IMBSIR 77-1278

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I DO NOT REMC”E

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' “***

The Air Density Equation and

the Transfer of the Mass Unit

Frank E. Jones

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

July 1977

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

. W t£M o' HBMi

NBSIR 77-1278

THE AIR DENSITY EQUATION AND

THE TRANSFER OF THE MASS UNIT

Frank E. Jones

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

July 1977

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Or. Sidney Harman. Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Acting Director
'
The Air Density Equation and the Transfer of the Mass Unit

by

Frank E. Jones

Ah ST RAC

T

A new formulation of the equation used for the calculation of air

density has been developed. The Quinn, Colclough and Chandler

value of the gas constant, currently accepted values of the atomic

weights, and recent determinations of the abundances of the various

constituents of air have been used. The abundance of carbon dioxide

has been treated as a variable and a factor enabling convenient

adjustment of the apparent molecular weight of air for deviation

of carbon dioxide abundance from a background value has been derived.

A new table of compressibility factor for the range of pressure and

temperature of interest in standards laboratories has been calculated

using recently determined values of virial coefficients. The enhancement factor, which has usually been ignored in air density equations,

has been explicitly included in the equation; a table of enhancement

factor has been calculated using a simple equation fitted to values

in the range of pressure and temperature of interest. A table of the

saturation water vapor pressure has been included; a simple equation

for the calculation of saturation water vapor pressure has been fitted

Uncertainties, random and systematic, in the parameters and in the

measurement of environmental variables and consequent uncertainties

in calculated air density have been estimated.

Application of the equation to air buoyancy determination and the

transfer of the mass unit at the various national standards

laboratories has been made.

Key Words: Air buoyancy, air density, mass unit transfer, real gas

equation.

The Air Density Equation and the Transfer of the Hass Unit

by

Frank E. Jones

ABSTRACT

A new formulation of the equation used for the calculation of air

density has been developed. The Quinn, Colclough and Chandler

value of the gas constant, currently accepted values of the atomic

weights, and recent determinations of the abundances of the various

constituents of air have been used. The abundance of carbon dioxide

has been treated as a variable and a factor enabling convenient

adjustment of the apparent molecular weight of air for deviation

of carbon dioxide abundance from a background value has been derived.

A new table of compressibility factor for the range of pressure and

temperature of interest in standards laboratories has been calculated

using recently determined values of virial coefficients. The enhancement factor, which has usually been ignored in air density equations,

has been explicitly included in the equation; a table of enhancement

factor has been calculated using a simple equation fitted to values

in the range of pressure and temperature of interest. A table of the

saturation water vapor pressure has been included; a simple equation

for the calculation of saturation water vapor pressure has been fitted.

Uncertainties, random and systematic, in the parameters and in the

measurement of environmental variables and consequent uncertainties

in calculated air density have been estimated.

Application of the equation to air buoyancy determination and the

transfer of the mass unit at the various national standards

laboratories has been made.

Key Words: Air buoyancy, air density, mass unit transfer, real gas

equation.
.

I
The Air Density Equation and the Transfer of the Mass Unit

by

Frank E. Jones

ABSTRACT

A new formulation of the equation used for the calculation of air

density has been developed. The Quinn, Colclough and Chandler

value of the gas constant, currently accepted values of the atomic

weights, and recent determinations of the abundances of the various

constituents of air have been used. The abundance of carbon dioxide

has been treated as a variable and a factor enabling convenient

adjustment of the apparent molecular weight of air for deviation

of carbon dioxide abundance from a background value has been derived.

A new table of compressibility factor for the range of pressure and

temperature of interest in standards laboratories has been calculated

using recently determined values of virial coefficients. The enhancement factor, which has usually been ignored in air density equations,

has been explicitly included in the equation; a table of enhancement

factor has been calculated using a simple equation fitted to values

in the range of pressure and temperature of interest. A table of the

saturation water vapor pressure has been included; a simple equation

for the calculation of saturation water vapor pressure has been fitted.

Uncertainties, random and systematic, in the parameters and in the

measurement of environmental variables and consequent uncertainties

in calculated air density have been estimated.

Application of the equation to air buoyancy determination and the

transfer of the mass unit at the various national standards

laboratories has been made.

Key Words: Air buoyancy, air density, mass unit transfer, real gas

equation.

The Air Density Equation and the Transfer of the Mass Unit

by

Frank E. Jones

ABSTRACT

A new formulation of the equation used for the calculation of air

density has been developed. The Quinn, Colclough and Chandler

value of the gas constant, currently accepted values of the atomic

weights, and recent determinations of the abundances of the various

constituents of air have been used. The abundance of carbon dioxide

has been treated as a variable and a factor enabling convenient

adjustment of the apparent molecular weight of air for deviation

of carbon dioxide abundance from a background value has been derived.

A new table of compressibility factor for the range of pressure and

temperature of interest in standards laboratories has been calculated

using recently determined values of virial coefficients. The enhancement factor, which has usually been ignored in air density equations,

has been explicitly included in the equation; a table of enhancement

factor has been calculated using a simple equation fitted to values

in the range of pressure and temperature of interest. A table of the

saturation water vapor pressure has been included; a simple equation

for the calculation of saturation water vapor pressure has been fitted.

Uncertainties, random and systematic, in the parameters and in the

measurement of environmental variables and consequent uncertainties

in calculated air density have been estimated.

Application of the equation to air buoyancy determination and the

transfer of the mass unit at the various national standards

laboratories has been made.

Key Words: Air buoyancy, air density, mass unit transfer, real gas

equation.
m
1 . INTRODUCTION

The transfer of the mass value from one object, such as the Internat Lonal

Prototype Kilogram, to another object is accomplished by comparison of the

objects by means of a balance. The difference in buoyant force on the two

objects is proportional to the difference in their displacement volumes

and to the air density. The air density is conventionally calculated using

an equation based on the equation of state of an air-water vapor mixture.

A new formulation of the air density equation is developed below.

2. DEVELOPMENT OF THE AIR DENSITY EQUATION

The total pressure, P, the total volume, V, and the absolute temperature,

T, of a mixture of ideal gases are related by the ideal gas equation,

PV = nRT , (1)

where n is the number of moles of the mixture and R is the universal gas

constant. In terms of density, P, rather than volume, Eq . (1) becomes

p=pfr, (2)

where is the apparent molecular weight of the mixture.

For a mixture of dry air (indicated by subscript a) and water vapor (subscript e), M is the apparent molecular weight of the air-water vapor

mixture

,

Since

m + m M = ” = _§ £

n n + n

a e

(3)

where m is the mass of the mixture and n is the number of moles of the

mixture

.

M = n M + n M

a a e e n + n

a e

/ n M

M

a l 1 +

nffl!

1 +

(4)

By introducing the water vapor mixing ratio, r:

mass of water vapor m

e n

eM

e > m n M

a a a mass of dry air

r

(5)
Eq. (4) becomes

M = M (1 + r) a

1 +

M /M

e a M

By designating the ratio — by e, Eq . (6) becomes

a

( 6 )

M (1 + r)

M = — , (1 + -)

e

(7)

and by substituting Eq . (7) in Eq. (2), we find that

p - iT (rr? ( 8 )

The effective vapor pressure, e' , of water in moist air is defined [1] by

therefore

,

r

( e + r)

P (9)

and Eq. (8) becomes

ee' r " (P - e')

P Hr1

a

1

_1 + (e-1)

( 10 )

(ID

Eq. (11) is the ideal gas equation for a mixture of dry air and water

vapor with a water vapor pressure of e'. If the air-water vapor mixture

behaved as a mixture of ideal gases,

ofr1

a |_1

f—+ (e- 1)

= Z = 1, ( 12 )

where Z is the compressibility factor. Since Z is not,, in general, equal

to 1, the mixture is not ideal. The magnitude of the non-ideality is

reflected in the departure of the value of Z from 1 and Eq. (12) becomes

2
( 13 )

Eq. (13) is the real gas equation for a mixture of dry air and water

vapor. By rearranging Eq. (13), the expression for density is

PM

a RTZ (14)

Eq. (14) is the form of the equation to be used to calculate air density.

