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NBS TECHNICAL NOTE 1029

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

100

.U5753

No. 1029

1980
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

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2 Some divisions within the center are located at Boulder, CO 80303.
RATIONAL BUREAU OF STANDARDS The Thermodynamic Properties LIBRARY

of Helium II from K

> US'753

to the Lambda Transitions %

Robert D. McCarty

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

f ¥ S

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1980
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1029

Nat. Bur. Stand. (U.S.), Tech. Note 1029, 64 pages (December 1980)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $3.75

(Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

Abstract 1

1. Introduction 1

2. Survey of the Literature 2

3. Vapor Pressure 3

4. Density of the Saturated Liquid and Gaseous Phases 6

5. The Mathematical Model of the Equation of State 6

6. The Derived Thermodynamic Properties 12

7. Computer Programs 13

8. Estimated Accuracy of the Thermodynamic Properties 14

9. Property Tables 27

10. References 40

Appendix A. Computer Program Listings 41

m

THE THERMODYNAMIC PROPERTIES OF HELIUM II FROM K TO THE

LAMBDA TRANSITIONS

Robert D. McCarty

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

The equation of state of He-II is modeled by an equation of state

explicit in pressure as a function of density and temperature. The

equation of state is divided into three regions of temperature to

.8 K; .8 to 1.2 K and 1.2 to the lambda temperature for which similar

functional forms are used with different adjustable parameters. The

combined functions are valid over the entire PT range of the superfluid

and may be used for all classical thermodynamic properties. Comparisons between calculated and experimental data are presented. A

computer program for calculation of thermodynamic properties (PVT,

isochoric heat capacity, isobaric heat capacity, internal energy,

enthalpy, entropy, and velocity of sound) is included.

Key words: Computer program; equation of state; deviation plots;

helium II, mathematical model; superfluid; thermodynamic properties.

1. Introduction

A reliable mathematical model of the equation of state of a fluid has become

an important tool in nearly all applications of the classical thermodynamic

properties. Until now, no such model has been available for superfluid helium II,

Because of this situation the NASA-Ames Research Center issued a contract to the

National Bureau of Standards in Boulder, Colorado to develop such a model. As a

result, a model has been developed which is valid for all of the thermodynamic

properties over the entire range of pressure and temperature of the superfluid

helium II. It is not valid however for the prediction of the anomalous behavior
of the superfluid transition (the lambda line). The purpose of this report is to

present the results of the study conducted for NASA.

2. Survey of the Literature

A search of the world's scientific literature began with a computerized

search of about 120,000 articles at the Cryogenic Data Center of the National

Bureau of Standards in Boulder, Colorado. This search produced about 250

references. Each of these articles was considered for possible applicable

information and leads to articles not revealed in the original computer search.

The literature search produced an extensive set of experimental data which

covered the entire pressure and temperature range of He II, and in some cases

regions of overlap between the data sets.

Two extensive sources of data were found in the literature; the data by

Brooks and Donnelly [3] and the data by Maynard [7], The Brooks and Donnelly [3]

reference contains extensive tabulations of thermodynamic properties over the

entire PT range of the superfluid phase and is truly a monumental effort on the

part of the authors. The Maynard reference [7] also contains tabulations of the

thermodynamic properties of the superfluid but only for temperatures of 1.2 K and

above. The tabulated properties in these two references are of sufficient

quality and quantity to satisfy almost any need for tabular properties. However,

neither of the references contain a satisfactory mathematical model of the

equation cf state which is the primary purpose of this work. For reasons

discussed later the data of Brooks and Donnelly [3] were selected as the primary

source of data for the least squares estimation procedures. In addition, the

1958 helium temperature scale [2] and the saturated liquid densities by Ker and

Taylor [5], were used as input data.
3. Vapor Pressure

An expression for the 1958 helium temperature scale from McCarty [8] was

used to predict the vapor pressure of helium between .8 and 2.172 K. The

expression is

lr\ P = £ B.T U~1;

(1)

i=l

n

Table 1. Coefficients for Eq. 1 (Vapor Pressure)

B] = - 49.510540356

B

2 = 651.9236417

B

3 = - 3707.5430856

B

4

= 12880.673491

B 5 = -30048.545554

B6

= 49532.267436

By = -59337.558548

B 8

= 52311.296025

B9

= -33950.233134

BlO = 16028.674003

Bll = - 5354.1038967

B 12 = 1199.0301906

Bl3 = - 161.46362959

B 14 = 9.8811553386

where P is in urn Hg = 133.322 Pa and T is in kelvin. The range of validity for

eq (1) is 0.5 to 2.172 K. The coefficients to eq (1) are given in table 1. The

vapor pressures in table 7 below 0.8 K are calculated via the thermodynamic
conditions for the coexistence of two phases in equilibrium assuming the ideal

gas law for gaseous helium and using the Gibbs free energy (G) of Brooks and

Donnelly [3] for the liquid phase. The agreement between the vapor pressures

calculated from eq (1) and those calculated as described above between 2.172 and

0.5 K is good, with typical differences of 0.1 percent or less. Table 2 gives a

detailed comparison between the 1958 helium scale and pressures calculated via

the phase rule using the Brooks and Donnelly [3] data for the saturated liquid

and the PVT surface of McCarty [8] for the saturated vapor. The agreement (as

shown in table 2) between the 1958 scale and this work is a little surprising

since the real gas contribution at 2.0 K on the vapor side amounts to a 3.8

percent of the pressure (i.e., the difference between using an ideal gas equation

of state and the one from McCarty). At 1.5 K this real gas contribution has

fallen to 1.1 percent and at 0.8 K it has disappeared entirely. Below 1 K the

value of the Gibbs energy of the saturated liquid has become so small that it has

little effect on the solution and the vapor pressure is determined almost

entirely by the choice of L , the latent heat of vaporization at K. A value

of 59.62 joules/mole was used in the 1958 helium vapor pressure scale and a value

of 59.60202 joules/mole was used here. In using the Brooks and Donnelly [3]

tables for this calculation, both the tabulated G and the tabulated H-TS were

used for comparative purposes. The results are similar but the procedure of

using the tabulated G produced results slightly closer to the 58 scale. The G

obtained by forming H-TS agrees quite well with the tabulated G down to 0.5 K but

begins to differ from the tabulated G below 0.5 K and the two values actually

have opposite signs at 0.2 K and below. Since the values of G are so small at

these temperatures the resulting vapor pressure is the same for either the - or

the + value of G.
Table 2. Comparison of 1958 He Vapor-Pressure-Temperature

Scale and Values From This Work.

2.10

2.00

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

p, ba r 1958 Scale (eq 1)

0.41906 X 10" 1

0.31688 X I0"

1

0.23303 X I0"

1

0.16618 X I0'

1

0.11451 X I0"

1

0.75857 X ID" 2

0.47985 X lO"

2

0.28736 X o2

0.16111 X o2

0.83320 X o3

0.38952 X O"

3

0.15999 X lO"

3

0.55431 X io4

0.15258 X

!0"4

0.30376 X o5

0.37487 X o6

0.21770 X o7

P, 15a r %

This Work

0.41780 X IO"

1 0.29

0.31692 X IO"

1 0.00

0.23281 X IO"

1 0.09

0.16616 X IO"

1 0.01

0.11455 X IO"

1 -0.04

0.75889 X IO"

2 -0.04

0.48012 X IO"

2 -0.05

0.28756 X O"

2 -0.07

0.16123 X IO"

2 -0.08

0.83357 X 0"3 -0.07

0.38967 X IO"

3 -0.04

0.16001 X IO"

3 -0.01

0.55436 X IO"

4

-0.01

0.15258 X 0"40.00

0.30385 X IO"

5 -0.01

0.37496 X 0"6 -0.02

0.21792 X O"

7 -0.10

0.34626 X 0"9

0.42918 X 0" 12

0.10086 X O"

17

0.48395 X 0" 34
4. Density of the Saturated Liquid and Gaseous Phases

The density of the saturated liquid Dh">se was taken from Ker and Taylor [5]

and is given by

V

sat liq " *1 + VT - Txl +*3lT-Tx l*n|T.-Tx

| (2)

Table 3. Coefficients for Eq. 2 (Saturated Liquid)

£-| = 3.31007

i2 = 0.00742434913

£3 = - 0.0059164737553

where V is in liters/mole, T is in kelvin and the £, , l~, £3, and T, are given

in table 3.

The density of the saturated gaseous state was calculated using the equation

of state from McCarty [8] between 0.8 and 2.172 K. Below 0.8 K the density of

the saturated gas was calculated using the vapor pressure (as described in the

previous section) and the ideal gas equation of state.

5. The Mathematical Model of the Equation of State

It is desirable to model the equation of state of a fluid with a single

mathematical function. A single function insures thermodynamic consistency and

eliminates potential mathematical continuity problems going from one function to

another. Unfortunately a single function for the helium IV equation of state was

not found; instead it was necessary to divide the surface into three regions:

one from 0.0 to 0.8 K (Region I), one from 0.8 to 1.2 K (Region II), and from 1.2

to the lambda line (Region III). Although mathematical continuity is not

achieved at the boundaries, the fit is good enough at the boundaries of each
region to minimize the thermodynamic inconsistencies. For the region of to

0.8 K the function is

P = [P (Ap) + F

s (Ap,T)] 1.01325 (3)

where

A i P n (AP) = E a. Ap

1 (4)

i=l

1

and

F

s (Ap,T) = - F s>1

D

2

T

4

/4 - F $j2

D

2

T

5

/5 - F Sj3

D 2T

6

/6

- 2FSj4 (Ap)D

2

T

3

/3 - 2F

$j5

(Ap)D

2

T

4

/4

- 2F

$j6

(Ap)D

2

T

5

/5 - 2F SJ (Ap)D

2

T

6

/6 - 2F

s>8

(Ap)D

2

T

7

/7

- 3F Sj9 (Ap)

2

D

2

T

3

/3 - 3F SJ0 (Ap)

2

D

2

T

4

/4 - 3F"S jj (Ap)

2

D

2

T

5

/5

- 3F SJ2 (Ap)

2

D

2

T

6

/6 - 3F SJ3 (Ap)

2

D

2

T

7 /7 - 4F

$ J4 (Ap)

3

D

2

T

3

/3 (5)

- 4FSJ5 (Ap)

3

D

2

T

5

/5 - 4F SJ6 (Ap)

3

D

2

T

7 /7 - 4F $jl7 (Ap)

3

D

2

T

9

/9

- 4FSJ8 (Ap)

3

D

2

T

11 /H - 5F SJg (Ap)

4

D

2

T

3

/3

- 5FSj20

(Ap)

4

D

2

T

5 /5 - 5F

s>21

(Ap)

4

D

2

T

7 /7

- 5F

Sj22

(Ap)

4

D

2

T

9

/9 - 5F

$ ^(ApjVT1 Vll

where P is the pressure in bar, D is the density in moles/liter, T is temperature

in kelvin and Ap = D -36.27877. The coefficients for Region I are given in

table 4.
Table 4. Coefficients to Equations for Region I (0 to 0.8 K)

Coefficients for eq 4 (Region I)

(note Ap for Region I is defined as

D - 36.27877)

a] = 2.281877372

a

2

= 0.16820886

a 3

= 0.005277884968

Coefficients for eq 5 (Region

s.l

s,2

s,3

s,4

s,5

s,6

s,7

s,8

s,9

s,10

= - .776003592103 x

.516985343553 x

= - .185460414352 x

.993150555179 x

= - .372729528003 x

.905240314118 x

= - .263138088468 x

.210133644446 x

= - .251675888508 x

.246805662352 x

F ,, = - .235766906295 x

s , I I

s,12

.636877273619 x

F c n = - .464000281660 x

S,l 3

s,14

s,15

.133504455025 x

.441252121325 x

F e lc = - .390205136440 x

S , I

s,17 .569946840678 x

F

s 18

= - .219762939629 x

, to 0.8 K)

O"

4

-4

>-5

>-6

>-5

,-4

,-3

>-3

,-6

,-5

>-4

>-4

>-4

t -7

,-6

>-5

,-5

*-5
Table 4. Continued

F lg

= - .581270462264 x 10~ 9

F on = - .191711245461 x 10* 7

F

s 21

= .144897551106 x 10"6

F. oo = " .793219612515 x 10*7

S ,2Z

F .,= - .390940913081 x 10"7 S,23

The model for Regions II and III is given by

P = [VP(T) + P Q

(AD) + P r (AD,T)] 1.01325 (6)

where P is pressure in bar, D is density in moles/liter, T is temperature in

kelvin and VP(T) is given by eq (1). The P (AD) is of the same functional

form as eq (4) but with different adjustable parameters and a new definition of

the independent variable which is AD = D - D . ,. . D is as before the

density in moles/ liter and D ca+ ,. is 1/V". -,. which is given by eq (2). sac i iq Sal i iq

The new a. are given in table 5. The P (AD,T) is defined as

P r (AD,T) = f

p (AD,T) + f

s (Ap,T) (7)

where

f

p

(AD,T) = F pJAD

3

T

3 + F Pj2

AD

3

T

2,5

+ F Rj3

AD

3

T

2

+ F R)4

AD

2

T

3 + F Pj5AD

2

T

2

(8)

+F

P,6

ADt3+F

P,7

ADt2 * 5+FP,8ADt2

The AD and T are as above and the F

p

. are given in table 5.
The f (Ap,T) in eq (7) is of the same functional form as eq (5) with

different F<. . . All of the parameters for Regions II and III are given in

table 5.

