15.739 4.139 1.77 139.56 [7]“' cyclopropane C^HgCg) 6. 100 2.285 2.33 55.59 [1]^

cyclobutane C4Hg(g) 7.769 2.942 2.92 70.86 [1]^

cyclopentane CjHioO) 3.966 4.101 0.82 83.11 [8]

cyclohexane CeHi2(g) 11.490 4.600 3.00 105.25 Cl]'' ethyne C2^(g) 5.063 0.985 -1.59 44.06 [1]'' ethene C2H4(g) 4.880 1.024 1.50 42.78 [9]

propene Cg^Cg) 7.277 1.858 0.80 64.41 [10]

benzene CgHgCg) 9.435 3.371 2.46 85.58 [11]

he 1 i urn He(g) 2.500 0.000 0.00 20.78 [1]*^ argon Ar(g) 2.500 0.000 0.00 20.78 [l]'^ oxygen 02(g) 3.520 0.044 0.28 29.37 [12]®

hydrogen H2(g) 3.433 0.155 -0.74 28.83 [12]®

nitrogen N2(g) 3.502 0.006 0.00 29.13 [12]

hydrogen sulfide H2S(g) 4.070 0.118 0.23 34.10 [12]

sulfur dioxide SO^Cg) 4.707 0.439 0.13 40.06 [12]

carbon monoxide C0(g) 3.503 0.009 0.09 29.15 [12]

carbon dioxide CO^Cg) 4.324 0.580 -0.65 37.12 [14]

^Recalculated from the partition coefficient of the molecule, using procedures and

molecular parameters in the reference.

'^Calculated using the procedure in the reference.

QUnpublished calculations from the molecular partition function.

= 2.5 S for the monatomic gas at temperatures such that kT is well below the first

eitited electronic state.

^Either taken directly from the table in the reference or recalculated from the partition function using the same parameters as used for the tables.

References for this table are given at the end of the appendix.
103

Table A7b. Enthalpies of combustion of selected hydrocarbons

and sources of the data

Substance

methane

ethane

propane

n“butane

2-methyl propane

n-pentane

2-methyl butane

2 ,2-dimethylpropane

n-hexane

2-methylpentane

-A^H(298.15 K)^

kJ mo 1 ^

890.31 ± 0.31

1559.84 ± 0.46

2219.90 ± 0.54

2877.25 ±1.2

2868.72 ±2.0

3535.77 ±0.5 (g)

3509.04 ±0.5 (£)

3528.87 ±0.4 (g)

3503.64 ±0.4 (£)

3514.60 ±0.6 (g)

3492.22 ±0.6 (1)

4194.75 ±0.6 (g)

4163.12 ±0.6 (£)

4187.64 ±1.0 (g)

4157.68 ±1.0 (2)

Data Sources^

from 17, 61, (19,30,41,69,70),

48, C: =2, =3, =28, 7^45, =77

from 17, (23,70), C: =2,

=3, =28, 7^45, =77

from 17, 56, (23,70), C: =2,

=3, =28, 7^45, =77

from 17, 50, 48, C: =2, =3,

7^28, 7^45, =77

from 18b, 57, 50, (70),

C: =2, =3, 7^28, 7^45, =77

from 32, 47, 52, ^ (vap), 56,

60, C: 7^2, 7^3, =18a, =28, =45,

ini

from 39, 63(vap), 60,

62, C: 7^2, 7^3, ^-28, =45, ill

from 39, C: 7^2, 7^3,

7^28, -v45, ill

from 18b, 31, 52, 34, ^(vap),

71(vap), (68, =74, 65) C: =2,

=3, =28, =45, =76, -^77

from 18b, M(vap), 71(vap),

C: =2, =3, =28, =45, =76, ^77
104

Table A7b (cont'd)

Substance

3-methyl pentane

2 ,2-di methyl butane

2,3-dimethyl butane

cyclopropane

cyclobutane

cyclopentane

cyclohexane

ethyne

ethene

-A^i/(298.15 K)^

kJ mol

^

4190.32 ±0.9 (g)

4159.94 ±0.9 (A)

4176.34 ±1.0 (g)

4148.52 ±1.0 (:^)

4184.17 ±0.9 (g)

4154.92 ±0.9 U)

2091.37 ±0.5

2745.16 ±0.5 (g)

2720.52 ±0.5 (£)

3319.59 ± 0.8 (g)

3290.93 ± 0.8 (jZ)

3952.96 ± 0.7 (g)

3919.86 ±0.7 (£)

1299.59 ±0.8

1410.97 ±0.4

Data Sources*^’^’^’®

from 18b, 51, 44(vap), 71(vap),

C: =2, =3, =28, =45, =76, <11

from 18b, 51, M(vap), 38(vap),

C: =2, =3, =28, =45, =76, ni

from 18b, 51, M(vap), Tl(vap),

(69, 70), C: =2, =3, =28, 5^45,

=76, HI

from 17, -W, (21, 23, 29) C: =:

=28, =45

from 18b, ^ = 26, ^(vap),

C: 5^3. #28, #45, #77

from 17 & 18b, 8(vap),

35, 64, (68 = 74), C: =2,

=3, =28, #45, =77

from 17 & 18b, 31, 33,

44(vap), -^35, ''-64, M3, (68 =

75, 59, 66, 67), C: =2, '-3,

=28, =45, =77

from 17, ^

= 25, (21, 23, 41,

42), C: =2, =3, #28, #45, =77

from 17, 58, 53 = 36, -'-78, 79 !

(19, 21, 23, 29, 30, 41, 42) |

C: =2, =3, #28, #45, =76, #77
105

Table A7b (cont'd)

Substance

propene

benzene

-A^H(298.15 K)^

kJ mol

^

2058.44 ±0.6

3301.51 ±0.5 (g)

3267.71 ± 0.5 (£)

Data Sources^’^’^’^

from 18b, 58, 73 , ^51, (21,

23, 69, 70), C: =2, =3, ^28 ,

5-^45, =77

from 2 & 17, 31, 49, 44(vap), 27,

C: =2, =3, =28, =45, =77

^The uncertainty is that given in the source from which the value has been taken.

See Appendix 7b for those assigned here.

*^References are given in the list at the end of this appendix.

"References are marked to show how data in them were used or correspond to the

selected values "from 10" - the value was taken from reference 10. 17 (double

underline): used in making the selection (in the source cited); recent

value, not used but to be considered in new evaluations; 31 (unmarked):

examined but not used: =2; same value; '^ 7 : approximately the same; 5^3; not

the same value; (17): not used, of historical interest only; C: compilations of data follow this sign.

The entry for n-pentane should be read as follows. The selected value was

taken from [32]. It was based on data in [32, 47, 52 and 44], the last being

for the vaporization enthalpy. References [56] and [60] were not used. The

present value agrees with the compilations [18a], [28] and [45], while it

differs from [2], [3] and [77].

°Some references not cited in the sources quoted have been added during the

preparation of this table. They are mainly for enthalpy of vaporization and

for compilations of data.
106

Table A7b (cont'd)

^Three references are to the API project 44 tables and successors:

[76], (1967); [77], (1981). The 1967 version is cited only of the

are significantly different.

[2], (1953)

other two
107

Table A7c. Heating value of methane - sources of data

Experimental Measurements -A^/f°(298. 15 K)/kJ mol

A. Andrews, 1848 [19] [883.6] a

B. Favre and Silberman 1852 [30] [864.1] a

C. Thomsen, 1880, 1886, [69,70] 887.0 ± 3 b

D. Berthelot, 1881, [21] 892.3 ± 9 b

E. Mixter, 1901, [41] [888.6] c

F. Rossini, 1931, [55] 890.31 ± 0.6 d

G. Roth and Banse, 1932, [61] 891.8 ± 2. e

H. Pittam and Pilcher, 1972, [48] 890.71 ± 0.8 f

Notes for Table A7c

^Not comparable to the later measurements. Corrected from ''^15 °C by comparison to authors' measurements of

^As recalculated by Rossini [55] to 30 °C and corrected by us to 25 °C.

From heat of decomposition, in the presence of acetylene. Corrections

are approximate.

^Uncertainty is precision only, 2a. Corrected from 30 °C to 25 °C.

®As recalculated by Cox and Pilcher [28].

'Uncertainty is precision only, 2a.
108

i

I I

Table A7d. Data points and statistics for studies of methane

by Rossini and by Pittam and Pilcher

Author -A^//°(298.15

1

K)/kJ mol’^

Individual Points Averages

Rossini [55]^ 891.823*^

1

1

Pittam and Pilcher [48]^ 891.23

Pittam and Pilcher 891.17 Ave (P&P): 890.71 ± 0.43^ I

Rossi ni 890.633 Ave (12 pts): 890.65 ± O. 37 I

Pittam and Pilcher 890.62

1

Pittam and Pilcher 890.61 Ave (R-6 pts): 890.56 ±

0.66J

Rossini 890.503 Ave (11 pts): 890.52 ± 0.27

Pittam and Pi 1 cher 890.36

Pittam and Pilcher 890.34

Rossini 890.340 Ave (R-5 pts): 890.31 ± 0.33fl

Rossini 890.061

1

Rossi ni 890.013

^A^//(298.15 K) = tabulated points in [55] -0.613 kJ mol

^Rejected by Rossini, a reasonable decision with only his set to consider,

but borderline for the combined set.