3. SPECIFICATION OF THE VALUES OF THE PARAMETERS IN EQ. (14)

3.1 Universal Gas Constant, R.

The value of the molar gas constant, R, listed in a compilation by Cohen

and Taylor [2] is 8.31441 + 0.00026 J mol K

-

^, where the uncertainty

corresponds to 1 standard deviation. Recently, Quinn, Colclough and

Chandler (QCC) [3] made a new determination of R by measuring the speed

of sound in argon by means of an acoustic interferometer. Their value is

8315.73 + 0.17 JK

- "*- kmol

-

^, where the uncertainty quoted is "one statistical

standard error in a fit of 95 degrees of freedom." Quinn et al concluded

that a real systematic difference exists between their value and the

Cohen and Taylor value, and that they were unable to point to any single

factor which could be unambiguously identified as the source of the

disagreement. Gammon [4] recently deduced a value of R from measurements

of the velocity of sound in helium. His latest reported value [5] is 8315.31 +

0.35 JK

_ kmol-1, which is in good agreement with the QCC value.

The difference between the Cohen and Taylor and the QCC values, if it were considered to be a systematic uncertainty, would correspond to a systematic

relative uncertainty in the calculated value of p of 159 p.p.m., a quite

significant contribution. One is thus confronted with the necessity to

choose between two significantly discordant values. The long-term alternative would be to await a new, definitive experimental determination of

R (plans are presently being made at NBS to make a definitive determination

of R) . We choose at present to use the QCC value, with the realization

that in the future it might be replaced by a new value, at which time the

formulation developed in the present work would be modified to accommodate

the new value.

3.2 Apparent Molecular Weight of Air, M

g

The apparent molecular weight of dry air, M

g

, is calculated using the

relationship

3
( 15 ) k

M = E M.x.

a . , 1 x i=l

k

E x. i=l

1

where each M-j is the molecular weight of an individual constituent

and x^ is the corresponding mole fraction. The molecular weights

and mole fractions of the constituents of dry air are tabulated in

cable 1. Other constituents are present in abundances which are negligible for the present application.

TABLE 1. NORMAL COMPOSITION OF CLEAN, DRY

ATMOSPHERIC AIR NEAR SEA LEVEL

Constituent Abundance

(mole fraction)

Molecular Weight

Nitrogen (N

2

) 0.78102 28.0134

Oxygen (0

2

) 0.20946 31.9988

Carbon Dioxide

(co

2

) 0.00033 44.0098

Argon (A) 0.00916 39.948

Neon (Ne) 0.00001818 20.179

Helium (He) 0.00000524 4.00260

Krypton (Kr) 0.00000114 83.80

Xenon (Xe) 0.000000087 131.30

Hydrogen (H

2

) 0.0000005 2.0158

Methane (CH^) 0.000002 16.0426

Nitrous Oxide 0.0000005 44.0128

(n

2 o)

The values of the atomic weights of the elements are taken from

reference [6] and are based on the carbon-12 scale. The molecular

weights are taken to be the sums of the atomic weights of the

appropriate elements.

The value for the abundance of oxygen is taken from reference [7].

4
L’he value for the abundance of carbon dioxide is taken from a recent

unpublished compilation of data on atmospheric concentration of carbon

dioxide at seven locations throughout the world. It must be emphasized

that 0.00033 is the mole fraction of CO

2 in the atmosphere and should

be considered to be a "background" value. The mole fraction of CO 2 in

laboratories, which is of course the value of interest here, is in general

greater than 0.00033 and is variable. For example, three samples of air

taken from a glove box in the Mass Laboratory at NBS had a mean value of

0.00043, and four samples of laboratory air taken at the National Center

for Atmospheric Research in Boulder, Colorado had a mean value of 0.00080.

Clearly, then, the optimum utilization of the air density calculation would

necessitate a measurement of C0

2 abundance on an air sample taken at the

time of the mass comparison.

One of the options one has in dealing with the variability of CO 2 abundance

is to select a reference level (for example, 0.00033 or 0.00043) and to

provide an adjustment to M a to account for known departures from the

reference level. Gluekauf [8], in discussing the variation of the abundance

of oxygen in the atmosphere, stated that "all major variations of the 02

content must result from the combustion of fuel, from the respiratory

exchange of organisms, or from the assimilation of CO2 in plants. The

first process does not result in more than local changes of O 2 content,

while the latter two processes, though locally altering the CO 2 /O 2 ratio,

leave their sum unchanged." The assumed constancy of the sum of the O 2 and CO 2 abundance simplifies the adjustment of Ma to account for departures

from the CO

2 reference level and simplifies the estimation of the uncertaint

in air density due to an uncertainty in CO 2 abundance. The constancy of the sum is expressed by the equation (for convenience, the subscript i has been

replaced by the chemical symbol):

x_ + x = constant = 0.20979 . lo

2 o2

(16)

The contribution of 0

2 and C02 to the apparent molecular weight of dry air

is

m

o

2 x

o

2 + m

co

2 x

co

2 - 31 - 9988 x

0

2 + 44 - 0098 X

C0

2

• (17)

From Eq . ( 16) ,

xn

= 0.20979 - xrn . (18)

z LU2

The right side of Eq . (17) becomes

co

2 31.9988 0.20979 - x + 44.0098 x (19)
thus

,

M

°2

X0

2 + M

C0

2 Xc°

2

' 12 ’ 011 Xc0

2

+6 ' 7130 ' <20)

Therefore,

S(M ) = 6 a M

°2

X

°2 + M<:0

2 XC

°2

= 12.011 6/x

co )

, (21)

that is, the variation in Ma due to a variation in CO 2 abundance is

equal to 12.011 (the atomic weight of carbon) multiplied by the variation

in CO2 abundance. The variation in Ma due to the difference between

the reference levels 0.00033 and 0.00043 is thus 0.0012 g mol

- *- which

corresponds to a relative variation of 41 p.p.m. in Mg and a corresponding relative variation of 41 p.p.m. in air density.

The adjustment of Ma to account for a departure of the CO 2 abundance

from the reference level becomes

12.011 ( 22 )

where the subscript o indicates the reference level. The adjusted Ma for a reference level of 0.00033 becomes

M

a M + 12.011 x

a033C°2

0.00033 , (23)

where M

ar>33

frac t ion of

is the apparent molecular weight of dry air with a CO mole

0.00033 (as indicated by the subscript 033).

The value for the abundance of argon in dry air, 0.00916, is that

calculated from the mass spectroinetric determination of the ratio

of argon to argon and nitrogen by Hughes [9].

The value for the abundance of nitrogen was arrived at by the usual

practice of inferring nitrogen abundance to be the difference between

unity and the sum of the mole fractions of the other constituents.

The abundances of the constituents neon through nitrous oxide in

table 1 were taken to be equal to the parts per volume concentration

in U.S. Standard Atmosphere, 1962 [10].

From the data of table 1, the apparent molecular weight of dry air with

a CO 2 mole fraction of 0.00033 is calculated by Eq . (15) to be 28 . 963 .

6
3.3 Compressibility Factor, Z

The compressibility factor is computed using the virial equation of

state of an air-water vapor mixture expressed as a power series in

reciprocal molar volume,

Z Pv

RT

1 +

B . mix

v

C . mix

j V2

* (24)

and expressed as a power series in pressure,

Z Pv

RT

1 + B' . P + C

1

. P

2 + . mix mix (25)

where _v is the molar volume, Bmix and B' mix are second virial coefficients

and Qn-jy and C 'mjy are third virial coefficients for the mixture. The virial

coefficients of the pressure series are related to the virial coefficients

of the volume power series by

B . t> i = mix

mix RT (26)

and

C' mix-f mix mix

(RT)

2 (27)

Each mixture virial coefficient is a function of the mole fractions of

the individual constituents and the virial coefficients for the constituents. The latter are functions of temperature only.