Table 5. Coefficients to Equations for Regions II and III

Coefficients for eq 4 (Regions II and III, 0.8 to Tx)

a] = 2.241456

a 2

= 0.1757482

a 3

= 0.00470035

Coefficients to eq 5 (Region III, 1.2 to Tx)

>-3

s,l

s,2

s,3

s,4

s,5

s,6

s,7

s,8

s,9

s,10

= - .299775895293 x

.261528116001 x

= - .451073420829 x

= - .179926805218 x

.268760818966 x

.153832317516 x

= - .162726148595 x

.477756675722 x

= - .356060361531 x

.407625370109 x

F. n = - .713769173335 x

s ,1

s,12

s,15

.449456804718 x

F 10 = - .913635541095 x

s, lo

F ,. = - .975555037829 x

S ,14

.121659779679 x

F lc = - .528306039117 x s,16

-3

>-4

.-3

>-3

,-4

>-3

,-4

,-4

,-3

,-3

>-3

,-4

>-5

,-4

>-5

10
Table 5. Continued

s,17

s,18

s,19

.311573112016 x

.613299771434 x

.506000325098 x

F 9n = - .612590386700 x

S ,20

.230922759488 x

.327499222785 x

s ,21

s,22

F- o? = - .515534867647 x

S ,23

Coefficients for eq 5 (Region II, 0.8 to 1.2 K)

,-2

s,l

s,2

s,3

s,4

s,5

s,6

s,7

s,8

s,9

s,10

s,ll

.108660418499 x

= - .217871751436 x

.102911648479 x

.189253572751 x

= - .674364748289 x

.798926642309 x

= - .344107467055 x

.299781633163 x

= - .214084674667 x

.335439600940 x

.559092792724 x

F 10 = - .119903558078 x

S ,1 C

,-6

>-7

,-6

>-6

>-6

>-8

-8

-2

,-2

,-2

,-2

>-2

>-2

,-3

>-3

.-3

.-3

,-2

11
Table 5. Continued

F

$

> 13

= .526331681180 x 10" 3

F. -,„ = .118775501632 x 10'4

s,14

s ,15

s,16

s,17

s,18

- .459408808154 x 10"4

.519701003921 x 10"4

- .196070771338 x 10"4

.731453369826 x 10" 6

F. ln = - .526985760908 x 10"6

F on

= .169561251135 x 10"5

S,£0

F o

, = - .131795348291 x 10"5

S,2 I F. oo = - .714287537326 x 10" 7

F s,23

— .258759130915 x 10

u

Coefficients for eq 9 (Regions II and III, 0.8 to l"x)

f

p,i

= - .320783527549

F P,2

= .580145141306

F

P,3

= - .294344361744

F

P,4

= .290449403103

F P,5

= .801446582474

F p,6

= - .175703015761

F P,7

= .400129303603

F P,8

= - .255176262894

6. The Derived Thermodynamic Properties

All of the He II property values tabulated here have been calculated using

the equation of state, as described in the previous sections, applied to the

following relationships

12
AS = - /

p sat p W dp (9)

AH = /'

sat

ap

8T dp + P/p (10)

AC. -Usat

o 2 W

)

dp (11)

s

= c v* (£)

2

'fe) T

^

(12)

W = ) (*)J

1/2

(13)

The S, H and C of the saturated liquid state were taken from Brooks and

Donnelly [3]. The tabulated reference values of Brooks and Donnelly [3] and

those of Maynard [7] must be adjusted to be consistent with the published tables

of McCarty [8], The adjustment of 59.869851 joules/mole must be added to the

enthalpy. The saturated vapor values tabulated here were calculated using the

equation of state by McCarty. Computer program listings for all of the He-II

properties are given in Appendix A.

7. Computer Programs

The listings of the computer programs for the properties of He-II in

Appendix A are the programs used to calculate the various thermodynamic properties in tables 7 and 8. In general the Fortran functions have been named using

the first letter or two to describe the property and the last three letters to

describe the fluid. For example FUNCTION CVHE2(D,T) is the routine for specific

13
heat capacity at constant volume and the fluid is He-II. In addition to the

subprograms used in the example in Appendix A, there are a few other routines

which are not used in the calculation of the properties but have been included

here for the reader's convenience. These routines are PMELT2(T), FINDTD(D),

FINDTP(P), DENLAM(T) and PRSLAM(T). The PMELT2(T) routine calculates the melting

pressure of He-II according to the work of Grilly [4]. The other four routines

mentioned above give the PVT of the lambda line. FINDTP(P) calculates the

temperature given a pressure; PRSLAM(T) calculates the pressure given a

temperature; DENLAM(T) calculates the density given a temperature; and

FINDTD(D) calculates the temperature given a density. The PVT relations of the

lambda line are from the work of Kierstead [6], The remainder of the routines

are used in the calculation of the He-II properties.

8. Estimated Accuracy of the Thermodynamic Properties

The accuracy of the thermodynamic properties in tables 7 and 8 depends

almost entirely upon the accuracy of the data used in developiny the mathematical

models. Figures 1 through 9 illustrate the deviations between properties

calculated using the methods described in this work and data from the two primary

sources of Brooks and Donnelly [3] and Maynard [7]. In viewing these deviation

plots, it should be kept in mind that the Brooks and Donnelly [3] data were used

in the fit, over the entire temperature range but comparisons to the Maynard data

are included to illustrate the differences (or agreement) between the two major

sources in their range of overlapping pressures and temperatures.

No attempt was made to force the equation of state to reproduce the

asymptotic behavior of He-II as the lambda line is approached. However, as may

be seen in table 6, with the exception of the specific heat capacities, the

performance of the equation of state along the lambda line is comparable with the

14
.1rIU

o

5-Hz

O .1f-

<

IU

o

-.t

09 -0-

O

a

1.3

• 25 bar

o 20

o15

-8—tr

o

A o

A

1.5

a -jo bar

a 5

v 2.5

o

T

1.7

o

A

9

O

A

a

1.9

2

2.1

—

r

odd • aOo*aO «do a

aOo

0.2 0.4 0.6 0.8 1.0

TEMPERATURE, K

Figure 1. Differences between calculated densities and those

of Brooks and Donnelly [3],

15
.5rS

°

AC

UJ -.5

.5

m

a o

*

-.5-

*

V

-0-

A

O

A

O

o

B

§ o

*

o

1.3

o

A

4>-

1.5

• 25 bar

o 20

a 15

$•

1.7

* 10 bar

o5

° 2.5

1.9

o

a

a

o

A

# *—i—*

B

o

a

0.2 0.4 0.6 0.8

TEMPERATURE, K

2.1

1.0

Figure 2. Differences between calculated entropy and those

of Brooks and Donnelly [3].

2

8

16
o 1 _ a

D

a

a

V

III

III s s a

D o V

0. A

A A A

0» A

• A

•

i A

I i Oi o

z

s

1 •o

o

1 • Oi O

o I I 3 o

o o

o •

(ft

-1

1.3 1.5

•1.7

• 1.9 2.1

o

<

>

III a

1r-

-iL

• 25 bar

o 20

o15

* '.0 bar

v 2.5

£o—Jo—go £o—£o £a 6 $ § S £

n „ o« n D D ••

V V V V V V

0.2 0.4 0.6 0.8

TEMPERATURE, K

Figure 3. Differences between calculated sound speed and

those of Brooks and Donnelly [3].

1.0

a

17
>

o

5 -2

o

<

>

LU

O

2r-

-2

t•2.8

A

s

t t 1

•2.8 «4.5 #5.6

03.9

O

X-4 *—8-

A

a

a

D

1.3 1.5 1.7

A3.6

1.9 * 2.1

TM" • 25 bar A 10 bar

A o 20 D 5

o

• a 15 V 2.5

a

Y •

A

o

a

A

•Y

o

B

n i

I V

i

i 6

V

a

O

A

i V

2

8

o

V

o

•

£

V

•O

6

a

i• o

•

I a

t

8

•

4

g

o

?" 3.8 • - 15

0.2 0.4 0.6 0.8 1.0

TEMPERATURE, K

Figure 4. Differences between calculated specific heat

capacity at constant value and those of Brooks and

Donnelly [3].

18
aO

5 -6

• 9%

o O

f A * O O O

-Q-++34 ^og^—*-*-*.

1.3 1.5 1.7

*

D V

1.9 2.1

III

O

I— A

-2

a d (j

• 25 bar a 10 bar

o 20 d5

15 v2.5

A

^T^tavad °

na

D

o

A m

O

0.2 0.4 0.6

JQ_ .a

• o • o

D

o

v

o o

0.8 1.0

TEMPERATURE, K

Figure 5. Differences between calculated specific heat

capacity at constant pressure and those of Brooks

and Donnelly [3].

19
4rQ.

o

ai

2-

o

>

LU

a

tf -4-

O D V

O °

#

O vA 0v D A

1 6- 1 2 1—*-^0 $-a-g $

o 5 v

D

$ * o

s • 25 bar a 10 bar

a o 20 Q 6

015 M

1.3 1.5 1.7 1.0

TEMPERATURE, K

2.1

Figure 6. Differences between calculated entropy and those of

Maynard [7]

20
0.4-

z

hi

a

0.2-

£ -0.2

Ul

a

# -0.4-

—

• •

•

• • •

•

- o ° o

A

a

a

i

o

A

o

a

o

o

O O

V

V A

A

d a

o o

* * * * V

A

A

D

1 1 1

i

• 25 bar

o 20

6 1 a 1

o 15

A 10

°5

—, v 1

1.3 1.5 1.7 1.9 2.1

TEMPERATURE, K

Figure 7. Differences between calculated density and those

of Maynard [7],

21
o

m

in

GL

(0

O

w

o

<

>

o

4r2-

-2

-4l_

o 25 bar

o 20

15

a 10 bar

v1

V

a

a o

a a s

i

v

t

A A

O O O O ° O O $

1.3 1.5 1.7 1.9

TEMPERATURE, K

Figure 8. Differences between calculated sound speed and

those of Maynard [7].

2.1

22
6

4 -

O

biO o

<

IAJ o

2 -

-2

-4

-6

• •

O D

A O

A O

-o-* 8 °, i °

d n

v - a v 8

• 25 bar

o 20

015

a 10 bar

5

v 1

1.3 1.5 1.7 1.9

TEMPERATURE, K

t l57o

2.1

Figure 9. Differences between calculated specific heat

capacity at constant pressure and those of Maynard [7],

23
rest of the surface. The PVT in table 6 are from Kierstead [6] and the entropies

and sound velocity are from Ahlers [1],

Table 6. Selected Properties of He Along the Lambda Line With

Comparison to the Equation of State in This Work.

P bar

T p

l % C

2

7

Cliff

S

2

%

K moles

liter

meters

second

joules

mole

1. 2.163 37.001 - .2 225.3 -3.4 6.1560 1.9

2. 2.154 37.451 - .2 233.4 - .3 6.0719 3.0

3. 2.143 37.900 .2 241.0 2.0 5.9879 4.0

4. 2.133 38.300 .15 248.1 2.5 5.9118 4.7

5. 2.122 38.675 .06 254.8 2.2 5.8398 3.8

6. 2.111 39.050 .07 261.0 1.5 5.7717 3.2

7. 2.099 39.399 0.04 266.9 0.7 5.7037 2.7

8. 2.087 39.724 0.06 272.5 0.1 5.6397 1.9

9. 2.075 40.049 .07 277.7 -0.4 5.5796 1.8

10. 2.063 40.349 .1 282.6 -0.9 5.5196 1.5

11. 2.051 40.649 .1 287.3 -1.2 5.4635 1.2

12. 2.038 40.923 0.15 291.8 -1.3 5.4075 1.3

13. 2.025 41.198 0.16 296.0 -1.2 5.3515 1.2

14. 2.012 41.473 0.2 300.1 -1.0 5.2994 1.3

15. 1.998 41.748 0.3 303.9 -0.7 5.2474 1.6

16. 1.985 41.998 0.3 307.6 -0.2 5.1954 1.6

17. 1.971 42.248 0.3 311.1 0.2 5.1473 1.8

18. 1.957 42.497 + .3 314.5 0.8 5.0953 1.9

19. 1.942 42.722 - .3 317.7 1.4 5.0473 2.3

24
Table 6. Continued

Pbar

T P

l % C

2

7/o diff S

2

%

K moles

1 i ter

meters

second

joules

mole

20. 1.928 42.947 0.3 320.7 1*9 4.9992 2.5

21. 1.913 43.172 0.4 323.6 2.5 4.9512 2.2

22. 1.897 43.337 .04 326.3 3.0 4.9032 2.5

23. 1.882 43.622 0.4 328.8 3.5 4.8552 2.3

24. 1.866 43.846 0.4 331.1 3.8 4.8071 2.4

25. 1.850 44.046 0.4 333.3 4.0 4.7591 2.6

When using the equation of state, the best accuracy will result when the

input variables are pressure and temperature rather than density and temperature.