^A^/f°(298. 15 K) = tabulated points in [48] -0.02 kJ mol

^Uncertainties are 95% confidence level for the mean but for precision only.

(
109

Table A7e. Enthalpy of combusti on of methane as an ideal gas and relative

atomic mass scales given in compilations of thermochemi cal data

Compi 1 ati on Atomic ^’Sss

Seal

e^

-A^if°(298.15 K)^^

kJ mo 1 ^

International Critical Tables (1929) [87] 1923 (G) 882. 37

NBS Circular C461 (1947) [88] 1941 (F) 890. 35

N6S Circular C500 (1952) [89] 1941 (F) 890.35

Rossini et al . (1953) [2] (API Project 44) 1941 (F) 890.35^'

Landol t-Bornstei n (1961) [90] 1953 (E)^^ 890.36"

NBS Tech. Note 270-3 (1968) [16] 1961 (D) 890.36

Stull, Westrum, Sinke (1969) [3] 1941 (F) 890.34-

Cox, Pilcher (1970) [28] 1961 (D) 890.32

JANAF Thermochemical Tables (1971) [81] 1961 (D) 890.33

ASTM DS4A (1971) [97] 1969 (C) 890. 36"

Domalski (1972) [17] 1961 (0) 890 31

NBS Tech. Note 653 (1974) [91] 1969 (C) —

NBS Tech. Note 684 (1976) [92] 1969 (C)

ASTM D-3588 (1977) [93] 1971 (3) 890. 36'*

Pedley, Rylance (1977) [45] 1971 (3) 890.7

GPA Publication 2145-77 (1977) [94a] 1961 (D) 890.40^’®

CCDATA Key Values (1978) [95] 1969 (C) —

Robie, Hemmingway, Fisher (1978) [95] 1971 (B) 890.36®

GPA Publication 2145-SI-80 (1980) [94b, c] 1969 (C) 890. 33®’®

GPA Engineering Data Book (1980) [96] 1969 (C) 890.33®’®

This work 1981 (A) 890. 31

^Year the scale was established and the code given in Table A5a.

^Values reported in other units or at other temperatures have been convertea using

factors in this document.

^Value for methane taken from another compilation.

'^Physical scale ^^0 = 16 indicated without elaboration. Atomic mass scale may be

earlier than 1953. Reported enthalpy has been multiplied by 4.184/4.1855 to con- vert to current absolute joules.

0 Calculated from values at 288.15 K or 288.71 K from enthalpies of combustion in MJ

m

^ using thermodynamic properties developed in this work. Values in MJ kg ^ in the

_same .sources are in error.
110

Table A7f. Vapor pressure of water, pCH^O), used in this document

r/K p(H20)/Pa^

273.15 611.213

288.15 1705.32

288.706 1767.34

298.15 3168.74

'^Using the 1968 International Practical Temperature Scale and the Vapor

Pressure Equation [84]:

n pCH^O) = ^ ^ 68

1=0

where

= -0.29912729 x 10^, = -0.60170128 x lo"^,

g^ = 0.1887643854 x 10^, g^

= -0.28354721 x lO'^,

g^ = 0.17838301 x 10*^, g^

= -0.84150417 x lo'^,

= 0.44412543 x lO'^^, and g^ = 0.2858487 x 10^.
Ill

Appendix 7b. Uncertainties in thermodynamic quantities

Uncertainties in the standard enthalpies of combustion at r = 298. 15 K

Cox and Pilcher [28], Good [32], Oomalski [18a], Pedley and Rylance

[45] and Chao [80] have estimated uncertainties in some or all of

the standard enthalpies of combustion or the enthalpies of formation at

T = 298.15 K of the hydrocarbons listed in Table 4. Their estimates ai^e

given in Table A7g. Where necessary these have been converted to kilojoules per mole.

The selected uncertainties s and relative ur.certai nti es s /A H° for

c c c this work are listed in the right hand columns of Table A7g. They are

considered to represent 95 percent confidence limits for the values of

A at 298.15 K. c When the enthalpy of formation is derived from the enthalpy of

combustion, the estimated uncertainty in the enthalpy of formation s^ is

related to the estimated uncertainty in the enthalpy of combustion s^ by

the relationship

Sf = + [as^CCO^)]^ + [(b/2)Sj:(H20)]^

where s^(C02) is 0.13 kJ mcl s^(H.,0) = 0.042 kJ mol ^ and a and b

give the stoichiometric coefficients in the equation of combustion.

+ (b/4 + 3)0^ = aCO^ + (b/2)H20

Because s^(C02) and s.p(H20) are substantially smaller than s^ or s^ for

the hydrocarbons the difference s^ - s^ is generally quite small. The

sign of the difference depends on whether is derived from s^ or vice

versa. Typical differences may be found in the column labeled C&P in

Table A7g. For the purpose of this discussion, little distinction will be

made between s^ and s^. However s^ is used preferentially if it is available.
112

Table A7a. Estimates of uncertainties s ^ in enthalpy of combustion or s^ in

enthalpy of formation by various authors

C&P G D P&R Chao This Work lO^s

s

.

s s s s s s A H°

f c c c c c c c

kJ mol

-1

methane 0.29 0.25 - 0.29(. 50) 0.4 0.50*^ 0.62 7.0

ethane 0. 50 0.^6 - 0.46 0.2^ 0.59*^ 1.01 6.5

propane 0.59 0.54 - 0. 54 0.3 0.63^ 1.02 4.6

n-butane 0.67 0.63 - 1.26 0.6 0.67^ 1.44 5.0

2-methlypropane 0.54 0.46 - 2.09(1.26) 0.5 1.44 5.0

n-pentane 0.63 (0.63) 0.46 0.46 0.7 0.46 1.3

2-methlybutane 0.63 (0.63) 0.46 0.46 0.6 0.46 1.3

2, 2- dimethyl propane 1.05 0-96 0.50 0.63 0.7^ 0.50 1.4

n-hexane 0.75 (0.75) - 0.63^ 0.4 0.63 1.5

2-methyl pentane 1.00 0.92 - 1.05^ 0.9 1.05 2.5

3-methlypentane 1.00 0.92 - 1.05^ 0.9 1.05 2.5

2 ,2- dimethyl butane 1.00 0.92 - 0.96^ 0.8^ 0.96 2.3

2 , 3-dimethyl butane 1.00 0.92 - 0.92 0.7^ 0.92 2.2

cyclopropane 0.59 0.54 - - 0.5 0.54 2.8

cyclobutane 0.59 0.50 - - 0.5^ 0.50 1.8

cyclopentane 0.84 0.71 - - 0.8^ 0.71 2.1

cyclohexane 0.63 (0.63) - - 0.3^ 0.63 1.6

ethyne 0.79 0.63 - - 1.0 0.63 4.8

ethene 0.42 (0.42) - - 1.2^ 0.42 3.0

propene 0.67 (0.67) - - 0.4 0.67 3.2

benzene 0.54 (0.54) - - 0.3^ 0.54 1.6

oxygen

carbon dioxide 0.13 -

water (liquid) 0.042^ -

Parentheses in the C&P column indicates that s c was set equal to s^ (for which

a value is given)

^Value for gas phase combustion is taken as the same• as for the liquid.

Indicates the value for s^ is used.

Code to heading: C&P, Cox and Pilcher [28]; G, Good [32]; 0, Domalski [18b]; P&R,

Pedley and Rylance [45], Chao [80].
113

The following procedure was used in arriving at the estimated uncertainties

s^ (this work) given in Table A7g. Two tests were made for the normal

hydrocarbons to C^. Both combine estimates of accuracy made by evaluators

with the spread between the means of the two sets of experiments. The

uncertainties of Domalski and Chao were averaged and added quadratical ly to

the difference between the average values of presented by Domalski

[18b] and by Pittam and Pilcher [48]. Domalski 's estimate of s^ for

methane was considered to be low as a result of the difference 0.4 kJ

mol ^ between the work of Pittam and Pilcher and the previously accepted

values for methane. Therefore, Chao's estimate alone was used in the quadratic addition.

Table A7h. Estimated total uncertainties - first test

Substance s /kJ mol ^ s /kJ mol ^ lO^s /A

c c c c Chao D A this work

methane 0.50 (0.29)0. 50 0.40 (0.56)0.64 7.2

ethane 0.59 0.46 0.86 1.01 6.5

propane 0.63 0.54 -0.73 0.94 4.2

n-butane 0.67 1.26 0.29 1.01 3.5

The results are summarized in Table A7h above. in which s«, and Chao ^D

sent the two estimated errors, A represents the difference between the

values selected by Domalski (these tables) and the results presented by

Pittam and Pilcher. The value of s is calculated as

c = [»=Chao ^ ,

This has been reduced to a relative uncertainty in the last column.

A second test was made, and this was applied also to 2-methylpropane.

In this test, the uncertainties listed by Domalski were summed quadratical ly

with the uncertainties listed by Pittam and Pilcher for their combustion

experiments and the values of A. This value is
114

Sc = (s2 . s% .

The data, the values of the square root of the sum of squares and the relative uncertainties are given in Table A7i.