The second interaction (cross) virial coefficient of moist air, B aw , is one of the contributors to and expresses the effects of interaction

between an air molecule and a water molecule. The values of BOTT used in dw

the calculation of Z are experimental values which strictly apply to

C 02 ~free air.

Using equations (34) and (35), below, and the virial coefficients [11, 15]

provided by Hyland, a table of compressibility factor, Z, for CC^-free air,

table 2, has been generated.

7
TABLE 2 COMPRESSIBILITY FACTOR, Z, FOR C02 -FREE AIR

Temperature

(Celsius

)

15.0

16.0

17.0

18.0

Pressure Relative Humidity in Percent

( pascals

)

(mm Hg) 0 25 50 75 100

70000 525.0 .99971 .99970 .99968 .99967 .99965

75000 562.5 .99969 .99968 .99966 .99965 .99963

80000 600.0 .99966 .99966 .99964 .99963 .99961

85000 637.6 .99964 .99963 .99962 .99961 .99959

90000 675.1 .99962 .99961 .99960 .99959 .99957

95000 712.6 .99960 .99959 .99958 .99957 .99955

100000 750.1 .99958 .99957 .99956 .99955 .99953

101325 760.0 .99958 .99957 .99956 .99954 .99953

105000 787.6 .99956 .99955 .99954 .99953 .99951

110000 825.1 .99954 .99953 .99952 .99951 .99949

70000 525.0 .99971 .99970 .99969 .99967 .99965

75000 562.5 .99969 .99968 .99967 .99965 .99963

80000 600.0 .99967 .99966 .99965 .99963 .99962

85000 637.6 .99965 .99964 .99963 .99961 .99960

90000 675.1 .99963 .99962 .99961 .99959 .99958

95000 712.6 .99961 .99960 .99959 .99958 .99956

100000 750.1 .99959 .99958 .99957 .99956 .99954

101325 760.0 .99959 .99958 .99956 .99955 .99953

105000 787.6 .99957 .99956 .99955 .99954 .99952

110000 825.1 .99955 .99954 .99953 .99952 .99950

70000 525.0 .99972 .99971 .99970 .99968 .99966

75000 562.5 .99970 .99969 .99968 .99966 .99964

80000 600.0 .99968 .99967 .99966 .99964 .99962

85000 637.6 .99966 .99965 .99964 .99962 .99960

90000 675.1 .99964 .99963 .99962 .99960 .99958

95000 712.6 .99962 .99961 .99960 .99958 .99956

100000 750.1 .99960 .99959 .99958 .99956 .99954

101325 760.0 .99960 .99959 .99957 .99956 .99954

105000 787 .6 .99958 .99957 .99956 .99954 .99953

110000 825.1 .99956 .99955 .99954 .99952 .99951

70000 525.0 .99973 .99972 .99970 .99968 .99966

75000 562.5 .99971 .99970 .99968 .99966 .99964

80000 600.0 .99969 .99968 .99966 .99964 .99962

85000 637.6 .99967 .99966 .99964 .99963 .99960

90000 675.1 .99965 .99964 .99962 .99961 .99959

95000 712.6 .99963 .99962 .99961 .99959 .99957

100000 750.1 .99961 .99960 .99959 .99957 .99955

101325 760.0 .99961 .99960 .99958 .99957 .99955

105000 787.6 .99959 .99958 .99957 .99955 .99953

110000 825.1 .99957 .99956 .99955 .99953 .99951

8
Table 2 (Continued)

Temperature Pressure Relative 1Humidity in Percent

(Celsius) ( pascals) (mm Hg) 0 25 50 75 100

19.0 70000 525.0 .99973 .99972 .99971 .99968 .99966

75000 562.5 .99972 .99970 .99969 .99967 .99964

80000 600.0 .99970 .99968 .99967 .99965 .99963

85000 637.6 .99968 .99967 .99965 .99963 .99961

90000 675.1 .99966 .99965 .99963 .99961 .99959

95000 712.6 .99964 .99963 .99961 .99960 .99957

100000 750.1 .99962 .99961 .99959 .99958 .99956

101325 760.0 .99962 .99960 .99959 .99957 .99955

105000 787.6 .99960 .99959 .99958 .99956 .99954

110000 825.1 .99958 .99957 .99956 .99954 .99952

20.0 70000 525.0 .99974 .99973 .99971 .99969 .99966

75000 562.5 .99972 .99971 .99969 .99967 .99964

80000 600.0 .99970 .99969 .99967 .99965 .99963

85000 637.6 .99969 .99967 .99966 .99964 .99961

90000 675.1 .99967 .99966 .99964 .99962 .99959

95000 712.6 .99965 .99964 .99962 .99960 .99958

100000 750.1 .99963 .99962 .99960 .99958 .99956

101325 760.0 .99963 .99961 .99960 .99958 .99956

105000 787.6 .99961 .99960 .99958 .99957 .99954

110000 825.1 .99959 .99958 .99957 .99955 .99953

21.0 70000 525.0 .99975 .99973 .99971 .99969 .99966

75000 562.5 .99973 .99972 .99970 .99967 .99964

80000 600.0 .99971 .99970 .99968 .99966 .99963

85000 637.6 .99969 .99968 .99966 .99964 .99961

90000 675.1 .99968 .99966 .99965 .99962 .99960

95000 712.6 .99966 .99965 .99963 .99961 .99958

100000 750.1 .99964 .99963 .99961 .99959 .99956

101325 760.0 .99964 .99962 .99961 .99959 .99956

105000 787.6 .99962 .99961 .99959 .99957 .99955

110000 825.1 .99960 .99959 .99958 .99956 .99953

22.0 70000 525.0 .99975 .99974 .99972 .99969 .99966

75000 562.5 .99974 .99972 .99970 .99968 .99964

80000 600.0 .99972 .99971 .99969 .99966 .99963

85000 637.6 .99970 .99969 .99967 .99964 .99961

90000 675.1 .99968 .99967 .99965 .99963 .99960

95000 712.6 .99967 .99965 .99963 .99961 .99958

100000 750.1 .99965 .99964 .99962 .99960 .99957

101325 760.0 .99965 .99963 .99961 .99959 .99956

105000 787.6 .99963 .99962 .99960 .99958 .99955

110000 825.1 .99962 .99960 .99958 .99956 .99954
Table 2 (Continued)

Temperature Pressure Relative 1Humidity in Percent

( Celsius

)