This is true of all equations of state in tht liquid region and is a consequence

of the nature of the liquid surface rather than the inadequacies of a

mathematical model.

Figures 1 through 9 illustrate a wide variation in the ability of the model

to reproduce some of the properties. Below 1.2 K the deviations in general

between calculated and input properties are noticably less than those above

1.2 K. The thermodynamic consistency of the data seems to be better at the lower

temperatures, and Brooks and Donnelly [3] estimate the accuracy of the data is

better at the lower temperatures. At the beginning of the project, Maynard [7]

was chosen as the primary source of data above 1.2 K. This decision was based on

preliminary analysis of the data themselves and on the claims of accuracy made by

the author. During the course of the investigation it became apparent that the

Brooks and Donnelly [3] data are the more internally consistent of the two sets

and these data were therefore chosen to be used in the fit over the entire

25
temperature range. The problem with the Maynard [7] data seems to be with the

density. Fitting the Maynard data by itself using a simultaneous data fitting

technique (which is necessary for this thermodynamic surface) where the PVT and

the entropy are used in the least squares estimation of the adjustable

parameters, produces a reasonably good fit to those two properties. Comparisons

to the other thermodynamic properties such as C and velocity of sound with

corresponding calculated values indicate large systematic deviations. Further

these deviation patterns are the same when comparing with the Brooks and Donnelly

data except for PVT, where there is a definite difference of about 0.3 percent

between Maynard and Brooks and Donnelly. Performing the same estimation

procedure as before (i.e., using PVT and entropy as input) but using the Brooks

and Donnelly data instead of the Maynard data produces a better overall fit to

both sets of data except again for the PVT of Maynard. This strongly suggests

that the Maynard PVT are inconsistent with the true thermodynamic surfaces.

Quite a lot of experimentation using various different combinations of the data

as input to the least squares estimation procedure indicated several other

points. First it is impossible to produce an equation of state for Helium-II

which is adequate to calculate all the thermodynamic properties using only PVT

data as input to the estimation procedure. This is true because the (9P/3T)

2 2

and (8 P/8T ) are so small that several orders of magnitude of precision in

the PVT would be required to determine them from the PVT alone. Second using

more than two types of data, for example, PVT, entropy and C simultaneously,

improved the C representation (but only marginally) over the case where only

PVT and entropy were used, however, the slight improvement of C degraded the

entropy representation noticably. Below 0.8 kelvin the entropy may be fit

independently of the PVT using eq (5) and then simply added to eq (4) without any

noticable effect on the representation of pressure.

26
An indication of the accuracy of the properties calculated from the equation

of state may be found by consulting the deviation plots, figs. 1 through 9.

9. Property Tables

The thermodynamic properties for He-II given here in tables 7 and 8, with

one exception, have been calculated using the computer programs given in the

appendix. The properties of the saturated vapor phase in tables 7 and 8 have

been calculated using the equation of state from McCarty [8]. To achieve

continuity between this work and the enthalpy of the author's 1973 work, a value

of 59.869851 joules/mole must be added to the enthalpy of the 1973 work. No

adjustment to the entropy is required.

The tables and equations given here are based on the 1958 helium vapor

pressure scale and the temperature adjustment made in the author's previous

helium work (1973) has not been made here.

The molecular weight of helium is taken to be 4.0026 and the value of R used

is .0831434 1 iter#bar/mole«kel vin.