Table A7i. Estimated total uncertainties - second test

substance -A H°

c

s c -A H° s c c

kJ mol

^

A

10^(1^

A H°

c relative

Domal ski Pittam and Pilcher

CH^ 890.31 0.29 890.71 0.33 0.4 0.62 7.0

^2^6 1559.84 0.46 1560.69 0.25 0.86 1.01 6.5

^3^8 2219.90 0.54 2219.17 0.46 -0.73 1.02 4.6

^"^4^10 2877.25 1.26 2877.54 0.63 0.29 1.44 5.0

'-Vio 2868.72

1.26

(2.09) 2869.00 0.59 0.28 1.42 5.0

This calculation gives a reasonably uniform set of absolute and relative

uncertainties and is adopted for the data presented for these compounds

j

in this document as shown in the right hand columns of Table A7g. Here

| again an adjustment was made to Domal ski's estimate of Sc for 2-methlypropane.|

He had given a large estimate based on disagreement between two sets of J

combustion measurements made by Rossini [57] and Prosen, Maron and ,[

Rossini [50]. The later results of Pittam and Pilcher

i, eliminate the earlier measurements by Rossini from further consideration. .!

i On this basis the uncertainty for Domal ski's average value for 2-methlypropane

was reduced to the same uncertainty for n-butane for this calculation. I

For the pentanes, the uncertainties estimated by Good [32] were considered to be appropriate and were adopted without change.

For the hexanes, the estimates of uncertainty made by Domal ski [18b] j

t are used. For the cyclic and unsaturated hydrocarbons the uncertainties

i given by Cox and Pilcher [28] are used. f
115

Uncertaifiti’es 1n heat capacities and enthalpies

For all of the alkane hydrocarbons, Scott, [7], presents values of

for which he gives the first uncertain figure to be at the hundredths

_1 _T -T -1

cal mol "^K . On this basis, we infer s(Cp) < 0.05 cal mol K which

would be s(C ) < 0.21 J mol^K Over the range 273.15 K to 298.15 K

P this amounts to an error of H{T) - 7f(298.15 K) < 25 x 0.21 J mol

^

(5.2 J mol ^). We adopt the conservative position that s{H{T) -

if(298.15)} = 5.2 J mol ^ for all the hydrocarbons.

In Scott [7] individual uncertainties in the heat capacities of the

hydrocarbons are given four ways: as s(C , observed), s(C , calculated),

P P A(C obs - C calc), and statistical uncertainty at 298.15 K calculated

P P

from the appropriate variance-covariance matrix.

Table A7j. Estimated uncertainties in of hydrocarbons from Scott's [7]

correlation of thermodynamic functions

Substance f/K Uncertainties at T/K at 298.15 K s(C ,obs) s(C ,calc) A(C ,obs-calc) s(C .matrix)

P P P P

2 -1 -1

10^ J mol

propane 293.15 17 8 71 8

n-butane 4

2-methylpropane 273.15 17 8 21 13

n-pentane 298.15 25 8 4 8

2-methyl butane 317.2 25 21 4 37

2 ,2- dimethyl propane 298.15 25 25 8 25

n-hexane 333.85 29 13 13 17

2-methyl pentane 325.10 29 17 46 17

3-methy‘' pentane 332.10 29 25 21 42

2, 2- dimethyl butane 341.55 33 17 8 29

2, 3-d i methyl butane 341.60 33 21 4 71

Scott recommends that the uncertainties in the heat capacities be taken as

twice the values listed. We have taken twice the values listed as the

entries in the column headed s(C°,obs), and these are listed in Table 15.
116

Table A7k. Comparative calculations of C°

Wilhoit [1] Scott[7] Gurvich[82] JANAF[81] Wagman[16] Angus[9 , 10]

g mol'^K"^

CH4 35.71 (35.68)* 35.695 (35.64) (35.31) 35.71

^ 2^2 44.06 44.036 (44.095) (43.93)

C 2H

4

42.88 42.882 (42.89) (43.56) 42.88

52.47 (52.59) 52. 486 (52.63)

''3^8 73. 5S (72.96)

cycl 0 <:4»8 70.86

n 98.44 (97.49)

i 96.59 (96.77)

cycl 0 83.11

n C5H12 119.96 (120.04)

1 -me ^5^12 118.95 (118.86)

2,2 C5H12 120.96 (120.83)

cycl 0 '' 6^12 105.25

n =6»14 142.62 (142.59)

2-me S«14 142.25 (142.21)

3-me 140.20 (140.12)

2,2 ^6«14 141. 55 (141.46) -

2,3 ^6^14 139.56 (139.41)

<^ 6«6 85.58

°2 29.37 29.378 (29.37) (29.35)

CO

2 37.12 37.135 (37.13) (37.11) 37.13

H.

c 28.83 28.836 (28.836) (28.824) 1

C 8.536 (8.527) (8.527)

Values in parentheses have been converted from other units.
117

There are slightly different calculated values of in various compendia. They are compared in Table A7k. The differences are due largely to

choices of molecular parameters, choices of energy differences between conformers and the calculational models. The differences are small and ignored

here.

Uncertainty in the vapor pressure of water

The uncertainty of the vapor pressure of water has several aspects:

(1) the uncertainty with which the vapor pressure has been experimentally

determined, which is taken by us as the uncertainty in the partial pressure

of water vapor in the water saturated ideal -gas fuels; (2) the deviation of

the vapor pressure of water from its standard value when in the presence of

hydrocarbons; (3) the degree to which true saturation is achieved in real -gas

mixtures measured under "saturated" conditions. The third aspect above is a

measurement problem encountered in the field that must be treated as a separate

issue. For the purposes of this document we assume uncertainty (1) to be

s(p,H20) = 1 Pa. While Wexler [84] does not specifically state estimates

of uncertainty, he compares his formulation with measurements by Douslin

[85] and Besley and Bottomly [86] and finds differences not exceeding

1.3 Pa in the range 273.15 to 298.15 K. At T = 273.15 K the triple point

pressure is not uncertain by more than 0.01 Pa.

Uncertainty in the relative partial pressure of fuel

Using criterion (1) for the uncertainty in the partial pressure of

water, we calculate the relative uncertainty s(p, fuel )/(p, fuel ) in the

-5

partial pressure of fuel at the reference states to be 1 x 10 , as

fol 1 ows.

Taking s(p,H20) = 1 Pa and values of s(p,fuel) ranging from a maximum

of 101.560 kPa - 0.611 kPa = 100.95 kPa at 273.15 K to a minimum of 100 kPa -

i 3.168 kPa = 96.83 kPa, s(p, fuel )/(p, fuel ) ranges from 9.9 x 10 ^ to 10.3
118

X 10 Since this range is much smaller than our knowledge of the uncertai

of p(H20 ), we assign a value of 10 x 10 ^ for s(p, fuel )/(p, fuel ) for water

saturated gas at all pressures and temperatures.
119

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65. Stohmann, F. ; Kleber, C. ; J. prakt. Chem. (1891), [2], 43, 1-21.

66. Stohmann, F. ; Langbein, H, ; J. prakt. Chem. (1892), [2], 45, 305-356.

67. Stohmann, F. ; Langbein, H. ; J. prakt. Chem. (1893), [2], 48, 447-453.

68. Swietoslawski , W. ; J. Am. Chem. Soc. (1920), 42, 1092-1110.

69. Thomsen, J. ; Ber. (1880), U, 1321-1334.

70. Thomsen, J. ; "Thermochemi sche Untersuchungen" ; J.A. Barth, Leipzig

(1886).

71. Waddington, G. ; Smith, J.C.; Scott, D.W. ; Huffman, H.M. ; J. Am. Chem.

Soc. (1949), 71, 3902-3906.

72. Wagman, D.D.; Kilpatrick, J.E.; Pitzer, K.S.; Rossini, F.D.; J. Res.

Nat. Bur. Stand. (U.S.) (1945), 467-496.

73. Wiberg, K.B.; Fenoglio, R.A. ; J. Am. Chem. Soc. (1968), 90, 3395-3397.

74. Zubov, P. ; Zhur. Russ. Fiz. Khim. Obshchestva, Chast Khim. (1901), 33 ,

708-722.

75. Zubov, P. ; Zhur. Russ. Fiz. Khim. Obshchestva, Chast Khim. (1913), 45,

240-251.

76. Zwol inski, B.J.; Wilhoit, R.C.; "Heats of formation and heats of

combustion"; in "American Institute of Physics Handbook", Am. Inst.

Phys. , New York (1972), ch. 4, sect. 41, pp. 4-316 to 4-342. Data in

this reference are drawn from API project 44 and Thermodynamic

Research Center (Texas A&M Univ.) loose leaf tables, (ca. 1967).
125

References (cont'd)

77. Thermodynamic Research Center Hydrocarbon Project; "Selected values of

physical and thermodynamic properties of hydrocarbons and related compounds"; Thermodynamic Research Center, Texas A&M Univ. , College

Station, Texas. These tables are the successor to those of API project

44. Current (1981) loose leaf tables for compounds of interest are:

Cl - C5 hydrocarbons, Table In, (1955); C6 hydrocarbons , Table

23-2-(1.201)n, (1975); cycloalkanes. Table 23-2-(3. 100)n , (1968);

alkenes. Table 23-2-(5. 1101)n, (1981); alkynes. Table 25n, (1952);

normal alkyl benzenes. Table 23-2-(33. 1100)n , (1962).

78. Kistiakowsky , G.B.; Nickle, A.G.; Disc. Faraday Soc. (1951), 10 ,

175-187.