(pascals) (mm Hg) 0 25 50 75 100

23.0 70000 525.0 .99976 .99975 .99972 .99969 .99966

73000 562.5 .99974 .99973 .99971 .99968 .99964

80000 600.0 .99973 .99971 .99969 .99966 .99963

85000 637.6 .99971 .99969 .99967 .99965 .99962

90000 675.1 .99969 .99968 .99966 .99963 .99960

95000 712.6 .99968 .99966 .99964 .99962 .99959

100000 750.1 .99966 .99964 .99962 .99960 .99957

101325 760.0 .99965 .99964 .99962 .99960 .99957

105000 787.6 .99964 .99963 .99961 .99958 .99956

110000 825.1 .99963 .99961 .99959 .99957 .99954

24.0 70000 525.0 .99977 .99975 .99973 .99969 .99965

75000 562.5 .99975 .99973 .99971 .99968 .99964

80000 600.0 .99973 .99972 .99970 .99967 .99963

85000 637.6 .99972 .99970 .99968 .99965 .99962

90000 675.1 .99970 .99969 .99966 .99964 .99960

95000 712.6 .99968 .99967 .99965 .99962 .99959

100000 750.1 .99967 .99965 .99963 .99961 .99957

101325 760.0 .99966 .99965 .99963 .99960 .99957

105000 787.6 .99965 .99964 .99962 .99959 .99956

110000 825.1 .99964 .99962 .99960 .99957 .99954

23.0 70000 525.0 .99977 .99976 .99973 .99970 .99965

75000 562.5 .99976 .99974 .99971 .99968 .99964

80000 600.0 .99974 .99972 .99970 .99967 .99963

85000 637.6 .99973 .99971 .99968 .99965 .99962

90000 675.1 .99971 .99969 .99967 .99964 .99960

95000 712.6 .99969 .99968 .99965, .99962 .99959

100000 750.1 .99968 .99966 .99964 .99961 .99958

101325 760.0 .99967 .99966 .99963 .99961 .99957

105000 787.6 .99966 .99964 .99962 .99960 .99956

110000 825.1 .99965 .99963 .99961 .99958 .99955

26.0 70000 525.0 .99978 .99976 .99973 .99970 .99965

75000 562.5 .99976 .99975 .99972 .99968 .99964

80000 600.0 .99975 .99973 .99970 .99967 .99963

85000 637.6 .99973 .99971 .99969 .99966 .99961

90000 675.1 .99972 .99970 .99967 .99964 .99960

95000 712.6 .99970 .99968 .99966 .99963 .99959

100000 750.1 .99969 .99967 .99964 .99961 .99958

101325 760.0 .99968 .99966 .99964 .99961 .99957

105000 787.6 .99967 .99965 .99963 .99960 .99956

110000 825.1 .99966 .99964 .99961 .99959 .99955

10
Table 2 (Continued )

Temperature Pressure Relative Humidity in Percent

(Celsius ) (pascals) (mm Hg) 0 25 50 75 100

27.0 70000 525.0 .99979 .99977 .99974 .99969 .99964

75000 562.5 .99977 .99975 .99972 .99968 .99963

80000 600.0 .99976 .99974 .99971 .99967 .99962

85000 637.6 .99974 .99972 .99969 .99966 .99961

90000 675.1 .99973 .•99971 .99968 .99964 .99960

95000 712.6 .99971 .99969 .99966 .99963 .99959

100000 750.1 .99970 .99968 .99965 .99962 .99958

101325 760.0 .99969 .99967 .99965 .99961 .99957

105000 787.6 .99968 .99966 .99964 .99960 .99956

110000 825.1 .99966 .99965 .99962 .99959 .99955

28.0 70000 525.0 .99979 .99977 .99974 .99969 .99964

75000 652.5 .99978 .99976 .99972 .99968 .99963

80000 600.0 .99976 .99974 .99971 .99967 .99962

85000 637.6 .99975 .99973 .99970 .99966 .99961

90000 675.1 .99973 .99971 .99968 .99965 .99960

95000 712.6 .99972 .99970 .99967 .99963 .99959

100000 750.1 .99970 .99968 .99966 .99962 .99958

101325 760.0 .99970 .99968 .99965 .99962 .99957

105000 787.6 .99969 .99967 .99964 .99961 .99956

110000 825.1 .99967 .99965 .99963 .99959 .99955

In the absence of values for virial coefficients for air-C02 mixtures in

the temperature range of interest, the effect of the variability of CO 2 abundance on Z has been estimated, by two methods, to be of the order of

1 p.p.m. and, therefore, negligible. Also, the effect of the variability

of CO

2 abundance on Z due to the interaction of CO 2 with water vapor has

been estimated to be of the order of 0.1 p.p.m. and, therefore, negligible.

Consequently, table 2 is applicable to air containing reasonable amounts of

co

2

.

For temperatures and/or pressures outside the range of table 2, the

table of compressibility factor of moist air (also C02~free) in the

Smithsonian Meteorological Tables [14] can be used, with some loss of

precision since the listing there is just to the fourth decimal place.

3.4 Ratio of the Molecular Weight of Water to the Molecular Weight of

Dry Air, e

The molecular weight of water is 18.0152 [6], The ratio, £, of the molecular

weight of water to that of dry air is, therefore, 0.62201 for dry air with

a CO 2 mole fraction of 0.00033.

11
4. UNCERTAINTY IN CALCULATION OF AIR DENSITY

In estimating uncertainties we shall report them as 1 standard deviation

and we shall follow the suggested practice of Eisenhart [12, 13] in

stating separately the random and systematic components.

4.1 Uncertainty in R

The estimates of the uncertainties in the molar gas constant are based

on the work of Quinn, Colclough and Chandler [3]. Their quoted random

uncertainty is + 0.17 JK~1 kmol“l, which is + 2.0 x 10~^. The systematic

uncertainties combined in quadrature results in an "overall systematic

uncertainty of 17 p.p.m."

4.2 Uncertainties in Ma The uncertainty in the O2 abundance (mole fraction) of air [7] is separable

into a random component and a systematic component. The random uncertainty

is estimated to be + 0.00001, the systematic component is + 0.00006.

The random component of the uncertainty in the argon abundance is inferred

from the precision of Hughes' measurements [9] to be + 0.00001. No assignment of systematic uncertainty was made by Hughes.

The uncertainty in the CO 2 abundance has been mentioned earlier with

respect to the variability of the CO 2 abundance in the laboratory. For

a sample of air taken at the time a mass comparison is made, the uncertainty in the subsequent mass spectrometric determination of CO2 abundance

is _+ 0.00003 at the 0.00033 level. Since the measurements made by the mass spectrometric method are considered to be very precise, the estimated

uncertainty is considered to be systematic.

Since the N^ abundance is the difference between unity and the sum of

the mole fractions of the other constituents, the random component of

the uncertainty in the N

2 abundance is found by combining by quadrature

the random components of the uncertainties of the other three major components to be 1 x 10~ . For the systematic component, however, the practice

of finding the N

2 abundance by difference provides for very significant

cancellation of uncertainties. This is shown in the following treatment.

Eq. (15) can be written:

M

a M

0„ CL + M

C0 o CO,

m

a

xa

+ V X

N,

k

E

±=5 M.

1 (28)

12
The mole fraction of N

2 is, by difference.

x. - Z x. A

1=5 1 (29)

If we ignore the last term in Eq. (28), differentiate and go to finite

di f ferences

,

A <Vx.

" M

0

AX

0, + M

C0

,

AX

C0 + M

A

Ax

A

+ V( ' Ax

0

? " Ax

CC> 2 ' Ax

A>

1 2 2 z z zz z (30) = (M0

2 - + (MC0

2 - m

n

2

)Axco

2 + (ma - V

Ax

a

•

With the substitution of the appropriate systematic uncertainties and

the molecular weights into Eq. (30), the systematic component of the

relative uncertainty in Ma due to uncertainties in abundance of the

constituents is found to be 3 x 10“'*. The random component of the

relative uncertainty is found to be + 2 x 10

-

~* by combining by quadrature

the product of the molecular weight and the random component of uncertainty

in abundance for each of the four major constituents and dividing by M a

.

The uncertainty in M a due to the uncertainty in the value of the atomic

or molecular weights of the i-th constituent is

A(Ma ) M . = x. AM.

1 1

(31)

The uncertainties in the values of the atomic or molecular weights are inferred from reference [6]^. For the four major constituents they are: for 0^, _+ 0.0002; for CO^ , _+ 0.0005; for A, +_ 0.001; and for N 2, + 0.0001. These uncertainties are considered to be systematic. The

sum of the four terms represented by Eq . (3.1) is _+ 0.0001, corresponding

to a relative uncertainty in M a of 4 x 10“ .

The random component of the overall relative uncertainty in M a is, by

quadrature, + 2 x 10“^. The systematic component of the overall relative

uncertainty in M ais + 3 x 10“^, the sum of the component due to the

systematic uncertainty in abundance and that due to the uncertainty

in atomic or molecular weight. The corresponding random and systematic

components of the relative uncertainty in p are + 2 x 10“^ and + 3 x 10“-*,

respectively

.