27
Table 7 . Preipertles of Coexisting Liquid and Vapor.

PRESSURE TEMP DENSITY ENTHALPY ENTROPY CV CP SOUND SPEED

BAR K MOLES/LITER JOULES/MOLE JOULES/MOLE* K KETERS/SECONO

.4707E-01 2.15 .2777E+00 .1021E + 03 .4846E+02 .1278E+02 .2239E+02 83.7

.4707E-01 2.15 .3647E+02 .1023E+02 .5656E+01 .3317E*02 .3329E+02 238.0

.4133E-01 2.10 .2486E+00 .1013E+03 .4910E+02 .1276E+02 .2227E+02 82.9

.4133E-01 2.10 .3643E+02 •8903E+01 .4959E+01 .2657E+02 .2664E+02 238.5

.3605E-01 2.05 .2212E+00 .1005E+03 .4977E+02 .1274E+02 .2214E+02 82.1

.3605E-Q1 2.05 .3640E+02 .7664E+01 .4369E+01 .2361E+02 .2365E+02 238.7

.3123E-01 2.00 .1957E*00 •9962E+02 .5050E+02 .1272E+02 .2202E+02 81.3

.3123E-01 2.00 .3638E+02 •6518E+01 .3804E+01 .2076E+02 .2080E+02 238.5

.2687E-01 1.95 .1720E+00 .9875E+02 .5126E+02 .1270E+02 .2191E+02 80.4

.2687E-01 1.95 .3635E+02 .5525E+01 .3305E+01 .1805E+02 .1808E+02 238.1

.2295E-01 1.90 .1502E*00 .9787E+02 .5207E*02 .1268E+02 .2180E+02 79.6

.2295E-01 1.90 .3633E+02 .47 06E+01 .2887E+01 .1545E+02 .1547E+02 237.6

.1945E-Q1 1.85 .1303E+00 .9698E+02 •5293E+02 .1266E+02 .2169E+02 78.7

.1945E-Q1 1.85 .3631E+02 .3985E+01 .2507E+01 .1333E+02 .1335E+02 237.1

.1635E-01 1.80 .1122E+00 .9608E+02 •5384E+02 .1264E+02 .2159E+-02 77.8

.1635E-01 1.80 .3630E+02 •3354E+01 .2166E+01 .1171E+02 .1172E*02 236.5

.1363E-01 1.75 .9591E-01 .9517E + 02 .548 0E+02 .1262E+02 .2150E + 02 76.8

.1363E-01 1.75 •3628E+02 .2803E+01 .1860E+01 .1018E+02 .1019E+02 236.0

.1126E-01 1.70 .8131E-01 .9424E+02 .5582E*02 .1260E+02 .2141E+02 75.8

.1126E-01 1.70 .3627E+02 .2318E+01 .1582E+01 .8745E+01 .8753E+01 235.6

.9205E-02 1.65 .6833E-01 .9331E+02 .5690E+02 .1258E+02 .2133E+02 74.8

.9205E-02 1.65 .3626E+Q2 •1910E+01 .1342E+01 .7395E+01 .7401E + 01 235.2

.7449E-02 1.60 .5688E-01 .9236E+02 .5805E+02 .1257E+02 .2126E+02 73.8

.7449E-02 1.60 .3625E+02 .1572E+01 .1137E+01 .6200E+01 .6204E + 01 234.9

.5958E-02 1.55 .4686E-Q1 •9140E+02 .5927E+02 .1255E+02 .2119E>02 72.7

.5958E-02 1.55 .3624E+02 .1290E+01 .9605E+00 .5266E+01 •5268E+01 234.7

.4707E-02 1.50 .3818E-01 .9044E+02 .6057E+02 .1254E+02 .2112E+02 71.6

.4707E-02 1.50 .3623E+02 .1Q46E+01 .8028E+00 .4532E+01 .4534E+01 234.6

.3667E-02 1.45 .3071E-01 .8946E+02 .6196E+02 .1252E+02 .2107E+02 70.5

.3667E-02 1.45 .3623E+02 .8313E+00 .6591E+00 .3879E+01 .3880E+01 234.5

.2815E-02 1.40 •2438E-01 .8848E+02 .6345E+02 .1251E*02 .2102E+02 69.3

.2815E-Q2 1.40 .3622E+02 .6483E+00 .5320E+00 .3236E+01 .3236E+01 234.6

.2125E-02 1.35 .1906E-01 .8748E+02 .6505E+02 .1250E*02 .2097E+02 68.1

.2125E-02 1.35 .3622E+02 .5017E+00 •4270E+00 .2629E+01 .2629E+01 234.7

.1576E-02 1.30 .1466E-01 •8648E+02 •6676E+02 .1249E*02 .2093E+02 66.9

.1576E-02 1.30 . 3622E+02 .3860E+00 .3411E+00 .2105E + 01 .2105E+01 234.8

.1145E-02 1.25 .1106E-01 .8548E+02 .6862E+02 .1248E+02 .2090E1-02 65.6

.1145E-02 1.25 .3622E*02 .2916E+00 .2677E+00 .1685E+01 .1685E+01 235.0

.8134E-03 1.20 .8180E-02 .8447E+02 .7062E+02 .1248E+02 .2087E+02 64.3

.8134E-Q3 1.20 .3622E+02 •2160E+00 .2069E+00 .1340E+01 .1341E+01 235.2

.5635E-03 1.15 .590 8E-02 .8345E+02 .7280E+02 .1247E+02 .2085E+02 63.0

.5635E-03 1.15 .3624E+02 •1570E+00 .1572E+00 .1043E+01 .1043E+01 232.0

.3794E-03 1.10 .4156E-02 •8243E+02 •7517E+02 .1247E+02 .2083E+02 61.7

.3794E-03 1.10 .3624E+02 .1113E+00 .1170E+00 • 7931E*00 .7932E+00 232.3

.2474E-03 1.05 .2837E-02 •8140E+02 .7777E+02 .1247E+02 .2081E+02 60.3

.2474E-03 1.05 •3624E+02 .7737E-01 .8588E-01 .5864E+00 .5864E+00 232.6

.1554E-03 1.00 .1871E-02 .8037E*02 .8062E+02 .1247E+02 .2080E*02 58.8

.1554E-03 1.00 .3624E+02 .5289E-01 .6228E-01 .4204E+00 .42 05E+00 232.8

.9358E-04 .95 .1185E-02 .7934E+Q2 •8378E+02 .1247E+02 .2079E+02 57.3

.9358E-04 .95 .3624E+02 .3578E-01 .4493E-01 .2802E+00 .2802E+00 233.0

.5365E-Q4 .90 .7173E-03 .7830E + Q2 .8728E+02 .1247E+02 .2079E+02 55.8

.5365E-04 .90 .3624E+02 .2401E-01 .3234E-01 .1905E+00 .1905E+00 233.2

.2905E-Q4 .85 .4111E-03 •7727E+02 .9120E+02 .1247E+02 .2079E+02 54.2

.2905E-04 .85 .3624E+02 .1611E-01 .2341E-01 .1271E+00 .1271E + 00 233.2

.1526E-04 .80 .2294E-03 .7623E+02 .9529E+02 .2079E+02 .2910E+02 48.2

.1526E-04 .80 .3624E + 02 •1085E-01 .1709E-01 .8428E-01 .8431E-01 233.3

.7145E-05 .75 .1146E-03 .7519E+02 .1003E+03 .2079E+02 .2910E*02 46.7

.7145E-05 .75 .362 8E+02 .7223E-02 .1260E-01 .5521E-01 .5523E-01 240.3

.3038E-05 .70 .5220E-04 .7415E+02 .1059E+03 .2079E+02 .2910E+02 45.1

.3038E-05 .70 .3628E+02 .4973E-02 .9495E-02 .3659E-01 .3660E-01 240.3

.1148E-05 .65 .2125E-04 .7311E*02 • 1125E+-03 .2079E + 02 .2910E+02 43.5

28
Table 7. (Continued)

.1148E-05 .65 .3628E+02 • 3467E-02 • 7249E-02 .2501E-01 .2502E-01 240.3

.3750E-06 .60 .7517E-05 .7207E+02 .1201E+03 .2079E*02 .2910E+02 41.8

.3750E-06 .60 .362 8E+02 .2428E-Q2 .5569E-02 .1764E-01 .1765E-01 240.3

.1018E-06 .55 .2227E-0F .7103E+02 •1292E+03 .2079E+02 .2910E*02 40.0

.1018E-06 .55 •3626E+02 . 16 88E-02 .4260E-02 .1283E-01 .1284E-01 240.3

.218DE-07 .50 .5243E-06 .6999E+02 .1400E+03 .2079E+02 .2910E+02 38.1

.2180E-07 .50 .3628E+02 .1151E-02 .3210E-02 .9454E-02 .9456E-02 240.3

.3405E-08 .45 .9101E-07 .6896E+02 •1532E*03 .2079E+02 .2910E+02 36.2

.3405E-Q8 .c*5 .3628E+02 .7607E-03 .2357E-02 .6898E-02 .6899E-02 240.3

.3463E-09 .40 .1041E-07 •6792E+02 .1698E+03 •2079E+02 .291CE + 02 34.1

.3463E-09 .40 .3628E+02 . <»840E-03 .1671E-02 .4892E-02 .4893E-02 240.3

.1917E-10 .35 .6588E-09 .6688E + 02 .1911E+03 .2079E+02 .2910E*02 31.9

.1917E-10 .35 .3628E+02 .2955E-03 .1130E-02 .3319E-02 •3319E-02 240.3

.4293E-12 .30 .1721E-10 •6584E+02 .2195E+03 .2079E+02 .2910E+02 29.5

.4293E-12 .30 . 3628E+02 .1748E-03 .7190E-03 .2118E-02 .2118E-02 240.3

.2287E-14 .25 .1100E-12 .6480E+02 .2592E*03 •2079E*02 .2910E*02 27.0

.2287E-14 .25 .3628E+02 .1039E-03 •4202E-03 .1243E-02 •1243E-02 240.3

.1009E-17 .20 .6066E-16 .6376E+02 .3188E+03 .2079E*02 .2910E+02 24.1

.1009E-17 .20 „3628E*02 .6696E-04 .2170E-03 . 6*»46E-03 .6447E-03 240.3

.3181E-23 .15 .2551E-21 .6272E+02 .4181E*03 .2079E+02 .2910E+02 20*9

.3181E-23 .15 .3628E+02 .50 82E-Q4 .9239E-04 .2751E-03 .2751E-03 240.3

.4839E-34 • 10 .5821E-32 .6168E+02 .6168E+03 .2079E+02 .2910E+02 17.1

.4839E-34 .10 .3628E+02 .4464E-04 .2757E-04 .8244E-04 .8244E-04 240.3

29
Table 8. Properties of Superfluld Helium.

1. BAR ISOBAR

TEMP OENSITY ENTHALPY ENTROPY CV CP SOUNO SPEED

K MOLES/LITER JOLLES/MOLE JOULES/MOLE»K METERS/SECOND

2.16 .36916E+02 .1378*»E*02 .60i*«*2E + 0i .<+7138E*02 .i*7350E*02 233.09

2.0 • 3680'»E + 02 .9233*»E*01 .38383E + 01 .209*»7E*02 .20987E+02 237.85

1.95 .36781E+02 .82*»A>7E*01 .333i*i*E + 01 .18197E+02 .18227E+02 238.65

1.90 .36762E+02 .7i*296E+01 .29137E+01 .15573E*02 .15596E+02 239.27

1.85 • 367i*i*E + 02 .67125E+01 .25300E+01 .13i»i*3E*02 .13i*60E + 02 239.76

1.80 .36729E+02 .60850E+01 .2186 3E + 01 .11807E*02 .11821E+02 2i*0.H*

1.75 •36716E+02 .55369E+01 •18776E+01 .10270E*02 .I028 0E+02 21*0.1*5

1.70 .36705E+02 .5 05i»6E+0i .15973E+01 .88219E*01 .88297E+01 21*0.71

1.65 •36695E+02 .**6i»88E + 01 ,1355i*E + Ql .7i*637E+01 .7*»696E + 01 21*0.91*

1.60 •36686E+02 .43126E+Q1 .lii*8i*E + 0i .626 09E+01 .62652E+01 Zkl.li*

1.55 .36679E+02 .i*0315E*01 .97Q26E+00 .5319^E+01 .53225E+01 21*1,33

1.50 .36673E+02 .3788i»E*01 •81092E+00 .l*579l*E*01 .i*5816E + 01 21*1.51

i.i*5 .36668E4-02 .357«*3E*Q1 .66568E+00 .39210E+01 .39225E+01 21*1.70

l.<t0 .36663E+02 .33916E*01 .53716E+00 .32723E+01 .32 73 3E + 01 21*1.88

1.35 .3666QE+02 .32i*52E*01 .i*3Q93E*00 .26610E+01 .26616E+01 21*2.08

1.30 .36658E+02 .31295E+01 .3<*387E + 00 .21325E+01 .21328E+01 21*2.27

1.25 .36656E+02 .30350E*0i .26952E+00 .17083E+01 .1708<*E + 01 21*2.1*7

1.20 .3665tfE + 02 .29593E+01 .20776E+00 .136Q8E + 01 .13608E+01 242.67

1.15 .3668l*E + 02 .28983E+Q1 .15756E+00 .10610E+01 .10610E+01 21*1. i*i* 1.10 . 3668<*E+02 .2852i*E+01 .11676E+00 ,803i*OE*G0 .803i*OE + 00 21*1.67

1.05 .3668<»E + 02 .28185E+01 .85279E-01 .59168E+00 .59168E*00 21*1.90

1.00 .3668i*E + 02 .279i*lE*01 .61507E-01 .1*2251*E*00 .l»2251»E*0Q 21*2.10

.95 •3668^E+02 .27771E+01 .i*i*109E-Qi .28050E+00 .2805QE+Q0 21*2.25

.90 .3668i*E + 02 .2765^E*01 .31526E-01 .19003E+00 .1900i»E+0Q 21*2.36

.85 .36685E+Q2 .27576E+01 .22621E-01 .12659E+Q0 .12660E+00 21*2.1*3

.80 .36685E+02 .2752<tE+01 .16325E-01 .8i»2i*lE-01 .8*»25*»E-01 21*2.1*9

.75 .36696E+02 .27«»73E + 01 .11933E-01 .5i»126E-01 .54138E-01 21*7.71*

.70 .36697E+02 .27i*51E + 01 .89057E-02 .35^3E-01 .35l*53E-0i 21*7.77

.65 .36697E+02 .27i»37E*0i .67<*29E-02 .239 09E-01 .23916E-01 21*7.78

.60 •36697E+02 .27*»27E+01 .51i»72E-02 .16650E-01 .16655E-01 21*7.79

.55 .36697E+02 .27i*20E+01 .39187E-Q2 .11987E-01 .1199QE-01 21*7.80

.50 .36698E+02 .27i*15E*0i .29^15E-02 .8 7629E-Q2 .87650E-02 21*7.80

.1*5 .36698E+Q2 .27M1E + 01 .21536E-Q2 .63590E-02 •63602E-02 2t*7.80

,k0 .36698E+02 .27i»09E*01 .15224E-02 .i*li903E-02 .l*i»910E-02 21*7.80

.35 .36698E + 02 • 27l*07E*01 .10272E-02 .3035i*E-02 .3Q357E-02 21*7.79

.30 .36698E+02 .27<»06E*01 .65160E-03 .19305E-02 .19307E-02 21*7.79

.25 .36698E+02 .27i*05E+0i .37975E-03 .11285E-02 .11285E-02 21*7.79

.20 .36698E+02 .27i*05E*01 .19550E-03 .58332E-03 .58333E-03 2<*7.79

.15 .36698E+02 .271*05E*01 .83009E-Q1* .2i*795E-03 .2*»795E-03 21*7.79

.10 .36698E+02 .27^05E*01 .2i*711E-0i* .73982E-0<* .73982E-Q1* 21*7.79

30
Table 8. (Continued)

2. BAR ISOBAR

TEMP DENSITY ENTHALPY ENTROPY CV CP SOUNO SPEED

K MOLES/LITER JOULES/MOLE JOULES/MOLE.