79. von Wartenberg, H. ; Krause, G. ; Z. physik. Chem. (1930), 151A ,

105-108.

80. Chao, J. ; private communication (1981).

81. Stull, D. R. ; Prophet, H. ; editors, "JANAF thermochemical tables"; 2d

ed. , Nat. Stand. Ref. Data ser. , Nat. Bur. Stand. (U.S.) (1971),

82. Gurvich, L.V. ; Veits, I.V.; Medvedev, V.A. ; Khachkuruzov , G.A.;

Yungman, V.S.; Bergman, G.A. ; editors, "Termodi namicheskie svoistva

individual ' nykh veshchestv" ; [Thermodynamic properties of individual

substances], Glushko, V.P.; general editor "Nauka", Moscow (1978), 1;

(1979), 2; (1981), 3.

83. Armstrong, G.T.; "Calculation of the heating value of a sample of high

purity methane for use as a reference material", Nat. Bur. Stand.

(U.S.) Tech. Note 299 (1966).

84. Wexler, A.; J. Res. Nat. Bur. Stand. (U.S.) (1976), 775-785.

85. Douslin, D.R.; J. Chem. Thermodynamics (1971), 3, 187-193.

86. Besley, L. ; Bottomley, G.A. ; J. Chem. Thermodynamics (1973), 5,

397-410.
126

References (cont'd)

87. Washburn, E.W. ; editor, "International critical tables of numerical

data, physics, chemistry and technology"; McGraw Hill Book Co., Inc.,

New York (1926).

88. Rossini, F.D.; Pitzer, K.S.; Taylor, W.J.; Kilpatrick, J.E.; Ebert,

J.P.; Beckett, C.W. ; Williams, M.G.; Werner, H.G.; "Tables of selected

values of properties of hydrocarbons"; Nat. Bur. Stand. (U.S.) Circ.

(1947), C461 .

89. Rossini, F.D.; Wagman, D.D.; Evans, W.H.; Levine, S. ; Jaffe, I.;

"Selected values of chemical thermodynamic properties"; Nat. Bur.

Stand. (U.S.) Circ. (1952), C5QQ .

90. Auer, W. ; "Organische Substanzen" ; in "Landolt-Bornstein Zahlenwerte

und Funktionen aus Physik, Chemie, Astronomie, Geophysik, und Technik"

(Sechste Auflage), Springer-Verl ag, Berlin (1961), II. Band, 4. Teil

table 24132, p. 261. See also p. XI (atomic weights) and pp. 1,15

(energy conversion factors).

91. Goodwin, R.D.; "The thermophysical properties of methane, from 90 to

500 K at pressures to 700 bar"; Nat. Bur. Stand. (U.S.) Tech. Note

(1974), 653 .

92. Goodwin, R.D.; Roder, H.M.; Straty, G.C.; "Thermophysical properties

of ethane from 90 to 600 K at pressures to 700 bar"; Nat. Bur. Stand.

(U.S.) Tech. Note (1976), 684 .

93. Mason, D.M. ; "Research report on standard method of calculating

calorific value and specific gravity of gaseous fuels"; ASTM Research

Report, Am. Soc. for Testing and Materials, Philadelphia (1977), no.

D03-1000

,

(Documentation for ASTM D3588).
127

Fleferences (cont'd)

94. (a). "Standard table of physical constants of paraffin hydrocarbons

and other components of natural gas"; Gas Processors Association,

Tulsa (1977), publication 2145-77 . (b). , (1980), publication

2145-80 . (c). , (1981), publication 2145-81 .

95. Robie, R.A. ; Hemingway, B.S.; Fisher, J.R.; "Thermodynamic properties

of minerals and related substances at 298.15 K and 1 bar (105 pascals)

pressure and at higher temperatures"; Geological Survey (U.S.) Bulletin

(1978), 1452 .

96. "Engineering data book"; Gas Processors Suppliers Association, Tulsa

(1981).

97. "Physical Constants of hydrocarbons to ; ASTM Data Series,

American Society for Testing and Materials, Philadelphia (1971),

Publication PS 4A .

98. Somayajulu, G. R. ; Wilhoit, R. C. ; Hall, K. R. ; "Enthalpies of formation of the lower alkenes"; manuscript, received May 1982.
128

Appendix 8. Relationships between thermodynamic quantities

in Tables 3 and 4.

Standard Enthalpy of formation from the elements

The enthalpy of formation of a compound in Tables 3a and 3b from

its elements is the enthalpy change of a process such as is illustrated

below for H 20 (l) and C 02 (g).

H2(g) + I 02 (g) = H20(£); (A8.1)

C(c, graphite) + O^Cg) = CO^Cg); (A8.2)

Similar reactions can be written for the other substances. The enthalpy

change, A^° , is for the process with the substances in their standard

states. For example, the first reaction at 298.15 K is represented as

fol 1 ows

A./f°(H20, £, 298.15 K) = £, 298.15 K)

g, 298.15 K) - I

g, 298. 15K), p = 101.325 Pa (A8.3)

Information in parentheses specifies the conditions, and if the conditions

are clearly stated elsewhere some of this information may be omitted as

being understood. The individual E° terms are not measurable, only changes

in them are, e.g. A^° , A^lf° or [H(r^) -

The enthalpy of reaction (including formation) at another temperature

can be calculated from that at 298.15 K using a cycle of reactions

A+B^C+D (298.15 K)

ti (A8.4)

A + B C + D
129

= AH(1) + A//(2) + A^i/(298.15 K)

+ A//(3) + A//(4) .

For reaction A8.1, the formation of water in its standard state, this

, becomes:

A^°(H20,1,7’2) = [H°(298.15 K) -

+ 0.5[/f°(298.15 K) - (A8.5)

+ A^°(H.0,1 14 ,298.15 K) - [//(298.15 K) - (H.0,1) . 4 4 Note that all enthalpy differences have been written with T = 298.15 K first.

The degree sign indicates standard state conditions. When T - 288.15 K

equation A8.5 gives one of the conversions among reference conditions needed

to calculate Table 3.

Inserting values from Table 3a and 3b into equation A8.5 we find:

A/f^CH^O, 1, 288.15 K) = -285.830 kJ mol"^

-0.753 kJ mol"^ + 0.288 kJ mol'^ + |

x 0.293 kJ mol’^. (A8.6)

-A^°(H20, 1, 288.15 K) = 286.1485 kJ mol'^.

This reproduces the value found in Table 3.

By an analogous process the enthalpy of combustion of a hydrocarbon

can be determined at a different temperature. The combustion of a gaseous

hydrocarbon is represented by the following equation:

CjHjjCg) + (a + |) O^Cg) = a CO^Cg) + |

H^OCi), (A8.7)

where the stoichometric coefficients depend upon the composition of the

hydrocarbon. The standard enthalpy of combustion at temperature T -

298.15 K represents the enthalpy of combustion with each reactant and each

product of combustion in its standard state at the stated temperature.

The standard enthalpy of combustion depends on the temperature. The

conversion of the standard enthalpy of combustion from one temperature to

another will be illustrated for the combustion of methane.
130 T

CH^(g) + 2 O^Cg) = CO^Cg) + 2 H20(J^). (A8.8)

In this case a = 1; b = 4; b/2 = 2; and a + b/4 = 2; giving the stoichiometric

coefficients for carbon dioxide, water, and oxygen, for the combustion of

one mole of methane. The standard enthalpy of combustion at r = 288.15 K

can be calculated from the following relationship (which is a rearrangement

of equation A8.4):

A^/f°(298.15 K) - A^//°(288.15 K) =

a[/f°(C02, g, 298.15 K) - g, 288.15 K)]

+ H, 298.15 K) - £, 288.15 K)] (A8.9)

- (a + b/4)[H°(02, g, 298.15 K) - g, 288.15 K)]

- IH°(C H. , g, 298.15 K) - H°(C H. , g, 288.15 K].

for which the required data are found in Tables 3 and 4. Using methane

to illustrate the calculation we have:

[A^//°(CH^, g, 298.15 K) - A^/f°(CH^, g, 288.15 K)]/kJ mol'^

= 0.369 + 2 X 0.753 - 2 x 0.293 - 0.355 = 0.934. (A8.10)

Taking A^7/°(CH^, g, 298.15 K)/kJ mol ^ = -890.31 from Table 4 we find

A^H°(CH^, g, 288.15 K)/kJ mol’^ = -890.31 - 0.934 = 891.244 which rounds

to -891.24.

This process is illustrated in Figure 1.

Relationship between standard enthalpy of formation, A^° and Tor

a hydrocarbon

The general reaction A8.7 can be used as the basis for

calculating the standard enthalpy of formation from the elements of a

hydrocarbon compound A^°(C^H^), if the enthalpy of combustion of the

hydrocarbon is known. The relationship is:

A^/f"(C^Hb) = a A^°(C02 , g) + (b/2 )A^°(H20 , £)

- A^°(C^H^) - (a + b/4) g). (A8.11)
131

By the use of information in Tables 3b and 4b this equation may be used

to calculate the enthalpy of formation of methane as an example. The

temperature to which the values taken from the tables correspond must be the

same for all substances involved in the equation. We use T - 298.15 K

for the example.

g) = A^H°(C02 , g) + 2 1)

- A^//°(CH^, g) - 2 A^°(02, g). (A8.12)

Taking values from Tables 3b and 4b at T* = 298.15 K, we substitute:

- 890.31 kJ mol'^ = -393.51 kJ mol"^ + 2(-285.830) kJ mol"^

- g) - 2(0.0). (A8.13)

By rearrangement and addition

A^if°(CH^, g) = (+890.31 - 393.51 - 2-285.830) kJ mol'^

= -74.86 kJ mol. (A8.14)

Note that all of the enthalpies of formation in this example have negative

values, meaning that heat is evolved in the reaction of formation from the

elements.