4.3 Uncertainty in Z

The uncertainty in Z, the compressibility factor, is estimated from the

various uncertainties in Z due to the uncertainties in the virial coefficients [15]. The dominant uncertainty is that assigned to B aa (one of

the three used in the calculation of B mix of Eq . (24) [11]), the second

virial coefficient expressing the effects of interaction between two air

1 The uncertainty assigned in the reference is divided by 3 to estimate

the uncertainty at 1 standard deviation.

13
molecules. The estimated relative uncertainty in Z at 293. 15K, 101325

Pa (1 atmosphere) and relative humidity 50 percent is + 1.7 x 10

-

^ . The

corresponding relative uncertainty, taken to be systematic, in p is

+ 1.7 x 10" 5

.

4.4 Uncertainty in e

The uncertainty in Me , the molecular weight of water, is +_ 0.0005 [6] and

is treated here as systematic. The uncertainty in e, the ratio of Me to

M , is given by

3.

A G = M AM - M AM

a e e a

(M ) 2 a

(32)

-5

The random component of Ae thus calculated is + 1 x 10 and the systematic

component is + 8 x 10 . The corresponding uncertainties in the factor in

Eq. ( 14) involving e,

1 + (e-l)r

(33)

at 293. 15K, 101325 Pa and 50 percent relative humidity are _+ 1 x 10

-7 and

9 x 10 -®, respectively. The corresponding relative uncertainties in the

term and consequently in p are + 1 x 10“7 and + 9 x 10-8

, respectively.

4.5 Combined Relative Uncertainties in p Due to the Uncertainties in

M , R, Z and e

The random and systematic components of the relative uncertainty in P at 293. 15K, 101325 Pa and 50 percent relative humidity, due to the

uncertainties in M , R, Z and e are tabulated in table 3. a TABLE 3 RANDOM AND SYSTEMATIC COMPONENTS OF THE RELATIVE

UNCERTAINTY IN p CONTRIBUTED BY M , R, Z and e (at 293. 15K, 101325 Pa and 50 percent relative

humidity)

Parameter

M

a R

Z

e

Combined

Random

+ 2 x 10"5

+ 2 x 10"5

+ 1 x 10

+ 3 x 10

Systemat ic

+ 3 x 10

" 5

+ 1.7 x 10~5

+ 1.7 x 10~5

+ 9 x 10

~ 8

+ 6 x 10

14
The random components were combined by quadrature, the systematic

components were combined by addition. These components of uncer- tainty can be considered to represent the uncertainty "intrinsic"

to the air density equation, that is, that which is contributed by

the limitations on the present knowledge of M a , M e , R and Z. With

these uncertainties must, of course, be combined the uncertainties

in the environmental variables: P, T and relative humidity, and in

the knowledge of the carbon dioxide abundance.

5. MEASUREMENT OF P, T AND RELATIVE HUMIDITY

In order to estimate the uncertainties in p due to uncertainties in

the measurements of the environmental variables, P, T and relative

humidity, we shall estimate the uncertainties in these measurements

when made using the best applicable instrumentation and procedures.

Therefore, the estimated uncertainties in p will be those contributed

by the best possible measurements.

5.1 Pressure Measurement

The state-of-the-art in pressure measurement [16] permits the measurement

of the pressure in a laboratory with a random relative uncertainty of

less than + 2 X 10”^. Calibration of pressure measuring instruments against a primary standard of pressure contributes a systematic

relative uncertainty of about _+ 3 x 10”5. The corresponding Ap/p's

contributed by uncertainties in the measurement of pressure in a laboratory in the vicinity of a balance case are + 2 x 10

-

^ and + 3 x 10 ^

5.2 Temperature Measurement

The measurement of temperature in a balance case, that is the temperature

which determines the buoyant forces, is potentially the most critical

measurement in terms of its effect on the uncertainty in the calculated

air density. In the absence of experimental results, it is possible at

this time to make only a rough estimate of the temperature uncertainty

to be expected. If the balance case were instrumented with a network

of thermopile junctions, for example, the measurements would be expected

to have a standard deviation of about +_ 0.05K [17]. The systematic

uncertainty is estimated to be of the order of + 0.01K. At a temperature

of 293. 15K, these uncertainties correspond to relative uncertainties of

_+ 2 x 10

-<

^

, and + 3 x 10

-

^, respectively. The corresponding Ap/p's are

+ 2 x 10

-

^ and + 3 x 10

5.3

Humidity Measurement

The state-of-the-art in humidity measurement [18] permits the measurement

of humidity in a balance case with a random uncertainty of + 0.5 percent

relative humidity and a systematic uncertainty of + 0.3 percent relative

humidity. These uncertainties correspond to relative uncertainties in the

15
water vapor pressure factor [1 + (e-l)e'/P], in Eq . (14) and, therefore,

to A p/p, of + 4 x 10" 5 and + 3 x 10“-*at 293. 15K, 101.325 Pa and 50 percent

relative humidity.

Since e' is the effective vapor pressure of water in mo i s t air , a word of caution with regard to inferring e' from measurements of

relative humidity is in order. Relative humidity, RH, can be defined

[19] by

e

* RH = —

f

X 100 percent, (34)

s

where is the effective saturation vapor pressure of water in moist

air. Is greater than e

g

, the saturation vapor pressure of pure phase

(i.e., water vapor without the admixture of air or any other substance)

over a plane surface of pure ordinary liquid water, since the introduction

of a second gas (air in this case) over the surface of the water increases

the saturation concentration of water vapor above the surface of the

water. This "enhancement" of water vapor pressure is expressed by the

enhancement factor, f, which is defined by

f = ~ • (35)

e

s

The most recently published [15] experimentally derived value of f at 293. 15K

and 100000 Pa is 1.00400. Therefore, the common practice of inferring e' from measured RH and tabulated values of e

g introduces a significant error

in e

1 if f has been ignored. The corresponding relative error in p at

293. 15K, 101325 Pa and 50 percent relative humidity is about 1.7 x 10~5 . f Is a function of temperature and pressure.

In the present work, Hyland's values of f [15] have been fitted to a threeparameter equation in the pressure and temperature (t, °C) ranges of interest ir

national standards laboratories. The resulting equation is

f = 1.00070 + 3.113 x 10" 8P + 5.4 x 10~ 7

t 2

. (36)

Values of f calculated using equation (36), tabulated in table 4, are in

agreement with Hyland's values and are therefore appropriate for the

present application.

The expression for e' is found by combining Eqs. (34) and (35) to be

, _ RH . 8 100

f es (37)

The systematic relative uncertainties in p due to the uncertainties

assigned to f Cl5H and e

g [22j are approximately 1 x 10“^ and 2 x 10“^,

respectively, and would be negligible whether f and e g were treated as parameters or with the environmental variables.

16
TABLE 4. VALUES OF ENHANCEMENT FACTOR, f, CALCULATED

USING EQUATION (36)

Pressure

,

t, c pascals 15 20 25 30

70 000 1 .0030 1.0031 1.0032 1.0034

73 000 1.0032 1.0033 1.0034 1.0035

80 000 1.0033 1.0034 1.0035 1.0037

85 000 1.0035 1.0036 1.0037 1.0038

90 000 1.0036 1.0037 1.0038 1.0040

95 000 1.0038 1.0039 1.0040 1.0041

100 000 1.0039 1.0040 1.0042 1.0043

101 325 1.0040 1.0041 1.0042 1.0043

105 000 1.0041 1.0042 1.0043 1.0045

110 000 1.0043 1.0043 1.0045 1.0046

For the temperature range of interest in the present application, any of

several tables of e

g

, for example, references [21], [22], and [23], can

be used. Besley and Bottomley [24] have recently published experimental

values of e c in the temperature range 272.60 to 298. 04K. b

Values of e

g in the temperature range of interest for standards laboratories,

have been calculated using the formulation of Wexler and Greenspan [22]

and are tabulated in table 5.