K

METERS/SECONO

2.15 .37369E+02 .16013E+02 .58774E+01 .38667E*02 .38902E+02 232.63

2.0 .37244E+02 .12013E+02 .38774E*01 .21156E+Q2 .21215E+02 239.83

1.95 .37216E+02 .11013E+02 .33681E+01 .18363E+02 . 13406E+02 2i*l.i*7

1.90 .37193E+02 .10189E+02 .29429E+01 .15711E+02 .15743E+02 ZkZ.67

1.85 .37172E+02 .94651E+01 .25553E+01 .13559E+02 .13584E+02 244.06

1.80 .37155E+02 .88316E+01 .22083E+Q1 .11908E+02 .11927E+02 245.06

1.75 .37139E+02 .82784E+01 .18968E+01 .10358E+02 .10372E+02 245.91

1.70 .37126E + Q2 .77918E+01 .16140E*01 .88984E+G1 .89091E+01 246.63

1.65 .37115E+02 .73821E+01 .13698E+01 .75308E+01 .75388E+01 247.23

1.60 .37105E+02 .70426E+01 .11608E+01 .63198E+Q1 .63257E+01 247.75

1.55 .37096E+02 .67587E+01 .98082E+00 .53711E+01 .53754E+01 248.18

1.50 .37089E+02 .65130E+01 .81983E+00 .46248E+01 .46279E+01 248.55

1.45 .37083E+02 .62968E+01 .67311E+0Q .39608E*0i .39630E+01 248.88

1.40 .37078E + 02 .61122E+01 .54325E+00 .33071E+G1 .33087E*01 249.16

1.35 .37074E+02 .59642E+01 .43581E+00 .26914E+Q1 .26924E+01 249.41

1.30 .37071E+02 .58470E+01 .34767E+00 .21590E*01 .21596E*01 249.64

1.25 .37068E+02 .57512E+01 .27233E*Q0 .17313E + 01 .17316E+01 249.86

1.20 .37 066E+02 .56745E+01 .20969E+00 .138 07E*01 .13809E*01 250.06

1.15 .37098E+02 .56106E+01 .15914E+00 .10747E+01 .10748E+01 250.04

1.10 .37097E+02 .55641E+01 .11778E+00 .81470E+0Q .81477E+00 250.23

1.05 .37096E+02 .55297E+01 .85827E-01 .60070E+00 .60073E+00 250.40

1.00 .37096E+02 .55049E+01 .61667E-01 .42940E*Q0 .42941E+00 250.55

.95 .37095E+02 .54876E+01 .43970E-01 .28536E+Q0 .28536E+00 250.67

.90 .37095E+02 .54758E+01 .31174E-01 .19312E+00 .19312E+00 250.76

.85 .37096E+02 .54678E+01 .22137E-01 .12819E+00 .12820E+00 250.82

.80 .37 096E + 02 .54626E+01 .15780E-01 .84734E-01 .84741E-01 250.87

.75 .37092E+02 .54573E+01 .11396E-01 .53810E-01 .53818E-01 254.82

.70 .37092E + 02 .54551E+01 .8<t063E-Q2 .34735E-01 .34743E-01 254.85

.65 .37093E+02 .S4537E+01 .63028E-02 .23055E-01 .23061E-01 254.87

.60 .37093E+02 .54528E+01 .47759E-02 .15803E-01 .15807E-Q1 254.88

.55 .37093E+02 .54521E+01 .36174E-Q2 .11232E-01 .U235E-01 254.88

.5 .37093E+02 .54516E+01 •27060E-02 .81390E-02 .81407E-Q2 254.89

.45 .37093E+02 .54513E+01 .19764E-02 .58736E-02 .58746E-02 254.88

.40 .37093E + 02 .54511E+Q1 .13945E-02 .41334E-02 .41339E-02 254.88

.35 .37093E + 02 .545 09E*0i .93933E-03 .27877E-02 .27879E-02 254.88

.30 .37094E + 02 .54508E+01 .59476E-03 .17695E-02 .17696E-02 254.88

.25 .37094E+02 .54508E+01 .3^588E-03 .10321E-02 .10321E-02 254.88

.20 .37094E*02 .54507E+01 .17760E-03 .53195E*03 .53195E-03 254.88

.15 .37094E+02 .545 07E + 01 .75221E-Q4 .22524E-03 .22524E-03 254.88

.10 .37094E+02 .54507E+01 .22363E-04 .66931E-04 .66931E-04 254.87

31
Table 8. (Continued)

4. BAR ISOBAR

TEHP DENSITY ENTHALPY ENTROPY CV CP SOUNO SPEE

K MOLES/LITER JOULES/MOLE JOULES/MOLE *K METERS/SECON

2.13 •38244E+02 .20861E+02 .56718E+01 .3104 3E+Q2 .31451E+02 241.72

2.00 .38085E+02 .17533E*02 .39826E+Q1 .21684E*Q2 .21826E+02 21*9.79

1.95 .38043E+02 .16502E*02 .34577E+01 .18786E+02 .18890E+02 252.08

1.90 .38007E+02 .15654E*02 .30199E+01 .16064E+02 .16142E+02 254.06

1.65 .37977E+02 ,14910E*Q2 .26217E+01 .13860E+02 .13919E+02 265.78

1.80 •37950E+02 .14260E+Q2 .22656E+01 .12168E+02 .12211E+02 257.25

1.75 •37928E+02 .13693E*Q2 .19462E+01 .10583E*02 .10615E+02 258.52

1.70 .37908E+Q2 .13194E+Q2 .16565E+01 .90943E*01 .91183E+01 259.61

1.65 .37892E+02 .12774E+02 .14063E*0i .77010E+01 .77187E+01 260.5'*

1.60 •37878E+02 .12426E+02 .11920E+01 .64674E+01 .64802E+01 261.32

1.55 •37866E+02 .12135E*02 .18Q74E+01 .54986E+01 .55079E+01 261.96

1.50 .37855E+02 .11383E+02 . 84228E+00 .47345E+01 .47412E+01 262.49

1.45 .37847E+02 .11662E+02 .69196E+00 .40548E+01 .40595E+01 262.93

1.40 .3784QE+02 .11473E»02 .55895E+00 .33874E*01 .33907E+Q1 263.30

1.35 .37834E+02 .11321E+02 .44874E+Q0 •27596E+01 .27618E+Q1 263.60

1.30 .37829E*02 .11200E+02 .35817E+00 .22167E*01 .22183E+01 263.86

1.25 .37825E+02 .11102E*02 .28071E+00 .17801E+01 .17811E+01 264.08

1.20 •37822E*02 .11023E+02 . 21621E*Q0 .14219E+01 . 14226E + 01 264.28

1.15 .3785QE + 02 .10953E+02 .16471E+00 •11025E+01 .11Q3QE+01 265.31

1.10 .37848E+02 .109 05E+02 . 12212E+00 .84151E*00 .84181E+00 265.44

1.05 .37846E*02 .10869E+02 .88954E-01 .62520E+OQ . 62536E+00 265.54

1.00 •37845E*02 .1Q844E*02 .63674E-01 .45046E*00 . 45054E+0 265.63

.95 .37844E+02 .10625E4Q2 .45002E-Q1 .30216E + 00 .30219E+00 265.70

.90 •37844E+02 • 10813E + 02 .31424E-01 .20520E+00 .20520E+00 265.75

.$5 .37844E+02 .10804E+02 .21836E-01 -13549E+Q0 .13549E+00 265.80

.80 .37844E+02 .10799E*02 .15175E-01 .87621E-01 •87622E-01 265.83

.75 ,37823E*02 .1Q795E+02 .10650E-01 .54844E-Q1 .54847E-01 268.03

.70 .37823E+02 .1Q793E + 02 .76436E-02 .34351E-01 .34356E-Q1 268.06

.65 •37823E+02 .10792E*02 .55978E-02 .21989E-01 .21993E-01 268.08

.60 .37824E+02 .10791E*02 .41674E-02 .14521E-01 .14524E-01 268.09

.55 •37824E+02 .10790E*02 .31197E-Q2 .10Q02E-01 .10004E-01 268.10

.50 .37824E+02 .10790E*02 .23173E-02 .70922E-02 .70934E-02 268.10

.45 •37824E+02 .1079QE+02 .16859E-02 .50551E-02 .5055 8E-02 268.10

.40 .37824E*02 .10789E+02 .11869E-02 .35363E-02 .35366E-02 268.10

.35 .37824E+02 .10789E+02 .79811E-03 .23792E-02 .23794E-02 268.10

.30 •37824E+02 .10789E+02 .50433E-03 .15084E-02 .15084E-02 268.10

.25 .37824E+02 .10789E+02 .29242E-03 .87810E-03 .87813E-03 268. 09

.20 .37824E+02 .10789E«02 .14950E-03 .45Q76E-Q3 .45076E-Q3 268.09

.15 .37824E*02 .10789E+02 .63056E-04 .18951E-03 .18952E-03 268.09

• 10 .37824E*02 •10789E402 .18749E-04 .55869E-04 .55869E-D4 268.09

32
Table 8. (Continued)

3. BAR ISOBAR

TEMP OENSITY ENTHALPY ENTROPY CV CP SOUNO SPEED

,. K HOLES/LITER JOULES/MOLE JOULES/MOLE »K METERS/SECOND

2.14 .37816E+02 .18384E+02 .57551E+01 .33558E»02 . 33 863E+02 235.85

2.00 •37673E*02 .14777E+Q2 •39249E+01 .21398E+02 .21491E+82 244.82 , i.95 .37638E+02 .13J64E+Q2 .34086E+Q1 .18556E+02 .18625E+Q2 246.13