The converse process, the calculation of the standard enthalpy of combustion from known values of standard enthalpy of formation, uses equation

A8.11 and data from Tables 3 and 4 in the same way.

Molar volume of the ideal gas

All gases, when treated as ideal gases,

have the same volume per mole at the same temperature and pressure.

V|Jj^(r) = RT/p

where R is the gas constant, p is the pressure and T is the absolute temperature. With the temperature in kelvin, pressure in pascal and volume

in cubic meters, this equation becomes

= 8.31441 T/p . m The value for R is taken from Table 1. Numerically it is the same for

cubic meter-pascal per mole-kelvin as for joule per mole-kelvin.
132 >'.'• 'j'l ’ . ifv

The amount of substance (molar) density is the reciprocal of the

molar volume.

r i :

t- .>;• • .’ •.. r*J, V '"i V ''••

1 .

' .

, V ,' -'r.-i',-! '*' "

'

'

. • V'

t. u

e'i'%

'W

/ " ' .'

' W t* ft «’ V -

'. - f , -T*:'

':T“':'.\ 5> *

'V, •: t ix/a Jifm'

i , ;-.• -f> -U j . TS! ‘; 5. 5C . : ^ .T .

'..ft: .; -;

J,

,,L

'

,'

- ,. / -

a'Wl-. ,.t; -it, , . V i 'i»

^

». iSjA .

_ iXjL'

^ '' • •

’ ^ ' iffif-fRo tiff'd

."'- ;.v# -iv'

< j'.' ':ty '"®'-V no f J a'f', ';'C"i ••.To. t^'i ;.V - V A' .A" 'i-'y--'* ''*

, :'

-; >'• i ’ S ;

•' ; A-'\ t^yiS

V- '

\ .SI' T'CtiVi',, -tiiu.f,

:’S-^- f . t.v6-;,.Ai:d0p , '-'L

. '" ''

. '. 'll - -K ,< ; »

>1^, lt~ - -.

; ,;

'; S-rK,'" : , i 'H •' ^ I

' -:tT '# .‘TOt

• .rf^itV. . ... -

• . V . -2» ^

'y?'

(%:, c i'M 1
133

Appendix 9. Non-ideality effects and the viria! equation of state

Two topics are discussed in this appendix: the procedures used to

correct ideal-gas properties to those of the real gas and the selection of

the virial coefficients in Table 8 (footnotes and references). Propagation

of errors is discussed in Appendix 10.

The virial equation of state

The equation of state of a real gas is well represented by the so

called virial equation of state:

pV^/RT = 1 + B{T')/V^ + C(2’)/V^ + ... , (A9.1)

in which the second virial coefficient is B{T)\ the third virial coefficient

is C{T)\ is the real -gas molar volume. The second virial coefficient

is generally considered to represent the effects of interactions involving

two molecules; the third virial coefficient, interactions involving

three molecules, and so on. At the low pressures represented by the

reference conditions, the second virial coefficient is adequate to represent

the interactions that occur and so for this document the virial equation

is truncated at the B{T) term:

pV^/RT = 1 + B{T)/V^ (A9.2)

5, as well as the other virial coefficients, is a function of temperature.

The errors introduced Dy this procedure are considered to be less than

those due to the uncertainties in the values of B{T) for the various

substances.

The virial coefficient is used for two purposes in this document:

(a) to calculate the effect of non-ideality on enthalpy of the real gas

as compared to the ideal gas; and (b) to calculate the molar volume of

the real gas as opposed to the molar volume of the ideal gas. This is
134

used both in the calculation of volumetric enthalpies of combustion and

in correcting enthalpies.

(a) Enthalpic effect.

The difference H-H° in the enthalpy of the real gas and the ideal

gas is given by:

{Hmm- H°)/RT - {3 - T(d3/(iT)]/V (A9.3)

m For the purpose of determining fCdB/dr) it is convenient to have

equations representing BiT) as a polynomial in T. Equations of the form

B(T) = c + dT + eT^ (A9.4)

have been derived and the constants c, d, and e, are given in Table A9a.

(b) Volumetric effect.

Rearrangement of equation (A9.2) gives:

= (RT/p) {1 + BiT)/Vj , (A9.5)

a quadratic equation which has the solution:

^tn " * I

(A9.6)

i d

The negative solution is ignored as being physically meaningless. is

a constant for a given value of p and T, and so the equation can be

solved exactly if B{T) is also known.

Vi rial coefficients of mixtures

Any gas has a second virial coefficient. For pure substances B{T)

is a characteristic physical property of the substance. For a mixture,

B{T) is a function of the composition as well as the pressure and temperature.

Reflecting the probabilities of binary interactions between molecules,

the virial coefficient of a binary mixture is given as a linear combination

of terms:

2 2 S(mixture) = ^l'^2‘^12 ^2 ^22 (A9.7)
135

in which and X

2 are the mole fractions, respectively, of components 1

and 2 , for which the virial coefficients of the pure substance are

and ^

2 ? respectively. represents the effect of interaction between

unlike molecules.

For more complex mixtures equation (A9.7) can be generalized by

taking pairwise interactions of all combinations of molecules.

2 2 2 B(mixture) = + ^2^22 ^ ^3^33 ^

H 2x

^X2^^2 ^ 2x^X3S^3 +

+ 2X

2

X3523 + . . . (A9. 8 )

For calculation of enthalpic effects and volumetric effects 5(mixture)

obtained in this way is used to calculate the properties of any particular

mixture for which the composition (x^,X

2

...x^) is known.

Most of the virial coefficients for individual substances and for

binary interactions for substances and mixtures with which this document

is concerned have been measured and have been compiled by Dymond and

Smith [2] .

The virial coefficients S(T) for the substances in this document are

listed in Table 8a. For binary mixtures involving methane as one gas, the

interaction coefficients 5^2 given in Table 8b. Because the importance

of terms involving minor constituents is proportional to the products of

their mole fractions, the interaction terms not involving methane have

been omitted from the tables. Many of them can be found in Dymond and

Smith, if necessary.

•k References are given at the end of this appendix.
136

Selection of virial coefficients: footnotes to Table 8.

^ The uncertainties assigned to the virial coefficients from references

[1] and [2] in Table 8 are those suggested by the respective authors

unless otherwise stated.

^ Methane virial coefficients at the required temperatures were obtained

from a smooth curve through the values tabulated in reference [1].

Q The values given by Dymond and Smith [2] bracket those given by Level

t

Sengers et al . [1].

Ethene values given by Dymond and Smith [2] bracket those given by

Douslin and Harrison [5] which were plotted. The values of Douslin

and Harrison were used at 273.15 K, 288.15 K and 2S8.15 K. The value

at 288.71 K was interpolated from the graph.

0 Ethane values given by Dymond and Smith [2] bracket those given by

Douslin and Harrison [6]. The values given by Douslin and Harrison were

plotted as well as those by Dymond and Smith. The values of Douslin

and Harrison were used at 273.15 K and 298.15 K and values interpolated

from the curve were used for other values.

^ Propane values given by Dymond and Smith [2] were plotted and interpolated to give all values.

^ n-Butane has been studied frequently. The selected values are interpolated on a curve drawn by eye through the points given by Dymond and

Smith [2].
137

The data for isobutane are scattered. Two points at 273.15 K by Jessen

and Lightfoot [10] and by Das, Reed, and Eubank [11] were averaged;

although they do not convert well to higher temperature data. A rough

curve was drawn by eye through points from 288 K to 333 K by Kretschmer

and Wiebe [12], Mason and Eakin [9]; Strain, Lichtenthaler, Schramm, and

Schafer [13], and Das, et al . [11]. This curve was used for interpolation

at the other temperatures.

The data for n-pentane are not plentiful in the temperature region below

300 K. Dymond and Smith [2] recommend values down to 300 K, and a smooth

curve was drawn through them and extended to 273.4 K where a point is

given by Kapello, et al. [14] whose data unfortunately drops 100

3 -1

cm mol below the line at higher temperatures. The line passes near a

point presented by Mason and Eakin [9], and somewhat above two points

given by Ratzsch and Bittrich [15] and the lower limit of the smoothed data

evaluated by Das, et al . [11] and below data given by McGlashan and

Potter [16] and by Hajjar, et al. [17].

For 2-methyl butane, Dymond and Smith [2] recommended values from Das

et al. [8] and a curve was drawn and values interpolated.

For 2,2-dimethylpropane a smooth curve was drawn through the data

recommended by Dymond and Smith [2] and the values were interpolated.

Benzene values were interpolated from a curve drawn from data recommended

by Dymond and Smith [2].