TABLE 5 VALUES OF SATURATION WATER VAPOR PRESSURE, e

g

, CALCULATED USING FORMULATION OF WEXLER AND

GREENSPAN [22]

e s , pascals

Temperature, C

15 16 17 18 19 20 21 22 23 24 25 26 27

00 1705 1818 1938 2064 2197 2338 2487 2644 2810 2985 3169 3363 3567

05 1711 1824 1944 2070 2204 2346 2495 2652 2818 2994 3178 3372 3577

10 1716 1830 1950 2077 2211 2353 2503 2660 2827 3003 3188 3382 3588

15 1722 1836 1956 2083 2218 2360 2510 2669 2836 3012 3197 3392 3598

20 1727 1841 1962 2090 2225 2367 2518 2677 2844 3021 3207 3402 3609

25 1733 1847 1968 2097 2232 2375 2526 2685 2853 3030 3216 3413 3619

30 1738 1853 1975 2103 2239 2382 2533 2693 2861 3039 3226 3423 3630

35 1744 1859 1981 2110 2246 2390 2541 2701 2870 3048 3235 3433 3641

40 1749 1865 1987 2116 2253 2397 2549 2709 2879 3057 3245 3443 3651

45 1755 1871 1994 2123 2260 2404 2557 2718 2887 3066 3255 3453 3662

17
Table 5 (Continued)

es, pascals

Temperature, C

15 16 17 18 19 20 21 22 23 24 25 26 27

.50 1761 1877 2000 2130 2267 2412 2565 2726 2896 3075 3264 3463 3673

.55 1766 1883 2006 2136 2274 2419 2573 2734 2905 3085 3274 3473 3683

.60 1772 1889 2012 2143 2281 2427 2580 2743 2914 3094 3284 3484 1694

.65 1778 1895 2019 2150 2288 2434 2588 2751 2922 3103 3294 3494 3705

.70 1783 1901 2025 2157 2295 2442 2596 2759 2931 3112 3303 3504 3716

.75 1789 1907 2032 2163 2302 2449 2604 2768 2940 3122 3313 3515 3727

.80 1795 1913 2038 2170 2310 2457 2612 2776 2949 3131 3323 3525 3738

.85 1801 1919 2044 2177 2317 2464 2620 2785 2958 3140 3333 3535 3749

.90 1806 1925 2051 2184 2324 2472 2628 2793 2967 3150 3343 3546 3759

.95 1812 1931 2057 2190 2331 2480 2636 2801 2976 3159 3353 3556 3770

In the present work, the data of Besley and Bottomley in the temperature

range 288.15 to 298. 15K and calculated values [22] for the remainder of

the temperature range to 301. 15K, have been fitted to a two-parameter

equation. The resulting equation is

e = 1.7526 x lO^xp (-5315 . 56/T) . (38)

s Values calculated using equation (38) are sufficiently close to experimental

values and values calculated by the more complex formulation, within + 0.1

percent, to be used in the present application.

5.4 Random and Systematic Components of the Relative Uncertainty in P Due to Anticipated Uncertainties in State-of-the-Art Measurements

of P, T and Relative Humidity

The random and systematic components of the relative uncertainty in P at

293. 15K, 101325 Pa and 50 percent relative humidity due to anticipated

uncertainties in state-of-the-art measurements of the environmental

variables are tabulated in table 6.

TABLE 6: RANDOM AND SYSTEMATIC COMPONENTS OF THE RELATIVE

UNCERTAINTY IN p DUE TO ANTICIPATED UNCERTAINTIES

IN STATE-OF-THE-ART MEASUREMENTS OF P, T AND

RELATIVE HUMIDITY (at 293. 15K, 101325 Pa and 50

percent relative humidity)

Variable

P

T

Relative Humidity

Combined

Random Systematic

+2 x io

-4

±3 X io5

+ 2 x io

-4

±3 X 10~ 5

+4 x io'

5 ±3 X

-5

10

+ 3 x io~

4

+9 X

-5

10

18
The random components were combined by quadrature, the systematic components

were combined by addition.

It is clear from inspection of tables 3 and 6 that the dominant uncertainty

in the calculation of air density is that contributed by the measurement

of pressure, temperature and relative humidity, even when the measurements

are made using the best instrumentation and procedures. Considerable care must therefore be taken in making measurements of these environmental

variables in order to attempt to approach the precision and accuracy

represented in table 6.

To illustrate the effect of errors in measurement of the environmental

variables, at 293. 15K, 101325 Pa and 50 percent relative humidity (RH) an error of 0.1 percent in calculated air density results from an error of

0.29K in temperature measurement or a 101 Pa error in pressure measurement

or an 11.3 percent RH error in the measurement of RH.

5.5 Carbon Dioxide Abundance

As stated in section 3.1, the CO2 abundances in laboratory air and

consequently in the air in balance cases is in general variable. A

variation of 0.0001 in CO2 mole fraction is equivalent to a relative

variation of 4 x 10“^in calculated air density. Consequently, for

optimum utilization of the air density calculation, the CO2 abundance

should be known. Eq (23) enables convenient adjustment of Ma for

departures from the reference level, 0.00033, of CO2 abundance.

6. OVERALL UNCERTAINTY IN p DUE TO UNCERTAINTIES IN THE PARAMETERS

AND VARIABLES IN THE AIR DENSITY EQUATION

The random and systematic components of the relative uncertainties in P tabulated in tables 3 and 6 when combined, provide estimates of the

components of the overall relative uncertainty in p. The random

component, by quadrature, is + 3 x 10”^; the systematic component,

by addition, is + 2 x 10“^. The relative uncertainty due to the

variation of CO 2 abundance (4 x 10~5 per 0.0001 in CO2 mole fraction) is

necessarily not included in this estimate.

These components of the overall uncertainty have been estimated

at 293. 15K, 101325 Pa and 50 percent relative humidity. They are based on the present knowledge of M , R, Z and £ and on estimates

of the anticipated uncertainties in the measurements of the environmental variables (P, T and relative humidity) using the best applicable

measurement instrumentation and procedures.

7. AIR DENSITY AT STANDARD CONDITIONS, 0 Q

The air density, Pq, at standard conditions (Pq, MaQ> R > Tq, Zq) , for

dry air is expressed by an equation of the form of Eq . (14) as

19
09 ) p ° RT

o

zo

By dividing E«] . (1.4) by Eq . (39) we arrive al

The gas constant, R, has in en eliminated but M a has not unless the

apparent molecular weight of the dry air in the standard state is

equal to M a . For example, if the Pq is determined by experiment,

is not necessarily equal to M a

.

(40)

11 an experimental value of of sufficient accuracy were available in

the determination of which R was not used, R and its associated

uncertainty would be eliminated. In the absence of such an experimental value, as is the present case, the use of an equation combining

Eq . (14) with Eq . (21) is preferred to an equation of the form of Eq

.

(40) .

If standard conditions are taken to be = 2 73. 1 5 K , = 101325 Pa, = 28.963, and relative humidity = 0, for which Z^ = 0.99940,

P is calculated by Eq . ( L 4 ) to be 1.2928 kg vn “ ^ . The same result

is obtained for M = 28.964 (i.e., for a COo mole fraction of 0.00043).

a

*•

8. AIR DENS LTV EQUATION

M

By combining Eq . (14) wiLh Eq . (37) and substitute

density equation developed in this work becomes

w

tor the air

P

PM

a RTZ

M \ w \ RH_

M / 100

a /

f e

s P (41)

By substituting the QCC value of R, the value 18.0152 for M w and by

substituting the appropriate value, 28.963, for '83033 in Eq . (23),

Eq. (41) becomes

where

p = 0.000120254

PM^

TZ~

1 - 1 - 18.0152 \ RH

f e

s M i 100 P

a (42)

M = 28.963 + 12.011 - 0.00033

sl I CO

^ i

(43)

20
When SI units of pressure (Pa) and temperature (K) are used, the SI unit

of density (kg m“3) results. (The 31 unit of density is, of course,

equivalent to 10“3g cm~3 > )

For T = 293. 15K, P = 101 325 Pa, RH = 50 percent and ^033 - 28.963 g mol

-

"*-, the air density calculated using Eq . (21) is 1.1991 kg m~3.