1.90 .37609E+02 .12929E+02 .29778E+01 .15872E*02 . 15923E+02 247.96

1.65 .37584E+Q2 .12196E+02 .25855E+01 . 13696E*02 .13735E+02 249.53

1.80 .37562E+Q2 .11555E*02 .22344E+Q1 .12026E+02 .12055E+02 250.87

1.75 .37543E+Q2 . 10996E+02 .19193E+01 .10460E+02 . 10482E+02 25 2.02

1.70 .37527E+02 .1Q504E*02 .16335E+Q1 .89874E*01 .90Q35E*01 253.00

1.65 .37513E+02 .1009QE+02 .13866E+01 .76082E*01 .76202E+01 253.82

1.60 .37501E+02 .97463E+Q1 .11752E+01 .63871E*01 .63959E»01 254.52

1.55 .37491E+02 .94590E+01 .99305E*00 .54295E*01 .54359E+Q1 255.10

1.50 .37483E+02 .92105E+01 . 8302 0E 4-00 .46754E+01 .4680BE+01 255.57

l.<»5 .37476E+02 .89918E+01 .68182E+00 .40044E+01 •40077E*01 255.97

1.40 •37469E+Q2 .88052E+01 .55049E+00 .33448E*01 .33470E+01 256.31

1.35 .37464E+02 .86553E*01 .44175E+QQ .27237E+01 .27252E+01 256.60

1.30 .37460E+02 .85367E+01 . 352<+6E*0Q .21866E+01 .21877E + 01 256.85

1.25 •37457E+Q2 .8<»396E + 0i .27610E+00 .17549E*iJI .17556E+01 257.07

1.20 •37454E+02 .83616E+01 .21255E+00 .14010E*01 . 14014E+01 257.27

1.15 .37485E+Q2 .82949E+01 .16160E+QO .10879E+01 .10882E+01 257.96

1.10 .37484E+02 .82477E+01 .11965E+00 .82735E*00 .82753E+00 258.11

1.05 .37482E + Q2 .82127E+01 .87133E-01 .61211E+00 . 61221E+0Q 258.25

1.00 .37481E+02 .81873E+01 .62455E-01 .439 07E+C0 .43911E+00 258.36

.95 •37481E+02 .81696E+01 .44313E-01 .29294E+0Q .29295E+00 258.45

.90 •37481E+02 .81574E+01 .31168E-01 .19845E*O0 .19845E+00 258.52

.85 .37481E+02 .8 1493E+01 .21892E-01 . 13131E+00 . 13131E+00 258.57

.80 .37481E+02 .81439E+01 .15409E-01 .85866E-01 .85869E-01 258.61

.75 .37467E+02 •81393E+01 .10974E-01 .54Q94E-01 .54Q99E-01 261.57

.70 •37467E+02 •81371E+01 .79888E-02 . 34393E-01 .34399E-01 261.60

.65 •37467E+02 .81358E*01 .59231E-02 .22425E-01 .22429E-01 261.62

.60 .37467E+02 .81348E+01 .44509E-02 .15897E-01 . 15101E-01 261.63

.55 •37468E*02 .81342E+01 .33525E-02 .10572E-01 .10575E-01 261.64

.50 •37468E+02 .81336E+01 •24992E-02 .75831E-02 .75846E-Q2 261.64

.45 .37468E+02 .81336E+01 .18217E-02 .54402E-Q2 .54410E-02 261.64

.40 .37468E+02 .81333E+Q1 .12837E-02 .38169E-02 .38173E-02 261.64

.35 •37468E+02 .81332E+Q1 .86373E-03 .257Q4E-82 .25706E-02 261.64

.30 .37468E+02 .81331E*01 .54622E-03 .16300E-02 .16301E-02 261.64

.25 .37468E+02 .81331E+Q1 .31713E-03 .94952E-03 .94955E-83 261.63

• 20 •37468E*02 .8133 0E+01 .16247E-03 .48830E-03 .48830E-Q3 261.63

.15 .37468E+Q2 .81330E+01 .6 8659E-04 .20601E-03 .20601E-Q3 261.63

.10 •37468E+Q2 .81330E+01 .20406E-04 .6 0969E-04 . 60969E-04 261.63

33
Table 8. (Continued)

6. BAR ISOBAR

TEMP DENSITY ENTHALPY ENTROPY CV CP SOUNO SPEEt

K MOLES/LITER JOULES/MOLE JOULES /MOLE •« METERS/SECONC

2.11 .39023E+02 .25953E*02 .55803E+01 .3Q491E+02 .31156E + 02 256.93

2.0 .38852E+02 .23021E+G2 .41277E+01 .22415E+02 .22691E+02 263.59

1.95 •38795E+02 .21947E*02 •35806E+01 .19368E + 02 .1956 8E+02 265.80

1.90 •38747E+02 .21065E«02 .31249E+01 .16547E+02 •16695E+Q2 267.68

1.85 •38706E+02 .2Q292E+02 .27119E*01 .14272E+02 .14382E+02 269.30

1.80 •38671E+02 .19619E+02 .23431E+01 .12523E+02 .12605E+02 270.69

1.75 •38641E+02 .19032E+82 .2Q126E+01 .10892E+02 .10953E+02 271.90

1.7 . 38616E+02 .18517E*02 .17134E+01 .93634E+G1 .94081E+01 272.96

1.65 •38594E+02 .18084E+02 .14548E+Q1 .79346E*01 .79672E+01 273.88

1.60 •38575E+02 .17724E+Q2 .12332E+01 .66690E+01 .66926E+Q1 274.66

1.55 .38560E+02 .17422E+02 .10422E+01 .5e714E+01 .56882E+01 275.31

1.50 .38546E+02 •17162E*02 .87155E+0Q .48813E+01 .48932E+01 275.85

i.45 .38535E*02 .16933E+Q2 .71644E+Q0 • 41783E+01 .41866E+01 276.29

1.40 •38526E+02 .16738E*02 .57931E+00 .349 03E+01 .34961E+Q1 276.66

1.35 •38519E+02 .16582E+02 .46560E+00 .28444E+01 .28484E+01 276.96

1.30 .38512E*Q2 .16457E*02 .37206E+Q0 .22861E+01 .22888E+01 277.21

1.25 •385Q7E+02 ,16356E*02 .29210E+00 .18363E*01 .18381E+01 277. 42

1.20 •38503E+02 .16274E+Q2 .22550E+00 .14673E*01 .14685E+01 277.60

1.15 •38526E+02 .16199E+02 .17237E+00 .11386E+01 .11394E+01 278.66

1.10 •38523E+02 .16149E+02 .12822E+00 .87465E+8Q .87517E+00 278.76

1.05 •38521E+02 .16112E+02 .9358 3E-01 .65498E+00 .65528E+00 278.83

1.00 .38519E+02 .16084E+02 .66941E-Q1 .47623E+00 .47639E+0Q 278.88

.95 .38518E+02 .16065E+02 .47Q62E-01 .32323E+00 .32330E + 00 278.92

• 90 •38517E+02 .16051E+02 .32484E-Q1 .22106E+00 .22108E+00 278.95

.85 •38517E+02 .16Q42E+02 .22147E-Q1 .14588E+D0 .14589E+00 278.98

.80 •38517E+02 .16036E+02 .15021E-01 .92703E-01 .92703E-01 279.00

.75 .38488E+02 .16036E+02 .1Q239E-01 .57341E-01 .57342E-01 280.18

.70 .38488E+Q2 .16034E*02 .71325E-02 .34953E-Q1 .34955E-01 280.20

.65 .38488E+02 .16Q33E*02 .5Q842E-02 .21590E-01 .21593E-01 280.22

.60 •38488E+Q2 .16032E+02 .3706 0E-02 .13698E-01 .13700E-01 280.23

.55 •38488E+02 .16031E + Q2 .27352E-Q2 .90977E-D2 .90990E-02 280.24

.50 .38489E+02 .16031E+02 .20154E-02 .62831E-02 .62 840E-02 280.24

.45 •38489E+02 .16031E+02 .14607E-02 .44117E-02 .44121E-Q2 280.24

.40 .38489E+Q2 .16Q30E+02 .10269E-02 .30667E-02 .30669E-02 280.24

.35 •38489E*02 .16030E+02 .69Q15E-03 .206Q7E-02 .20609E-02 28 0.24

.30 •38489E+02 .16030E+02 .43574E-03 .13072E-02 .13072E-02 280.24

.25 •38489E+02 .16Q3 0E*02 .25216E-03 .76087E-03 .76088E-03 280.24

• 20 •38489E+02 .16030E+02 .12846E-03 .38950E-03 .3895QE-03 280.24

.15 .38489E+02 .16030E+02 .540Q3E-04 .16272E-03 . 16272E-03 280.24

.10 .38489E+02 .16030E+02 .16096E-04 .47624E-04 .47624E-04 280.24

34
Table 8. (Continued)

10. BAR ISOBAR

TEMP OENSITY ENTHALPY ENTROPY CV CP SOUNO SPEE

K MOLES/LITER JOULES/MOLE JOULES/HOLE»K HETERS/SECON

2.06 .40313E+02 .35772E+02 .54414E+01 .310Q7E+02 .32182E+02 285.22

2.00 .40180E+02 .33914E*Q2 .45147E+01 .24677E+02 .25332E*02 289.49

1.95 .40099E+02 .32711E+02 .39028E+01 .21072E*02 .21518E*02 291.74

1.90 .40032E+02 ,31734E*02 .33976E+01 .17896E*Q2 .18212E+02 293.47

1.85 .39976E+02 .30887E+02 .29448E+01 .15381E+02 .15611E*02 294.80

1.80 .39928E+02 .30153E+02 .25430E+01 .13460E+02 .13630E+02 295.87

1.75 .39887E+02 .29517E+02 .218<»3E + Q1 .11698E+02 .11824E+02 296.79

1.70 .39852E+02 .28959E*02 .lo6Q5E*0i .10062E+02 .10155E+02 297.60

1.65 •39821E*02 .28489E+02 .15803E+01 .85417E+01 .86103E+01 298.32

1.60 .39796E + 02 .28Q98E+02 .13398E*01 .71950E+01 .72«»49E+01 298.94

1.55 •39 774E+02 .27771E+02 .11321E + Q1 .61238E+01 o61596E+01 299.47

1.50 .39756E+02 .27488E+02 .9^696E*0Q .52664E+01 .52917E+01 299.92

1.45 .39741E+Q2 .27241E*02 .77930E+00 .45019E*01 .45195E+Q1 300.30

1.40 •39728E+Q2 .27031E+02 .63143E+Q0 .37583E+01 .37704E+01 300.63

1.35 .39717E+02 .26862E+02 .50863E*00 .306 29E*01 .30712E+01 300.91

1.30 .39709E+02 .26727E+02 .40751E + 00 .24612E+01 . 24668E+01 301.14

1.25 .39702E+02 .26618E+02 .32130E+00 .19745E«0i .19783E+01 301.32

1.20 •39696E+02 .2653QE*02 .2<+963E+00 .15748E*01 .15774E+01 301.47

1.15 .39713E+02 .26443E+02 .19165E+00 .12396E+01 .12413E+01 301.53

1.10 .39710E+02 .26388E+02 .14334E+00 .95978E*Q0 .96077E+00 301.60

1.05 .39707E+02 .26347E+02 .10506E+00 .72620E+00 .72676E+00 301.65

1.00 .39705E+02 .26316E402 .75246E-01 .53481E+00 .53510E*0Q 301.68

.95 .39703E+02 -26294E+02 .52667E-01 .36996E+00 .37010E+00 301,69

.90 .39702E+02 .26279E+02 .35872E-01 .25644E*00 .25650E+00 301.70

.85 .39701E+02 .26268E+02 .23827E-01 .17036E*00 .17038E + 00 301.71

.80 .39701E+02 .26261E+02 .15537E-01 .10686E*00 . 10687E+0Q 301.72

.75 .39667E+02 .26269E+02 .10067E-01 .64699E-01 .64699E-Q1 301.97

.70 .39667E+02 .26267E+02 .66195E-02 .37923E-01 .37923E-01 301.98

.65 .39667E+02 .26265E+02 .4^538E-02 .22084E-01 .22084E-01 301.99

.60 •39667E+02 .26264E+02 .30912E-02 .13000E-01 . 13001E-01 302.01

.55 .39667E+02 .26264E+02 .22035E-02 .79908E-02 .79914E-02 302.01

.50 .39667E+02 .26264E+02 .15908E-02 .51870E-02 .51874E-0 2 302.02

.45 •39667E+02 .26263E+02 .11421E-02 .35083E-02 .35085E-02 302.02

.40 .39667E+02 .26263E+02 .80050E-03 .24 03E-02 .24004E-02 302.02

.35 •39667E+02 .26263E+Q2 .53772E-03 .16083E-02 .16083E-02 302.02

.30 .39667E+02 .26263E+02 .33920E-03 .10219E-02 .10219E-02 302.02

.25 .39667E+02 .26263E+02 .19574E-03 .59501E-03 .59502E-03 302.02

.20 .39667E+02 .26263E+02 .99195E-04 .30328E-03 . 30328E-03 302.02

.15 .39667E+02 .26263E+02 .i»1529E-04 .12544E-03 •12544E-03 302.01

.10 .39667E+02 .26263E+02 .12451E-04 .36373E-04 .36373E-04 302.01

35
Table 8. (Continued)

12. BAR ISOBAR

TEMP DENSITY ENTHALPY ENTROPY CV CP SOUND SPEEC

K MOLES/LITER JOULES/MOLE JOULE S/MOLE»K METERS/SECONO

2.04 •4Q865E+02 .40506E+02 .53391E+01 .30103E+02 .31499E+02 295.59

2.00 •40775E*02 .39333E+Q2 .47542E*01 .26324E+02 .27274E+02 298.68

1.95 .40679E*02 .38040E+02 •4Q959E+01 .22242E*02 .22856E+02 301.60

1.90 *40603E*Q2 .36998E+02 .35578E+01 .18770E*02 .19191E+02 303.71

1.85 .40539E+02 .36105E+02 .308Q2E+01 .1606i»E*02 .16364E+02 305.26

1.80 •4Q48 6E+02 .35335E402 .26585E+01 .l<f016E*02 .14235E+Q2 306.42

1.75 .40440E+02 .34669E+02 .22834E+01 .12164E+Q2 .12326E+02 307.39

1.70 .40401E+02 .34088E-9-Q2 .19456E+01 .10461E+Q2 .10581E402 309^21

1.65 .40368E+02 .33597E+02 .16532E+01 .88876E+B1 . 89759E+01 308.93

1.&0 •4034OE+O2 .33189E+02 .14Q18E+Q1 .74954E*01 .75599E+01 309.56

1.55 •40315E+02 .32846E*02 .11847E*01 .63837E+Q1 .64303E + 01 310.08

1.59 .4Q295E+02 .32551E+02 •99118E+00 .54894E*Q1 .55224E+01 310.52

l.i»5 •40278E+Q2 .32293E+02 .81621E+00 .46908E*0i .47140E+Q1 310.90

1.40 .40264E*Q2 .32074E+02 .66204E+00 .39158E+01 .39318E+01 311.23

1.35 .40252E+02 .31897E*02 .53388E+00 • 3192QE+61 .32029E+01 311.52

1.30 •40242E*Q2 .31757E+02 .42827E+00 .25650E+Q1 .25724E+01 311.75

1.25 •40234E+Q2 .31643E+02 .33837E+00 .20563E+01 -20612E*01 311.94

1.28 •40228E+02 .31551E+02 .26371E + 00 .16380E*01 .16414E + 01 312.09

1.15 .40246E+02 .31458E*02 .20286E+00 .13019E*01 .1304QE+Q1 311.59

1.10 .40241E*02 .314Q1E+02 .15202E+00 .i0Ii0E*01 .10123E+01 311.67

1.05 .40238E>02 .31357E*02 .11158E+Q0 .76749E+G0 .76821E+00 311.72

1.00 .4Q236E*02 .31325E+02 .79964E-01 .56738E+60 .56776E+00 311.75

.95 .40234E+Q2 .31301E+02 .55912E-01 .39499E*00 .39517E+00 311.76

.90 .40233E+02 .31284E*02 .37943E-01 .275C0E+00 .27508E+00 311.77

.85 •40232E+82 .31273E+02 .25001E-01 .18334E+0Q .18337E+00 311.78

• 80 .40232E+02 .31266E+02 .16075E-01 .115Q2E+00 .11502E+00 311.79

.75 .40197E+02 .31277E+02 .10202E-Q1 .69128E-Q1 .69129E-01 311.86

.70 .40197E+Q2 •31275E+02 .65370E-02 .40004E-01 „40004E-01 311.87

.65 .40197E+02 .31273E+02 .42 744E-02 .22772E-01 .22772E-01 311.88

.60 .40197E + 02 .31272E+02 .28894E-02 .12974E-Q1 .12975E-01 311.89

.55 •40197E+02 .31272E+02 .20189E-02 .76778E-02 .76782E-02 311.90

.50 .40197E+02 .31272E*02 .14400E-02 .48196E-02 .48199E-02 311.90

.45 •40197E*02 .31271E+Q2 .10278E-02 .31905E-02 .31907E-02 311.90

.40 .40198E*02 .31271E*02 .71893E-03 .21624E-02 .21625E-02 311.90

.35 •40198E+B2 .31271E+02 .48263E-03 .14457E-02 .14457E-Q2 311.90

.30 .4Q198E+Q2 .31271E+02 .3Q426E-03 .91863E-03 .91865E-Q3 311.90

.25 .4Q198E+02 •31271E+02 .17537E-03 .53456E-03 .53457E-03 311.90

.20 .40198E+02 .31271E*02 •88722E-04 .27188E-03 .27188E-03 311.90

.15 .40198E+02 .31271E+02 .37127E-04 .11214E-03 .11214E-03 311.90

.10 .4Q198E+02 .31271E+02 .11148E-04 .32511E-04 .32511E-Q4 311.90

36
Table 8. (Continued)

Id. BAR ISOBAR

TEMP DENSITY ENTHALPY ENTROPY CV CP SOUNO SPEEl

K MOLES/LITER JOULES/HOLE JOULES/MOLE»K METERS/ SECONl

2.01 .dl378E+02 .d51d8E*02 .52267E+01 .29756E+02 .31369E+Q2 303.18

2.00 .dl3d6E+02 .dd765E+02 .503d5E*0i .28dd6E*Q2 .29861E+02 30d.d6

1.95 .dl232E*Q2 .d33d9E*02 .d31d7E+0i .23691E+02 .2d551E*02 308. 7d

1.90 .dlld3E*02 .42229E+02 .37353E+01 .19811E+02 .20375E*02 311.71

1.85 .dl071E+02 .dl28QE*02 .32279E+Q1 . 168d9E+02 .17239E+02 313.81

1.80 .dl011E*02 •d0d68E*02 •27835E+01 .ld63dE*02 .ld913E+02 315.3d

1.75 .d0961E*02 .39770E+02 .23902E+01 .12670E*Q2 .1287dE+02 316.55

1.70 .d0918E+02 .39162E«02 .20371E+01 .10888E+02 .11038E+02 317.55

1.65 .d0881E+02 .38650E+02 .17316E+01 .925dlE*01 .936d7E*01 318. dO

1.60 .d0850E*02 .38223E*02 .ld688E*01 .78131E+01 .789dlE*01 319.11

1.55 .d0823E+02 .3786dE+02 .12dl5E*0i .66592E*fll .67179E+01 319.70

1.50 .d0801E+02 .37555E+02 .10391E + 01 .57268E*01 .57688E+Q1 320.19

l.d5 .d0782E*02 .37286E+02 .85631E+00 .d8932E*01 .d9228E+01 320.60

l.dO .dQ766E+02 .37057E+02 .6953dE*00 .dQ859E*01 .di06 3E + 0i 320.96

1.35 .d0753E*02 .36872E*02 .56136E+00 .33323E*Q1 .33d63E+0i 321.28

L.30 .d07d2E+02 .36725E+02 .d5085E+00 .26786E+01 .26880E+01 321. 5d

1.25 .d0733E+Q2 .366Q6E+02 .35689E+00 . 21d63E+01 .21527E+01 321. 7k

1.20 .d0726E+02 .36510E+02 .27891E+00 .17080E*01 .17123E+01 321.91

1.15 .d07d6E+02 .36dlQE*02 .21505E+00 .13700E+01 .13727E+01 320.98

1.10 .d07dlE+02 .36350E+02 .16id3E*00 .10672E*B1 .1Q689E+01 321.06

1.05 .dQ738E*02 .3630dE+02 .11862E+00 .81266E+00 .81358E+00 321.12

1.00 .d0735E+02 .36269E+02 .850d6E-01 .60252E*00 .60300E+00 321.16

.95 .d0733E+02 .362ddE+02 .59dl8E-01 .d2id6E*00 . d2169E*Q0 321.18

.90 .d0732E+02 .36227E+02 .d0219E-01 .29d26E*G0 . 29d36E + l»0 321.19

.85 .dO731E*02 .3621dE*02 .26352E-01 .19672E+00 . 19676E+00 321.20

.80 .d0731E+Q2 .36207E+02 .16766E-01 .12363E+Q0 . 1236dE+00 321.21

.75 .d0696E+02 .36222E*02 .10d52E-01 .7d012E-01 .7d013E-01 321.19

.70 .d0696EfQ2 .36219E+02 .65d2QE-02 .d2d38E-01 .d2d38E-01 321.20

.65 .d0696E*02 .36218E+02 .dl606E-Q2 .23703E-01 • 23703E-01 321.21

.60 .d0696E+02 .36217E*02 .27370E-02 .13113E-01 . 13113E-01 321.22

.55 .d0696E+02 .36216E+02 .18718E-02 .7d792E-02 .7d795E-02 321.23

.50 .d0696E*02 .36216E+02 .13173E-02 .d535dE-02 .d5356E-02 321.23

.d5 .d0696E+02 .36216E+02 .93d20E-Q3 .293dlE-02 .293d2E-02 321. Zk

.d0 .d0696E*Q2 .36216E+02 .65185E-03 .1968dE-02 .19685E-02 321. 2d

.35 .d0696E+Q2 .36216E+02 .d3719E-03 .13126E-02 .13126E-02 321.23

.30 .d0696E*02 .36216E+02 .27537E-03 .833d5E-03 . 833d6E-D3 321.23

.25 .d0696E+Q2 .36216E+02 .15855E-03 .d8d26E-03 .d8d26E-0 3 321.23

.20 .d0696E*02 .36216E+02 .8015 6E-0d .2d5 75E-03 .2d575E-03 321.23

.15 .d0696E+Q2 .36216E+02 .33575E-0d .10131E-03 .10131E-03 321.23

.10 .dQ696E+02 .36216E+02 .10076E-0d .29d96E-0d .29d96E-0d 321.23

37
Table 8. (Continued)

16. BAR ISOBAR

TEMP DENSITY ENTHALPY ENTROPY CV CP SOUND SPEEt

K MOLES/LITER JOULES/MOLE JOULES/MOLE.