For 2-methyl pentane the data of Waddington, et al. [7] were plotted and

the curve was extrapolated below 298.15 K. One other point was given by

3

Osborne and Ginnings [19] at 298.15 K but it was more negative by 40 cm

,-l

mol
138

For 3-methyl pentane the data of Waddington et al. [7] were plotted from 303

to 336 K and a curve below 300 K was extrapolated. One point at 298.15 K

was given by Osborne and Ginnings [19] but it appeared to be negative by

3 -1

about 40 cm mol

For 2,3-dimethylbutane the data of Waddington et al . [7] were plotted

from 295 to 331 K and a curve was drawn and extrapolated. One point

was determined from Osborne and Ginnings [19] but it appeared to be

3 -1

negative by about 40 cm mol

For argon the uncertainties and values were recommended by Level t Sengers,

et al . [1]. A smooth curve was drawn through their recommended values

and the points at temperatures other than 273.15 K were taken from the

curve. The value at 273.15 K also is from [1].

Argon plus methane, was fitted from data listed in [2] over the

range 250 to 323 K. The data of Thomas, et al . [23] and of Byrne,

3 -1

et al. [22] and of Belim et al . [20] fall about 6 to 8 cm mol below

the rest of the data. A smooth curve was drawn through the upper data

and the tabulated values were found by interpolation on the smooth

3 -1 curve. The value of may be as much as 8 cm mol more negative

than the tabulated values that are based on the work by Schafer and

colleagues [13, 18, 21].

The only data given in [2] are by Mason and Eakin [9].

Hydrogen data are smooth. A line was drawn through the data of [9] and

[36] and the values used were obtained by interpolation.

| A line was drawn through the point from [38,39] listed in [2] at

248.53 K and midway between points of [38,39] at 291.4 K and [9] at

288.7 K. The values listed were obtained from the curve by interpolation.
139

Ref. [2] lists only ref. [37] as a source. A slightly curved line was

drawn through the two points in the range 250 to 325 K and was extrapolated to obtain the values listed.

For ethene plus methane, is given in [2] by Mason [9], McMath and

Edmister [24], Lee and Edmister [25] which form a fairly consistent set,

plus unpublished work by Lee (with Seville) [26, 27] which fall about

3 -1

6 cm mol lower. The values cited are from a smooth curve through the

upper data.

A smooth curve was drawn by eye through all points found in [2] in the

range 250 to 325 K. The points were found by interpolation on the

curve.

All values listed for

^ in [2] in the range 250 to 325 K were plotted

and a smooth curve drawn by eye through the points. The data listed

were obtained by interpolation on the curve.

A curve was drawn through all values for listed in [2] in the

range 250 to 325 K. The data scatter badly. The curve was drawn with

slightly greater slope than for propane and methane, but otherwise of

a similar shape. The values listed were interpolated on the curve.

Only one point is given for iso-butane plus methane. A curve was drawn

through it and equally proportioned between the curves for methane plus

propane and methane plus n-butane.

aa For n-pentane plus methane values of B^^ scatter badly and the slopes from

different sources are not consistent. A curve with minimum slope was drawn.

The value at 273.15 K had to be extrapolated; otherwise points were

interpolated.

bb

Very bad scatter; a curve was drawn parallel to n-pentane, and a little less

negati ve.
140

For 2 ,2-dimethylpropane plus methane the data of Strain [13], Hamann [34],

and Belim [20] were plotted, a curve drawn through [13] and [20] fell

3 -1

about 25 cm mol more negative than for [34]. Other data by Baughman

[35] from 199 to 257 K was plotted and values were less negative. Values

were extrapolated from the curve drawn through data of [34].

For n-hexane plus methane the data of Pecsok [32] and Dantzler [29] were

plotted and a smooth curve was drawn by eye and values extrapolated.

For 2,2-dimethylbutane plus methane a smooth curve was drawn by eye through

the only two points given by Pecsok [32] and values were extrapolated.

3 -1

A uniform uncertainty of ±100 cm mol is used for the hexanes on the

basis of [2] and cyclohexane at 300 K. The uncertainty is probably

larger at 273.15 K because of lack of data.

gg

hh

i i

Ref. [2] does not list smoothed

at 298.15 K is from [4].

Uncertainty taken equal to that

Uncertainty taken as twice that

data below 300 K. The data point given

for CO^-free dry air.

2

of helium.
References for Vi rial Coefficient Data

1. Levelt Sengers, Klein, M. ; Gallagher, J.S.; ‘‘Pressure

Volume-Temperature Relationships of Gases; Virial Coefficients"; 0,£.

Gray; editor, American Institute of Physics Handbook, 3rd Edition,

1972, pp. 4-204 to 4-221.

2. Dymond, J.H.; Smith, E.G.; "The Virial Coefficients of Pure Gases

and Mixtures: A Critical Compilation"; Oxford University Press, 1S80.

3. Schafer, K. ; Z. Phys. Chem. (1937), B36 , 85.

4. Bottomley, G.A.; Reeves, C.G.; Seiflow, G.H.F.; Mature. (London)

(1958), 1^, 596.

5. Douslin, D.R. ; Harrison, R.H.; J. Chem. Thermodynamics (1976), 8, 301.

6. Douslin, D.R. ; Harrison, R.H.; J. Chem. Thermodynamics (1978), 5, 491.

7. Waddington, G. ; Smith, J.C.; Scott, D.W. ; Huffman, H.M. ; J. Am.

Chem.’ Soc. (1949), 71, 3902.

8. Das, T.R. ; Reed, C.O., Jr.; Eubank, P.T.; J. Chem. Eng. Data (1977). 22

9. Mason, D.M. ; Eakin, B.E.; J. Chem. Eng. Data (1961), 6, 499.

10. Jesson, F.W. ; Lightfoot, J.H.; Ind. Eng. Chem. Ind. Ed. (1938), 312

11. Das, T.R. ; Reed, C.O., Jr.; Eubank, P.T.; J. Chem. Eng. Data (1973), 18

253.

12. Kretschmer, C.B.; Wiebe, R. ; J. Am. Chem. Soc. (1951), 73, 3778.

13. Strein, K. ; Lichtenthaler, R.N.; Schramm, B. ; Schafer, K. ; Ber.

Bunsenges Phys. Chem. (1971), 75, 1308.
142

References (continued)

14. Kapello, W. ; Lund, N. ; Schafer, K. ; Z. Phys. Chem. N. F (Frankfurt).

(1963), 196.

15. Ratzsch, M. ; Bittrich, H.J.; Z. Phys. Chem. (1965), 2^, 81.

16. McGlashan, M. L. ; Potter, D.J.B.; Proc. Roy Soc. (1978), A267 , 478.

17. Najjar, R.F.; Kay, W.B.; Leverett, G.F.; J. Chem. Eng. Data (1969), 14, 377

18. Lichtenthaler, R.N., Schafer, K. ; Ber. Bunsenges. Phys. Chem. (1969), 73,

42.

19. Osborne, N.S.; Ginnings, D.C.; J. Res. National Bur. Stand. (1947), 453

20. Belim, J. ; Renche, W. ; Schafer, K, ; Schramm, B. ; Ber. Bunsenges.

Phys. Chem. (1974), 78, 282.

21. Hahn, R. ; Schafer, K. ; Schramm, B. ; Ber. Bunsenges. Phys. Chem.

(1974), 78, 287.

22. Byrne, M.A. ; Jones, M.R. ; Staveley, L.A.K. ; Trans Faraday Soc. (1968), 64,

1747.

23. Thomas, G. ; Steenwinkel, R. Van.; Stone, W. ; Mol. Phys. (1962), 5, 301.

24. McMath, H.G.; Edmister, W.C.; A.I.CH.E.J. (1969), lb, 370.

25. Lee, R.C.; Edmister, W.C.; A.I.CH.E.J (1970), 16, 1047.

26. Lee, J.W. ; Seville, G. ; unpub. [2] See also Lee, [27].

27. Lee, J.W. ; PhD - Thesis, University of London (1976).

28. Gunn, R.D.; M.S. Thesis, University California Berkeley (1958).

29. Dantzler, E.M. ; Knobler, C.M. ; Windsor, M. L. ; J. Phys. Chem.

Ithaca (1968), 72, 676.

30. Hoover, A.E.; Nagata, I.; Leland, T.W. , Jr.; Kobayashi , R. , J.

;

Chem. Phys. (1968), 48, 2633.
143

References (continued)

31. Wormald, C.J.; Lewis, E.J.; Hutchings, D.J.; J. Chem. Thermodynamics (1979), H, 1.

32. Pecsok, R. L. ; Windsor, M. L. ; Anal. Chem. (1968), 1238.

33. Massondi , R. ; King, A.D., Jr.; J. Phys. Ctiem. (1973), 77, 2016.

34. Hamann, S.D.; Lambert, J.A. ; Thomas, R.3.; Australian J. Chem. (1955), 8,

149.