8.1 Use of Constants in the Air Density Equation

In this section we shall investigate the effect on mass comparisons

of the use of appropriate constant values of f, Z and in Eq. (42).

The buoyant effect of the displacement of air by a mass artifact is

proportional to the density of the air, P, and the displacement volume,

Vm , of the artifact. We define here the buoyancy correction, m^, to be

added to the observed mass, by the following equation:

itl = pV = p — , (44)

‘b m pm

where m and are the mass and density, respectively, of the artifact.

To estimate the variation, Am^, in mb due to a relative uncertainty,

Ap/p, in air density, Eq. (44) is differentiated with respect to p and the

differentials are replaced by finite differences. The resulting equation

can be written

All > S- (^£) p . (45) * pm

p

By substituting a nominal value of p, 1.2 mg cm , in Eq. (45), we arrive

at

= 1.2 — (^) , (46)

Pm p m

where Amb ]S in milligrams. We shall return to this equation to estimate

the uncertainty in mass comparisons due to the use of certain constants

in Eq. (42) for the calculation of air density.

To estimate the effect of a variation in f about a selected constant value,

nominal values of P, T, Ma , Z, RH and e s are substituted in Eq . (42) and

the resulting equation is differentiated with respect to f. Nominal values

appropriate to the^Mass Laboratory of NBS are P

= 100000 Pa, T = 298. 15K,

M

a = 28.964 g mol , RH = 30 percent and e

g = 3169 Pa. The resulting equation

is

p = 1.1686 (1 - 0.003594 f). (47)

21

f
Differentiating Eq . (47) with respect to f, dividing the left and right

sides of the resulting equation by p and f, and going to finite differences

we find that — = -0.003609 ~ . (48)

P f The values of f in table 4 range from 1.0030 to 1.0046, therefore the

. O maximum Af/f for a nominal value of f of 1.0042 is equal to 1.2 x 10 . The corresponding Ap/ p is equal to 4.3 x 10“^.

To estimate the effect of a variation of Z about a selected constant value,

Eq. (42) is differentiated with respect to Z, the left and right sides of

the resulting equation are divided by p and Z and finite differences are substituted for the partial differentials. The result is

A_e = ^AZ

P z

' (49)

To estimate the expected variation in Z, it is necessary to estimate the

ranges of the environmental variables, P, T and RH. The mean pressure in

the Mass Laboratory at NBS is estimated to be 100106 Pa. The maximum and

minimum of the atmospheric pressure at NBS, taken from climatological

records of the National Weather Service of the National Oceanic and

Atmospheric Administration and adjusted to the elevation of the Mass

Laboratory, are 103850 and 96160 Pa. The expected ranges in temperature

and relative humidity are taken to be 18 and 28C and 10 and 50 percent,

respectively. For the nominal values given in this section, the expected

variations in the environmental variables result in a variation of Z between

0.99957 and 0.99971. These extremes in Z correspond to relative variations

in Z about the nominal value, 0.99966, of 9 x 10 and 5 x 10 , respectively.

The corresponding A_P's are -9 x

10”-* and -5 x 10”^.

P As was shown in section 3.2, a variation of 0.0001 in CO2 mole fraction

corresponds to a relative variation in M of 4 x

10”'* and to a Ap/p Q f -4 x 10”-*.

Combining by quadrature the maximum values of Ap/p due to expected variations

in f and Z and variation of 0.0001 in CO2 mole fraction, the result is 9.9 x 10 5. This is the value to be used in Eq . (46). Since the values combined

are maximum values, the combined value can be considered loosely to approximate

3 standard deviations.

For various materials of interest in the comparison of mass artifacts, Am^ in

micrograms (pg) has been calculated using Eq . (46) for 1 kg and 100 g artifacts

and tabulated in table 7.

22
TABLE 7 VARIATION IN BUOYANCY CORRECTION (mb ) DUE TO EXPECTED VARIATIONS IN f, Z AND

IN

s. MASS LABORATORY OF NBS

Material Pm> Amb , a v

-3

g cm V g M 8

1 kg 100 g

Plat inum-iridium 21.5 5.5 0.55

Stainless steel 8.0 15 1.5

Sil icon 2.2 54 5.4

Brass 8.4 14 1.4

A1 uminum 2.7 44 4.4

Tantal urn 16.6 7.1 0.71

Water 1 .0 119 12

We shall now apply Eq . (46) and table 7 to the comparison of mass artifacts.

In the comparison of two mass artifacts of nearly equal mass, m, but different

densities, and P 2 , the apparent difference in mass, m^ , due to the difference in buoyant force on the two artifacts is given by

m

d p(P1v

1

>

where is the displacement

air density.

A 1 x ^ A 1

( —) = cm (-

Pi P2 Pi

volume of the artifact of

(50)

density P^ and p_ is the

By differentiating Eq . (50) with respect to p and going to finite differences,

the resulting equation can be written

. ^ ,Ap. ,1 Am, = m(—) (—

d P p. -)p = m

P-i (^)P - ^

<^)p . (51)

Each of the terms on the right side of Eq . (51) is recognized to be of

Eq. (45). Therefore,

Am

d = (Am^) - (Am^^ , (52)

where the subscript numbers refer to materials of density P]^ and P 2 . Thus

the variation in the apparent difference in mass of two artifacts due to

the difference in buoyant force on them, resulting from the use of constant

values of f, Z and Mais approximated by the difference between the Amb'

s

in table 7 for the appropriate materials. For example, for a comparison of

kilogram artifacts of platinum-iridium and stainless steel the variation,

Am

d

, is 15 - 5.5 = 9.5 yg.

23
On the bases of the values of Ain^ calculated using table 7 and tne

precision of the balance used for mass comparisons, a judgment can be

made concerning the adequacy of t lie use of constant values of f, Z and

M a in Eq. (42). The precision of the balance used for the comparisons

of 1 kg mass artifacts at NBS is 25 Pg at the 1 standard deviation level.

Thus it can be concluded, for example, that constant values can be used

in the comparison of platinum-iridium and stainless steel artifacts, and

stainless steel and brass kilogram artifacts, but not for comparison of

platinum-iridium and silicon kilogram artifacts. The precision of the

kilogram balance at the Bureau International des Poids et Mesures (BIPM)

is 1.5 pg at the 1 standard deviation level, therefore it could be

concluded that the use of constants would not be appropriate in mass comparisons made using that balance.

For mass comparisons in the Mass Laboratory of NBS for which the values

in table 7 indicate that the use of constant values of f (1.0042), Z (0.99966) and Ma (28.964) in Eq . (42) is adequate, the resulting equation

i s

P 0.0034842

T

(P - 0.0037960 RH e

g

). (53)

in °C, Eq . (5 3)

, (54)

-3 where the SI units of density are equivalent to mg cm

For pressure in millimeters of mercury and temperature, t,

becomes

P '

(t +‘ 273U5) <P - 0-0037960 RH e.)

es In millimeters of mercury is cal c ul at ed us i ng an equation of the form

of Eq . (38) in which (t + 27 3.1 5) is substituted for T, and tiie preexponential

factor is modified. The resulting equation is

e s = 1.3146 x 109exp (-5315 . 56/ (t + 273.15)). (55)

Alternatively, the appropriate value of es in Pa could be taken from table 5 and multiplied by 0.00750062 to convert to millimeters of mercury.