METERS/SECONC

1.98 .<*1864E + 02 .i»c718E+02 .51Q62E+Q1 .29285E*02 .31137E+02 308.48

1.00 .41207E + 02 .41156E+02 .90552E-01 .640*»9E+0Q .64110E+Q0 330.02

.95 .41205E+02 .41130E+02 .63229E-01 .44974E+00 .<»500<»E^00 330.05

.90 • <*1203E + 02 .41111E+02 .42724E-01 .3145*fE*Q0 . 31467E+00 330.07

.85 •41202E+02 .4i098E*02 .27886E-Q1 .21069E+00 .21Q74E+00 330.09

.80 •41202E+02 .41089E+02 .17608E-01 .13277E*00 .13278E+Q0 330.10

.75 .41167E+02 .411Q9E + 02 .10815E-01 .79423E-01 .79427E-01 330.05

.70 .41167E+Q2 .<»ilQ5E*Q2 .66270E-02 .45261E-01 .4526 2E-01 330.06

.65 .41167E+02 .411B4E+02 .41032E-02 .24887E-01 .24887E-01 330.07

.60 .41167E+02 .4iiQ3E*Q2 .26250E-02 .13408E-01 .134Q8E-01 330.08

.55 .*»1167E*02 .41162E + 02 .17543E-02 .73795E-J2 .73797E-02 330.08

.50 .41167E+02 .41102E+02 .12165E-02 .43181E-02 .43183E-02 330.09

.45 .41167E+02 ,41102E+02 .85646E-Q3 .27254E-02 .27255E-02 330.09

.k0 .41167E+02 .41102E + 02 .59592E-03 .18082E-02 .18083E-02 330.09

.35 .41167E+02 .41102E*02 .39917E-03 .12020E-02 .12020E-02 330.09

.30 .<»1167E + Q2 .41102E+02 .25114E-Q3 .76212E-03 .76213E-03 330.09

.25 .«»1167E + 02 .41102E*fl2 .14449E-Q3 .44171E-03 .44171E-03 330.09

.20 .41167E + 02 .41102E+02 .73081E-04 .22367E-03 .22367E-03 330.09

.15 .41167E+02 .41102E«02 .30697E-04 .92414E-04 .92414E-04 330.09

.10 .41167E+02 .4UQ2E + 02 .91828E-05 .27170E-04 .27170E-04 330.09

38
Table 8. (Continued)

18. BAR ISOBAR

TEMP OENSITY ENTHALPY ENTROPY CV CP SOUND SPEED

K MOLES/LITER JOULES/MOLE JOULES yMOLE'K METERS/SECOND

1.96 .42330E+02 .54239E+02 .49873E+01 .28661E+02 . 30808E+02 312.02

1.00 .41654E+02 .4599 0E+02 .96570E-01 .68I47E+Q0 .68226E+00 338. 43

.95 .41652E+02 .45961E+02 .67417E-01 .48020E+00 .48Q58E+00 338. 48

.90 .41651E+02 .45941E*02 .45510E-01 . 33624E+Q0 .33641E*00 338.51

.85 .41650E+02 .45927E*02 .29635E-01 .22558E+00 .22565E+00 338.53

.80 .41649E+02 .45918E+02 .18616E-01 .14262E+00 .14265E+0Q 338.55

.75 .41614E+02 .45941E+02 .11297E-01 .8549QE-01 .85496E-01 338.48

.70 .41614E+Q2 .45938E+02 .67927E-02 .48552E-01 .48553E-01 338.49

.65 .41614E+02 .45936E+02 .40989E-02 .26361E-01 .26361E-01 338.50

.60 .41614E+02 .45935E+02 .25484E-02 .13869E-01 .13869E-01 338.51

.55 .41614E+02 .45934E+02 .16612E-02 .73748E-02 .73749E-Q2 338.52

.50 •41614E+02 .45934E+02 .11330E-02 .41587E-02 .41588E-02 338.52

.45 .41614E+02 .45934E*Q2 .79109E-03 .25550E-02 .25550E-02 338.52

.40 .41614E+02 .45934E+02 .5486QE-03 .16743E-02 .16744E-Q2 338.52

.35 .41614E+02 .45934E+02 .36688E-03 .11089E-02 .11089E-02 338.52

.30 .41614E+02 .45934E+02 .23053E-03 .70140E-03 .70141E-Q3 338.52

.25 .41614E+02 .45934E*02 .13258E-03 .405 07E-03 .40507E-03 338.52

.20 .41614E*02 .45934E*02 .67188E-04 .20470E-03 .20470E-03 338.52

.15 .41614E+02 .45934E+02 .28361E-04 .85065E-04 .85065E-Q4 338.52

.10 •41614E+02 .45934E+02 .84294E-05 .25414E-04 .25414E-04 338.52

39
10. References

[1] Ahlers, Guenter, Thermodynamics and Experimental Tests of Static Scaling

and Universality Near the Superfluid Transition in He4 Under Pressure,

Phys. Rev. A8> No. 1, 530-518 (Jul 1973).

[2] Brickwedde, F. G., Van Dijk, H., Durieu, M. , Clement, J. R., and Logan,

J. K. , The 1958 He 4 Scale of Temperatures, Part 1., J. Res. Nat. Bur.

Stand. (U.S.), 64A, No. 1 (Jan-Feb 1960).

[3] Brooks, J. S., and Donnelly, R. J., Thermodynamics of Superfluid Helium-4,

J. Phys. Chem. Ref. Data 6, No. 1, 51 (1977).

[4] Grilly, E. R., Pressure-Volume-Temperature Relations in Liquid and Solid

4He, J. Low Temp. Phys. JJ_, Nos. 1/2, 1973.

[5] Ker, E. C, and Taylor, R. D., The Molar Volume and Expansion Coefficient

of Liquid He7

, Ann. Phys. 26, 292 (1964).

[6] Kierstead, H. A., The Lambda Curve of Liquid He4

, Phys. Rev. T62, No. 1,

153 (Oct 1967).

[7] Maynard, J., Determination of the Thermodynamics of He-II From Sound

Velocity Data, Phys. Rev. B14, No. 9, 3868 (Nov 1976).

[8] McCarty, R. D., Thermodynamic Properties of Helium 4 From 2 to 1500 K, at

Pressures to 10 8 Pa, J. Phys. Chem. Ref. Data 2, 923 (1973).

40
APPENDIX A. Computer Program Listings

FUNCTION HE2<D,TT)

C MAIN PROPERTIES ROUTINE, FOR THE THERMODYNAMIC PROPERTIES

C OP HELIUM II.

C INPUT IS DENSITY IN MOLES/LITER, AND TEMPERATURE IN KELVIN

C OUTPUT IS PRESSURE, DPDT, DPDD, ENTROPY, ENTHALPY

C OR CV. OUTPUT UNITS ARE ATMOSPHERES, MOLES/LITER-KELVIN

C AND JOULES.

C

DIMENSION GC8I ,P(69> ,PR (23)

COMMON/DATA 2/PR,R,F

COMMON/DATA 3/G,RR

DATA(NR=0> ,<IH=0)

1 IF(NR.GT.0)GO TO 2

CALL OATA HE2

NT)s<»7

2 T*TT

IF(T.LT..l)T=.l

IF(T.GT.i.l99)G0 TO 3 Nfl*24

IFCT.GT..799)G0 TO < N0*1

GO TO <

3 ND*«»7

k I^CNR.NE.NOIGO TO 5

GO TO 7

5 NR=N0

NFsNR*22

J*0

DO 6 I=NR,NF

J*J*1

6 PR(J)sPd)

7 CONTINUE

GO TO (11, 12, 13, 14,15, 16) ,M

ENTRY PHE2

Msi

GO TO 1 ENTRY 0PDHE2

M*2

GO TO 1 ENTRY 0PTHE2

M=3

GO TO 1 ENTRY SHE2

Mel,

GO TO 1 ENTRY HHE2

IH=1

M*5

GO TO 1 ENTRY CVHE2

Me 6

GO TO 1 11 CALL PRESS2(P1,D,T)

IF(NR.LT.2*»)G0 TO 111

CALL PRESS3(P3,D,T)

HE2=P1*P3

IFdH.EQ.DGO TO 17

RETURN

111 HE2*P00(D)*P1

IF<IH.EQ.l)GO TO 17

RETURN

12 CALL DP002(P1,0,T>

IF<NR.LT.2UG0 TO 112

CALL DPD03IP3,D,T)

41
HE2=P1+P3

RETURN

112 HE2=Pl + OPOO<0>

RETURN

13 C*LL DP0T2<P1,D,T>

IF<NR.LT.2*»)G0 TO 113

CALL 0PDT3(P3,0 t T)

HE2=P1*P3

RETURN

113 HE2=P1

IF(TT.LT..1)HE2=HE2*TT*10.

RETURN

1*» DS = SATL2(T)

CALL 0SDN2<P1S,DS,T)

CALL DSDN2(Pi,D,T>

IF(NR.LT.2MGO T011<»

CALL DS0N3<P3S,QS,T)

CALL DSDN3(P3,D,T)

HE2MP1-P1S + P3-P3S + CVPHE2<T )» (

1

./D-l./DS) > *101.325«-S0 <T)

RETURN

11*» HE2=S0(T) 4CP1-P1S)*101.325

IP(TT.LT..1)HE2=HE2*TT*10.

RETURN

15 0S=SATL2(T»

GO TO 11

17 CONTINUE

CALL DSDN2(P1S,DS,T)

CALL PI02(P2S,0S,T>

CALL DS0N2CP1,C,T)

CALL PID2(P2,D,T>

IF<NR.LT.2<»>GO TO 115

CALL DS0N3CP3S,DS,T)

CALL PIC3(Pi»S,OS,T)

CALL DSCN3(P3»D,T>

CALL PI03(Pt»,D,TI

HE2=( (P1-P1S*P3-P3S*DVPHE2(T)*(1./D-1./DS) )*T*HE2/0

1+P2-P2S*P<4-P^S)*101.325*H0(T)

IH = RETURN

115 HE2=H0(T) UP1-P1S>*TMP2-P2S))*101.325MPOOH<D>+HE2/0) 101.3 25

HE2=HE2-P0 0H ( OS >*10 1.325

IH=0

RETURN

16 DS=SATL2<T>

CALL DS0T2(P1S,0S,T>

CALL DS0T2(P1,D,T)

IF(NR.LT.24)G0 TO 116

CALL DS0T3(P2S,DS,T)

CALL 0S0T3CP2,D,T)

HE2=<P1-P1S+P2-P2S)*101.325+CO (T) CVVP(0,T>

RETURN

116 HE2=(P1-P1S)*101.325*C0(T)

IF(TT.LT..1)HE2=HE2*TT*10.