35. Baughman, G.L. ; Westhoff, S.P.; Dincer, S.; Ouston, D.D.; Kidnay,

A.J.; J. Chem. Thermodynamics (1975), 2>

36. Mueller, W.H.; Leland, T.W. ; Kobayashi , R. ; A.I.C.H.E.J. (1961), 7, 267.

37. Rigby, M. ; Prausznitz, J.M. ; J. Phys. Chem. (1968), 72, 330.

38. Roe, D.R. ; Seville, G. ; Unpublished (see [2]).

39. Roe, D.R. ; PhD Thesis, University of London (1972).
144

Table A9a. Constants for quadratic equations for the virial coefficients

as functions of T in the range 273 to 300

2 5 = c + dT + eT

Substance c d e

-T -1 -1

cm"^ mol cm-^ mol K 10 cm"^ mol

methane -264.3 -1.0899 -1.166

ethane -1333.6 -6.4626 -8.763

propane -1618.0 -5.1270 -3.303

n-butane -7937.1 +42.5325 -61.636

2-methylpropane -18764.8 +117.552 -190.852

n-pentane -25054.4 +148.949 -232.017

2-methyl butane -10077.8 +52.7664 -75.838

2, 2-di methyl propane -11735.3 +63.9039 -92.816

n-hexane — — —

2-methlypentane -3577.8 -1.8440 +27.174

3-methlypentane -16491.6 +87.5927 -127.131

2, 2-di methyl butane — — —

2,3-dimethyl butane -15607.0 -85.6304 -128.881

ethyne — — —

ethene -912.2 +4.2046 -5.413

propene -1577.5 +5.5834 -4.873

benzene -17679.2 +97.4720 -144.782

carbon monoxide — — —

carbon dioxide -1280.8 +6.9745 -10.374

oxygen -110.8 +0.4066 +0.299

nitrogen -151.7 +0.7923 -1.009

hel i urn +2.0 +0.0773 -0.149

argon -185.3 +0.9461 -1.271

hydrogen +62.7 -0.3694 +0.696

air (C02 ~free, dry) -154.7 +0.7924 -1.009

water vapor — — —

^Data in Table 8 were fitted to the quadratic form. The main use is to

obtain dB/dr near 288 K. Because some points are extrapolated, these

expressions must be considered to be approximate.
145

Table A9b. Constants for quadratic equations for the second virial

coefficients of interaction, 5(l,i), as functions of temperature for

the range 273 to 300

5(l,i) = c + dT -I- eT^ (1 = methane)

Substance pair c d e

3 ,-l cm mol3 ,-l

cm mol K 10 ^ cm^ mol

methane + ethane -661.6 +3.2100 -4.358

(1)^ + propane -560.8 +2.0478 -2.044

(1) + n-butane -899.5 +3.7585 -4.478

(1) + 2-methyl propane -941.2 +4.3346 -5.627

(1) + n-pentane -1276.3 +5.6426 -7.076

(1) + 2-methlybutane -1449.9 +7.0162 -9.551

(1) + 2 , 2-dimethyl propane -3123.8 +18.7127 -29.534

(1) + n-hexane -3201.1 +18.7225 -29.901

(1) + 2-methyl pentane — — —

(1) + 3-meLhyl pentane — — —

(1) + 2 ,2-dimethlybutane -4491.4 +26.2291 -39.888

(1) + 2 ,3-dimethlybutane — — —

(1) + cycl opropane — — —

(1) cyclobutane — --- —

(1) + cyclopentane — — —

(1) + cyclohexane — — —

(1) + ethyne — — —

(1) + ethene -496.1 +2.2641 -2.738

(1) + propene — — —

(1) + benzene — — —

(1) + carbon monoxide — — —

(1) + carbon dioxide — — —

(1) + oxygen — — —

(1) + nitrogen -333.8 +2.0150 -3.198

(1) + hel i urn — — —

(1) + argon -368.1 +2.0854 -3.090

(1) + hydrogen +89.9 -0.7904 +1.769

(1) + air (C02 "free, dry) _ _ ^ _ _ ^ « « * (1) + water vapor -545.3 +2.4226 -2.700

^See footnote to Table A9a.

*^(1) = methane.
146

Table A9c. Enthalpy differences, H - H° ^ between the real and ideal gases

for pure substances p = 101.325 kPa at various reference temperatures

Substance H ~ H°

J mo 1

T/K 273.15 288.15 288.71 298.15

methane -18.0 -17.0 -17.0 -16.3

ethyne — — — —

ethene -51.9 -47.2 -47.0 -43.9

ethane -69.6 -61.9 -61.6 -56.6

propene -125. -121. -121. -118.

propane -142. -139. -139. -136.

cyclobutane — — — —

n-butane -353. -296. -294. -257.

2-methlypropane -478. -306. -299. -188.

cyclopentane — — — —

n-pentane -854. -626. -618. -475.

2-methyl butane -478. -404. -401. -355.

2, 2- dimethyl propane -517. -428. -424. -368.

benzene -767. -619. -613. -521.

cyclohexane — — — —

n-hexane — — — —

2-methylpentane -633. -647. -647. -657.

3-methylpentane -790. -656. -651. -568.

2, 2- dimethyl butane — — — —

2, 3- dimethyl butane -667. -537. -532. -451.

carbon monoxide — — — —

carbon dioxide -51.7 -42.7 -42.4 -36.5

oxygen -8.98 -8.72 -8.71 -8.54

nitrogen -7.74 -6.88 -6.85 -6.28

hel i urn 1.33 1.46 1.47 1.55

argon -9.18 -8.09 -8.05 -7.33

hydrogen 1.09 0.50 0.48 0.09

air (CO2 free dry) -8.05 -7.19 -7.15 -6.59

water vapor — — — —

i
Appendix 10. Propagation of uncertainties

Propagation of uncertainties

In calculating the propagation of uncertainties, we use the relation-

* ship, below, recommended by the lUPAC Commission on Thermodynamics [17] .

Where a quantity F is a function of variables x,y,z;

F = F(x,y,z) (AlO.l)

then the uncertainty Sp in F is related to the uncertainties in x, y, and

z by the equation:

(Sp)^ = OF/8x)2 + OF/3y)2 s ^ + ( 3 F/82 )^ (A10.2)

Strictly speaking, x, y and z must be independent, non-correl ated variables

for this equation to hold. Usually, when there is some correlation this

equation gives an upper limit. It is adopted here as an ^ hoc expression

to define the method of calculations.

In this document we use several functions of the type:

F = ax (A10.3)

F = ax + by - cz (A10.4)

wnere a, b, and c, are constants.

For equation (A10.3), by application of (A10.2) we obtain:

Sp = {3ax)/3x}^ s^ + {3(ax)/3a}^ s^ (A10.5)

Since a IS a can etant s =0 and the last term of (A10.5) is zero. As a

3

result:

Sp = a^s^ = (as (A10.6)

F X ^ x"^ or

Sp = as^ (A10.7)

* See general list of references.
148

For equation (A10.4), by application of (A10.2) we have, dropping the

zero terms as above:

Sp = {8(ax)/3x}^ s^ + {a(by)/3y}^ s^ + {3(-C2)/3z}^ s^ . (A10.8)

This reduces to:

7 7 7 7 7 7 7

Sp = a^s^ + b^Sy + (-c)'^s^ (A10.9)

Sp = (as^)^ + (bSy)^ + (cx^)^ (AlO.lO)

or:

Sp = {(as^)^ + (bSy)^ + (cs^)^]'^ (AlO.ll)

Uncertainties in enthalpies of combustion and enthalpies of formation

under different conditions

(a) Enthalpies of combustion and formation

The chemical equation for the combustion reaction:

+ (a + b/4)02 = aC02 + (b/2)H20 (A10.12)

leads to relationships between the enthalpies of formation Apif° and the

enthalpy of combustion A H°:

c A^H°(CaHb) = aApif°(C02) + (b/2) ApH^(H20) - (a+b/4)

-Apif°(C^H^) (A10.13)

which is a function of the type of equation (A10.4). By application of

# equation (A10.2) as illustrated in equations (A10.3 - 11) we obtain:

Sc(CaHb) = [{as^CCOj)}^ + {(b/2)s^(H20)}^ +

+ {(a + b/4)s^(02)!^ + {SfCCaHb)}^]*^ , (A10.14)

or alternatively:

Sf(CaHb) = [{as^CCOj))^ + {(b/2)s^(H20)|2

+ {(a + b/4)s^(02)}^ + {s^(CaHb)}^]^ (A10.15)

depending on whether A^ff°(C^Hj^) or Apif°(C^H|^) is taken as the dependent

quantity. In the above equation Apif°(02) = 0 and Sp(02) = 0.
149

(b) Adjustment from one temperature to another

For the adjustment of heating values from one temperature to another,

the following relationships are derived from equation (A10.12).

^ \^298.15K^^a^b^

+ a{(H° - i^298.15 ^^^7* ' ^298.15

- (a + b/4) {(r/° - ^298.15

' '^298.15 K^^^a^b^^

(A10.16)

Equation (A10.16) is also a function of the type of equation (A10.4). From

equation (16) we derive;

{s^(D}^ = {s^(298.15 K))2 + {aSj^(j.j(C02)i^ + {(b/2)Sj^^j,^(H20,l)}^

+ {(a + b/4)s^^j,j(02)}^ + ’ (A10.17)

in which s^(r) is the estimated uncertainty in enthalpy of combustion at

T, s^(298.15 K) is the estimated uncertainty in enthalpy of combustion

at r = 298.15 K (Table A7g) and is the estimated uncertainty in the

quantity (if° - ^293 15

subtance identified parenthetically

beside it.