9. SUMMARY AND CONCLUSIONS

An equation suitable for use in calculation of air density for air buoyancy

determination in the transfer of the mass unit by comparison of two mass artifacts on a balance has been developed. The Quinn, Colclough and Chandler

value of the gas constant, R, 8315.73 JK~^ kmol

-

^

, has been used in the

formulation. The apparent molecular weight of dry air, Ma , has been

calculated using currently accepted values of atomic weights and recent

determinations of the abundances of the various constituents of air. The

abundance of CO2 has been treated as variable. A "background" value of mole

fraction, 0.00033, has been used and a factor enabling convenient adjustment

of Ma for deviation of CO

2 abundance from the background value has been

24
derived. A new table of compressibility factor, Z, for the range of pressure

and temperature of interest in standards laboratories has been calculated

using recently determined values of virial coefficients. The enhancement

factor, f, which has usually been ignored in air density equations, has been

included explicitly in the equation. A table of f has been calculated using

a simple equation fitted to values of f in the range of pressure and temperature of interest. A table of the saturation vapor pressure of water, e s , based on a formulation of Wexler and Greenspan has been included. A simple

equation for the calculation of e s has been fitted.

Uncertainties, random and systematic, in the parameters R, M a , Z, and £, the ratio of the molecular weight of water vapor to the molecular weight

of dry air, and in the determination of the values of the environmental

variables, pressure, temperature and water vapor pressure, in the air

density equation have been estimated. The estimated random and systematic

relative uncertainties, Ap s contributed by the parameters are + 3 x 10

- ^ and + 6 x 10~^, respectively. The relative uncertainties anticipated to

be contributed by state-of-the-art measurements of the environmental

variables are estimated to be + 3 x 10 random and + 9 x 10 ^ systematic.

The effect of an ignored deviation of CO 2 abundance from the "background"

value would be an additional systematic relative uncertainty in P of

+ 4 x 10 ^ per 0.0001 variation of CO 2 mole fraction. The estimated

overall random and systematic relative uncertainties are + 3 x 1 0~^ and

+ 2 x 10“^, respectively.

At 293. 15K, 101325 Pa (1 atmosphere), and 50 percent relative humidity,

the estimated overall relative uncertainties in p correspond to

uncertainties in mass in the transfer between pi at i num- i r i d i urn and

stainless steel kilogram artifacts (volume difference of ^ 80 c.m J

) of

approximately 30 micrograms random and 20 micrograms systematic.

The following are recommendations concerning the transfer of the mass

unit at the various national standards laboratories:

(1) Eq. (42) should be adopted for use for all national standards

laboratories to provide both uniformity and the best available

calculation of air density.

(2) The CO 2 concentration should be treated as a variable and at

least a "background" value should -be determined for each of

the laboratories.

(3) Instrumentation and practices representing the state-of-the-art

in the measurement of the environmental variables should be applied.

10 . ACKNOWLEDGMENTS

The author gratefully acknowledges the invaluable contributions of

R. W. Hyland in several areas, and especially with the calculation of

25
the compress ib i 1 i ty factor; the considerable computational work of

C. L. Carroll, Jr.; and the typing of the manuscript by Kathy Durant.

11. REFERENCES

[1] List, R. J., Smithsonian Meteorological Tables, (The Smithsonian

Institution, Washington, D.C~ 1951) , 6th Revised Edition, p. 347 .

[2] Cohen, E. R. , and Taylor, B. N. , J. Phys. Chem. Ref. Data, 2, 663

(1973).

[3] Quinn, T. J. , Colclough, A. R. , and Chandler, T. R. D. , Phil. Trans.

Royal Society of London A283, 367 (1976).

[4] Gammon, B. E., J. Chem. Phys. 4, 2556 (1 976).

[5] Gammon, B. E., pr ivate commun i c at ion.

[6] International Union of Pure and Applied Chemistry, Inorganic Chemistry Division, Commission on Atomic Weights, Pure and Appl. Chem. 47,

75 (1976).

[7] Machta, L. , and Hughes, E., Science 168, 1582 (1970).

[8] Gluekauf, E., in Compendium of Meteorology, T. F. Malone, Editor,

(The American Meteorological Society, Boston, Mass., 1951), p. 4.

[9] Hughes, E. E., A Mass Spectrometric Technique for the Accurate

Determination of Atmospheric Argon, ACS 4th Mid-Atlantic Meeting,

Washington, D.C., Feb. 13, 1969, unpublished.

[10] U.S. Standard Atmosphere, 1962, (U.S. Government Printing Office

Washi ngton , D.C. , 1962) , p. 9.

[11] Hyland, R. W. , and Wexler, A., J

77A , 133 (1973).

. Res. Nat. Bur. St and . ( U. S . )

,

[12] Eisenhart

,

C. , J. Res. Nat . Bur. Stand. (U.S.), 67C

,

161 (1963).

[13] Eisenhart

,

C. , Science 160, 1201 (1968).

[14] List, R. J ., loc. cit., p. 333.

[15] Hyland, R. W. , J. Res. Nat. Bur. Stand . (U. S . )

,

79A, 551 (1975).

[16] Heydemann, P. L. M. , private communication.

[17] Roeser, W. F. and Wensel, H. T.

,

in Temperature, It s Measurement

and Control in Science and Industry , American Institute of Physic s

,

National Bureau of Standards, National Research Council (Reinhold

Publishing Corp., New York, 1941), p. 311.

[18] Wexler, A., Meteorological Monographs 11, 262 (1970).

26
[19] Harrison, L.P., in Humidity and Moisture, Vol. Ill, A. Wexler,

and W. A. Wildhack, Editors, (Rei nhol d Publisliing Corp. , New

York, 1964) , p. 51.

[20] Greenspan, L. , J. Res. Nat . Bur. Stand. (U .S.), 80A, 41 (1976).

[21] List, R. J. , loc. cit., p. 352.

[22] Wexler, A. and Greenspan,

75A, 213 (1971).

L. , J

.

Re s . Nat

.

Bur. Stand. (U.S.),

[23] Ambrose, D. and Lawrenson,

(1972).

I. J. , J. Chem. Thermody nami c s 4, 755

[24] Besley, L. , and Bottomley, G. A. , J. Chem. Thermodyaninic s 5, 397

(1973).

27
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLIC ATION OR RE PORT N'O. NBSIR 77-1278

2. Gov 't Arc t s s ion No.3. Recipient Ns Accession No.

4. TITLE AND SUBTITLE

The Air Density Equation and the Transfer of the Mass Unit

5. Publication Date

6. Performing Organization Code

7. AUTHORISE

Frank E. Jones

8 . Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Pro]ect/Task Work Unit No. 2130190

11. Contract Grant No.

12. Sponsoring Organization Name and Complete Address (Street. City. State. ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant in formation. If document includes a significant

bibliography or literature survey, mention it here.) A new formulation of the equation used for the calculation of air density has been

developed. The Quinn, Colclough and Chandler value of the gas constant, currently

accepted values of the atomic weights, and recent determinations of the abundances

of the various constituents of air have been used. The abundance of carbon dioxide

has been treated as a variable and a factor enabling convenient adjustment of the

apparent molecular weight of air for deviation of carbon dioxide abundance from -a background value has been derived. A new table of compressibility factor for the

range of pressure and temperature of interest in standards laboratories has been

calculated using recently determined values of virial coefficients. The enhancement factor, which has usually been ignored in air density equations, has been

explicitly included in the equation; a table of enhancement factor has been calculated

using a simple equation fitted to values in the range of pressure and temperature

of interest. A table of the saturation water vapor pressure has been included; a simple equation for the calculation of saturation water vapor pressure has been

fitted. Uncertainties, random and systematic, in the parameters and in the measurement of environmental variables and consequent uncertainties in calculated air density

have been estimated.

Application of the equation to air buoyancy determination and the transfer of the mass unit at the various national standards laboratories has been made.

17. KEY WORDS (six to twelve entries; alphabetical order, capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Air buoyancy; air density; mass unit transfer; real gas equation.

18. AVAILABILITY !XX Unlimited

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Springfield, Virginia 22151

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