Conversion of enthalpy from molar to mass basis

In calculating A H° we use the relationship:

0 w A = A H°/M (A10.18)

c w c m where M is the molar mass. From equation (A10.2) we have:

" {3(A^«»)/3M}2(s_^)2 (A10.19)

3(A^//°)/3(A^//°) = 1/M (A10.19a)

3(A 7f°)/8M = - A H°/M^ (A10.19b)

0 w 0 fn

Scm fo'f' This purpose is s^ from Table A7g; s^ is found in the discussion

on atomic weights in Section 13 and Appendix 5a,

s^ = (1/W)^(s + {(A H )/M^}'^ (s„)^

cw

^ ^ cm^ c m^ ’ ^ M

(A10.20)

In equations (A10.20 and A10.20a), s and s are the estimated uncerCW 0171 tainties of A H° and A H°. respectively, s,. is the estimated uncertainty in the
150

molar mass. We can obtain the estimated relative uncertainty (s /A H°) ^ cw c by dividing equation (A10.20) by (A H°)^ = (A We thus obtain:

0 w c fn

(s /A H°)^ = (s /A ^ cw c

^ cm c m^ ^ M

^

Conversion of enthalpies from molar to volumetric basis

(a) Uncertainty in ideal gas molar volume

In calculating we use the relationship: ^ m

^

v''^ = RT/p

m

In a manner similar to that used in earlier examples, we obtain:

(b) Conversion of enthalpy

In calculating A from A H° we use the relationships: ^ c V c m

^ A //° = A

(A10.20a)

(A10.21)

(A10.22)

c V c m m

(s^'^/A = (s''^/A cv c V ^ cm c m^ ^ V{T ,p) m

(A10.23)

(A10.24)

(c) Uncertainty in the ideal gas volume of water saturated gas

For the dry gas, the partial pressure of the fuel gas p for equation

(A10.21) is equal to the defined reference pressure p°. However for the

water saturated gas, requires that p(fuel) = p° - p(H20 ) at that temperature. The uncertainty s(p,fuel) in the partial pressure of fuel is

0 obtained from:

!s(p,fuel)!^ = {s(p»)}^ + {s(p,H20)i^ (A10.25)

Because p° is a defined constant s(p°) = 0 and:

(s(p,fuel)}^ = (s(p,H20)}^ (A10.26)

for the water saturated gas.

Uncertainties of Vi rial Coefficients

The estimated uncertainties s(S) in the virial coefficients B are given

in Table 8a for pure substances and in Table 8b for binary interactions

between substances. The uncertainty of the virial coefficient of a mixture

is dependent on the composition of the mixture as is discussed below.
151

We start with the general function:

B(mix) = B(B.y x.) (A10.27)

in which S(mix) is the virial coefficient of the mixture having n individual

components i or j, x^. is the mole fraction of component i and is the

virial coefficient corresponding to binary interactions between molecules

of i and of j. If j = i the virial coefficient is that of pure substance

i

If S(mix) has the functional form given in equation (A10.2S) it can be

shown by simple algebraic manipulation based on equation (A10.2) that the uncertainty s(S,mix) can be separated into two independent sets of contributing

terms, equation (A10.29) below, one of which depends on the uncertainties s(B

in the virial coefficients and the other depends on the uncertainties

s(x^. ) in the composition.

S(mix) = ^22^2 ^33^3 ^44^4

'*’

+ 25^2^i^2 ^^

13

^1^3 ^^14^1^4 ’

-t- -t- 2324X2X^ + . . .

2B34X3X^ -h ... (A1G.28)

{s(S,mix)}^ = I {3S(mix)/8B. {s(B. .)}^ I{3B(mix)/3x . }^{s(x. )]‘^

i j

1 (A10.29)

Because the x^. are variables that are particular to a gas mixture

and are measured parameters, except for illustrative purposes of defined

gas mixtures, the assignment of s(x.) is dependent on the measurement

process used, and is beyond the scope of this chapter. However the

relationship of s(B,mix) to s(x^ can be shown.

The coefficients in the first set of terms on the right hand side of

equation (A10.29) all have the form:

(3B(mix)/3B.j.) = x.x^. (A10.30)

The coefficients in the second set of terms in equation (A10.29) are more
152

complex because the x^. 's are not all independent but must sum to unity:

Ix. = 1 (A10.31)

2

If this condition is ignored the coefficient of each (s(x^.))

^ 1s:

(35(mix)/3xp^ = j = l,2,....n (A10.32)

When the condition A10.31 is introduced in equation A10.28 in the form:

x^ = 1 - X

2

- x^ - x^ - ... (A10.33)

2

the coefficients of the (s(x^. )) are replaced by terms of the form shown

in equations (A10.34) below.

six.)

2

coefficient: (3B(mix)/3x^.

)

s(x^) -

s(x2) C2(5i2-5ii) + 2(522-2B^2^5^^)x2 2(S^^-S^2'^13^^23)^3

+ 2(S^^-S^2’-®14'^^24^^4

+ 2(B^^-5^2'^15'^^25^^5 ' ' ’

S(X3) [2(5i3-5ii)

+ 2(S^^-5^2'^13'"^23^^2

+ 2(S^^-25^3+533)X3 + 2(S^^-B^3-5^^+53^)x^ (A10.34)

+ 2(Sii-Si3-Si5+S35)x5 + .

.

sCx^) + 2(S^^-S^2“^14'"^24^^2

+2(S^^-S^3-B^^+53^)x3 + 2(S^^-25^^+S^4)x^

+ 2(5^^-5^^-S^5+5^5)x5 + . . ]

S(Xj) .

.

The final result, for the real gas mixture in which the errors in composition

are constrained by the fact that 1 x. =1, is given by equation (A10.35).
153

{s(S,mix)}^ = (x^)^{s(S^^)}^ + (Zx^Xj)^

* (2x^X3)^ ! 5 (S

33

)}^ + (2x,x^) 2 {sfS,^)!^ * ..

+ (x^)^ {s(S22>}^ + (2X2X3)^ {s(B

23

)!^ +

+ (Xj)^ (5(333)}^ + (2X3X^)^ (5(53^)}^ + ..

+ (x2)2 {s(B44)}2 + ...

+ . . .

+ 2(B22-2S^2'^S^1^''2 ^

+ 2(S^j^~S^2"-®24"'®22^^4 ^ ••] (s(x2)}

+ {2(5^3-5jP + ^(^ir^l2*'^I3^^23^^2

^ 2^^1l'2^13^^33^^3

+ 2(5^^-S^3-B^2^+S3^)x^ + ...] [six^'j]

+ {2(5^^-S^^) + 2(B,^-B^2'^14'^^24^^2

^^^ll“'®13‘'®14'^^34^^3

+ Z(B^^-ZB^^-^B^^)x^ + ...]^ {s(x^)}-

(A10.35)

In practice, drastic simplification of the general result is required

(by the lack of data) and is warranted (by the small size of many terms).

* Usually, for the coefficients of s(B^. j) only terms involving x^ will be

large enough to matter. The situation is less clear for the coefficients

of s(xp unless that quantity itself is very small, because (1), only mole

fractions of the impurities appear, and, (2), some are likely to be

unknown. In some cases the entire set of terms in s(x^ will be small,

but each case must be tested.

Uncertainties in dB/6T

B(T) is fitted by equation of type (A10.36) for which constants and their

uncertainties are listed in Tables A9a and A9b.

B(T) = c + dT + eT^

dB/dT = d + ZeT

(A10.36)

(A10.37)
154

Using equation (A10.2) we calculated sCdB/dr) as follows:

[s(dB/dr)}^ = {8[dB/dr)ai?]^ {s(i))}^ + {8(d5/dn/3}^ (s(c)}^

+ {8(dS/dr)/8f}^ {s(T)]^ . (A10.38)

Because T is the independent variable, s(r) will be taken as zero. We obtai

{s(dB/dr)}^ = {2Ts{c)]^ (A10.39)
NBS»114A IREV. 2-80

U S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date REPORT NO.

BIBLIOGRAPHIC DATA

SHEET (See instructions) NBSIR 81-2401

4. TITLE AND SUBTITLE

Heating Values of Natural Gas and its Components

Conversion of values to measurement bases and calculation of mixtures

5. AUTHOR(S)

George T. Armstrong and Thomas L. Jobe, Jr.

6. PERFORMING ORGANIZATION (If joint or other than N'BS, see in struciion s)

national bureau of STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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SPONSORING organization NAME AND COMPLETE ADDRESS (Street. C/ty. State, ZIP)

Groupe Internationale des^ Importateurs de Gaz Natural Liquefie (GIIGNL)

10.

SUPPLEMENTARY NOTES

~l Document describes a computer program,; SF-185, FiPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document induces a S{ gn i ficant bibliography or literature survey, mention it here)

The standard enthalpies of combustion of the pure hydrocarbons C-, to

Cg at 298.15 K and 101 325 Pa selected from the literature, with reference

to a more detailed document, are used to derive heating values for the

ideal gas and the real gas on molar, mass, and volumetric bases at five

reference conditions involving other temperatures and pressures. Values

can be obtained at both dry and water saturated conditions. The second

virial coefficients and their first derivatives are used to calculate the

effects of non-ideality. Procedures are given and illustrated for calculating the heating values of mixtures from the composition and the date

given for the pure compounds. The relationships between the quantities

presented in the tables are illustrated by charts. A procedure is given

for estimating the uncertainties of the calculated results.

12.

KEY /lORDS (Six to twelve entries; alphabetical order; capital! ze only proper names; ana separate key wares oy semicolons;

Heating value; calorific value; enthalpy of combustion; hydrocarbon gases; gaseous

fuel mixtures; real gas; ideal gas; estimation from composition; reference

measurement conditions

13. AVAILABILITY

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