99 CONTINUE

RETURN

END

SUBROUTINE DATA HE2

SETS UP THE CONSTANTS FCR THE EQUATION CF STATE

DIMENSION G(8)»P(69) ,PR(23)

COMMON/DATA 2/PR,R,P

COMMON/DATA 3/GtRR

DATA(A0=2.2'»1'»56)

DATA(A1=.1757<»82)

DATA(A2=. 00^70035)

42
Rt. 08205616

PQsR

PC 1>= -.776003592103E-01*

PC 2)= .5169853««3553E-0«»

PC 3»= -.I85<»60%l*»352E-05

PC <> = .993150555179E-06

PC 5>= -.372729528003E-05

PC 6)= .9052*»03H»118E-0*4

PC 7)= -.263138088<»68E-03

PC 81= .2101336^i»«»6E-03

PC 9)= -.2516758885Q8E-06

PC10)= .246805662352E-05

PCli)= -.235766906295E-04

PC12)= .636877273619E-0««

PC13> = -.46*»000281660E-0<»

PCl<»> = .13350*»*»55025E-07

PC15>= .*»M252121325E-06

PC16)= -.390205136440E-05

PC 17)= .5699<»68<»0678E-05

PC18)= -.219762939629E-05

PC19I= -.581270*6226<»E-09

PC20)* -.1917il2<*5*.61E-07

PC21)= .H»**897551106E-06

PC22>= -.793219612515E-07

PC23)= -.3909<»0913081E-07

PC2«»> = .108660'»18'f99E-02

PC25)= -.217871751436E-02

PC26)= .1029116'«8*t79E-02

PC27>= .189253572751E-02

PC28)= -.67*»36*»7<«8289E-02

PC29)= .7989266<42309E-02

PC30»= -.3<»<»107«»67055E-02

PC31>= .299781633163E-03

PC32)= -.21<»08<»67'»667E-03

PC33I= .335W396009*»OE-03

PC3«»)= .55909279272t»E-03

PC35I= -.119903558078E-02

PC36)= .5263316C1180E-03

PC37)= .118775501632E-0«»

PC38>= -.1»59<»0880815«»E-0'»

PC39)= .519701003921E-0t»

PCt»0)= -.196070771338E-0<»

PC»i)= ,731<»53369826E-06

PC«»2I = -.526985760908E-06

PCi»3>= .169561251135E-05

Pikit)* -,1317953*»8291E-05

PC<»5)= -.71«»287537326E-07

PC*6)= .258759130915E-06

P(i»7)= -.299775895293E-03

P(i»8)= .261528116001E-03

PC<»9> = -.<»51073<»20829E-0*»

PC50) = -.179926805218E-03

PC51I= .268760818966E-03

PC52)= .153832317516E-0**

PC53)= -.162726i«»8595E-03

PC5«»> = .«»77756675722E-0i»

PC55)= -.3560603€1531E-0««

PC56)= .«f07625370109E-03

PC57)= -.713769173335E-03

PC58>= .4*»9tf5680U718E-03

PC59>= -.9136355m095E-0<»

PC60)= -.975555037829E-05

PC61> = .121659779679E-0**

PC62)= -.528306039117E-05

43
P(63)= .311573112016E-06

P(6<»> = .613299771<»3<iE-07

P(65)= .5Q6000325098E-06

P(66) = -.612590386700E-06

PC67)= .230922759<f88E-06

P<68>= .327t»99222785E-08

P(69)= -.5155348676*»7E-08

G( 11= -.3207835275*»9E*01

G( 2)= .580i«»5Hil306E*01

G( 3)= -.29<»3^3ei7i»«»E + 01

G( <> = .29 0<»U9A0 310 3E-01

GC 5)= .801«»<»65e2t»74E-01

G( 6)= -.175703015761E+00

G< 7) = .<»00129303603E + 00

G( 8)= -.25517626289*»E*00

RETURN

END

SUBROUTINE HE2E(P,Q,T>

C CALCULATE ALL PROPERTIES OF THE F SUB S PART OF THE EQUATION

C OF STATEt UNITS ARE THE SAME AS IN FUNCTION HE2

DIMENSION G<8> ,A<8)

C0MM0N/CATA3/G,R

OATA<A0 = 2.2«»ll»56>

DATA(A1 = .1757*»82>

0ATA(A2=. 00^70035}

1 N*l

IE=1

Q2=G*Q

Q3=Q2*Q

D0=SATL2(T>

002=00*00

003=002*00

OD0=DSATL2(T)

0002=000*000

D20=D2SATL(T)

0=Q

T2=T*T

T3=T2*T

T*» = T3*T

T5 = T***T

T6=T5*T

D*0-SATL2<T>

02=0**2

D3=D2*D

0^=03*0

GO TO (10,20, 30, ^0,50, 60), II

ENTRY PRESS3

11 = 1 GO TO 1 10 A(1)=A2*03*T3

A(2)=A2*03*T**2.5

A(3)=A2*D3*T2

A(M=A1*02*T3

A(5>=A1*D2*T2

A(6»=AO*0*T3

A(7)=A0*D*T**2.5

A(8) =AO*0*T2

2 SUM=0

DO 3 1=1,8

3 SUM=SUM+A{I)*G(I)

P=SUM+P0(G,T)+VPNNE2(T)

RETURN

ENTRY DS0N3

11=2

44
GO TO 1 20 ACi)=-A2*<Q2*T2*3/2.-3*Q*<000*T3+0 0*T2*3)+3*<2*0 0*ODO*T3

1 +3*002*T2)*ALOG<C)+(3*C02*CDO*T3+003*T2*3>/Q>

A<2)=-A2*(Q2*T**1.5*2.5/2.-3*Q*(DDO*T**2.5+0 0*T**1.5*2.5)

1 +3*ALOGfQI*C2*D0*CO0*T**2.5+2.5*D02*T**1.5>

2 +f3.*00 2*00 0*T**2.5+2.5*0 3*T**i.5)/Q>

A(3l=-A2*<Q2*T-3*Q*(OC0*T2+D0*T*2>+3*ALOG(Q>* <2*0 0*DDO*T2

i +2*D02*T)+< 3*0 02*00 0*T2+2*D0 3*T)/Q>

AU)*-Al*<3*T2*Q-ALOG(G)*(6*T2*D0 + 2*T3*DD0>

1 -(3*T2*O02+T3*OO0*O0*2)/Q)

A<5)=-A1*(T*Q*2-ALOG(Q)*U*T*DO+2*T2*DDQ>

1 -<2*T*D02+T2*OD0*D0*2)/Q>

A(6)=-A0*<3*ALOG(Q)*T2 (00 0*T3+T2+O0*3> /Q)

AC7)=-A0*<2.5*ALOGCQ)*T**1.5+(CO0*T**2.5*2.5*D0*T**1.5)/Q)

A( 8I=-AO*(2*ALOG<Q)*T+CCDO*T2+T*00*2)/Q>

SUM=-AO*ODO/Q+A1*2*DOO*AIOG(Q)+2*A1*00*COO/Q

1 -6*A2*00*DOO*AIOG(C)-3*A2*D0 2*00 0/Q+3**2*DOO*Q

00 22 1*1*8

22 SUM=SUM+A(II*G(I)

PfeSUM

RETURN

ENTRY DS0T3

11*3

GO TO 1 30 All)s-A2*<Q2*T*3-3*Q*<C20*T3 + DD0*T2*6 + D0*T*6>+3M2*OD02*T3

1 2*0 0*O20*T 3+6*0 0*CO0*T 2+6*0 0*DO*T2+6*D02*T)*ALOG(Q)

2 «3*D0*OO02*T 3*2 3*D02*D20*T 3*9*00 2*DO0*T2*9*D 02 *OO0*T2

3 +6*D03*TI/Q)*T

A(2J=-A2*(Q2*T**.5*3.75/2.-3*Q*<D20*T**2.5+00 0*T**1.5*2.5

1 +DO0*T**1.5*2.5+O0*T**.5*3.75)+3*ALOG<G)*(2*DD02*T**2.5

2 +2*D0*D2 0*T**2.5 +5*0 0*CDO*T*» 1.5+5*00 0*0 0*T** 1.5+3. 75*

3 02*T**.5)+<6*00 02*0 0*T**2. 5+3*002 »020*T**2. 5+ 7.5*002*

k ODO*T**1.5+7.5*DCO*002*T**1.5+3.75*003*T**.5) /Q)*T

A(3)=-A2*(Q2-3*Q*(O20*T2+OO0*T**»+00*2) + 3*ALOG (Q )* (2*D002*T2

1 00*020* T2*2 + 8*0 0» DO 0*T +2*0 02) +{ 3*0 2»0 2 0*T2 +6*000 2*0 0*T2

2 +12*O02*DO0*T+2*C03)/G>*T

A («»)x-Al*(6*T*Q-ALOG(Q)*(6*T2*OD0+12*T*C0 + 6*T 2*00 + 2*T 3*020

1 >-<6*00 0*0 0*T2+6*T*D02*6*T2*ODO*DO+2*T3*D2 0*00+2*T3*D0 02)/Q)*T

A (51 =-Al*(Q*2-ALOG«Q)*(e*T*DD0**i+D0+2*T 2*0 2 >-< 2*00 2 1+»*T*0 0*00 0+2*T2*C2 0*0 0*2*T2*00 02I/Q)*T

A(6)=-AO*(6*ALOG<Q)*T+(C2 0*T3+3*000*T2+6*T*0 0+3*T2*00 0)/G)*T

A(7)a-A0*(3.75*ALOG(G)*T**.5+ < D20*T** 2. 5+00 0* T**l .5*2.5

1 +2.5*000*T**1.5+3.75*DO*T**.5)/Q)*T

A(8)=-AO*C2*ALOG(Q) + C02C*T2 + 2*CCO*T+2*0 + 2*'r*000)/Q)*T

SUM=-A 0*020 /Q+Al*D2 0*ALOG(Q)*2+2*Al*(OD02+O0*C2 0)/Q

1 -6*A2*ALOGIQ)*(OD02+O0*D20)-3*A2*(D2 0*D0 2+2*DD02*D0)/Q

2 +3*A2*D20*Q

SUM=SUM*T

00 32 1=1,8

32 SUM=SUM+ACI)*GCI»

P* SUM

RETURN

ENTRY DP0C3

GO TO 1

t*0 A(1)=A2*02*T3*3.

A12»=A2*02*3.*T**2.5

AI3)sA2*02*T2*3.

AU)*A1*0*T3*2.

A<5>=A1*0*T2*2.

A(6)=A0*T3

AI7)sA0*T**2.5

A(8)=A0*T2

hi SUM=0

45
DO kZ 1=1,8

t*Z SUM=SUM+A<I)»G(I)

P=SUM*DP0<Q.,T>

RETURN

ENTRY DPDT3

IT=5

GO TO 1 50 A(1)=A2*<3*02M-00 0)*T3*3»T2»D3)

A<2)=A2M3»D2»<-OD01»T»*2.5 + 2.5»T»*i.5*C3>

A(3lsA2»(3»D2*(-0CQ)»T242*T»03>

AU)=A1*(2*0*<-ODO>*T3 3»T2»02>

A<5)=A1M2*OM-DDO)»T2 »2»T*D2>

A<6» = A0M<-DD0»*T3+3»T2«O ) A(7)=AO»((-ODO)»T»*2.5*2.5*T*»1.5*0)

A(8)=A0M(-DO0)»T2*2»T*C>

51 SUM=0

00 52 1=1*8

52 SUM=SUM*A(I)^G<I>

P=SUM+DVPHE2IT)*DP0OT(Q,T>

RETURN

ENTRY PI03

C INTEGRATE P/Q*»2 WITH RESPECT TO Q

11=6

GOTO 1 60 QINTGR= •5»Q**2-3*Q*00+3*ALOGCO)*DO**2+00**3/Q

A(1)=A2*T*»3.0*QINTGR

A(2)xA2»T*»2.5*QINTGR

A< 3) =A2»T**2. 0*QINTGR

QINTGR=Q-2»ALOG(Q)*D0-D0**2/Q

A(i»)=Al»T*»3.0*QINTGR

A(5>=A1»T»»2.0*QINTGR

QINTGR=ALCG(Q) +00/0

AC6J=A0*T*»3.0»QINTGR

AC7I=A0*T»»2-5»QINTGR

A(8)sA0»T»»2,0»QINTGR

SUM=0

00 62 1*1,8

62 SUM=SUM+A(I)»G(I)

P=SUM-VPNHE2(T)/Q*AO*ALOG(Q)*A1*Q-A1*2»CO»ALOG(Q»

A*A0*O0/Q-D0»»2*Al/Q+O0*»3*A2/Q

l*A2»Q»»2/2-A2»3*O0»Q*A2*3*O0**2*ALOG(Q)

RETURN

EWO

FUNCTION P0(OI,T)

C CALCULATE THE PRESSURE FOR THE PO PART OF THE EQUATION OF

C STATE, UNITS ARE THE SAME AS IN FUNCTION HE2

DATA(A0=2.2*»1^56)

0ATA<A1=. 17571.82)

OATA<A2=.00<»70035>

X=DI-SATL2CT)

P0=A0»X*A1*X»»2*A2»X»»3

RETURN

END

FUNCTION DPODT<OI,T>

C CALCULATE OPOOT, INITS ARE THE SAME AS IN FUNGTION HE2

OATAf A0 = 2.2«tl«»56>

0ATA(A1=.1757*»82)

OATA(A2=.00<»70035)

X*DI-SATL2IT)

0X0TS-DSATL2ITI

OPOOT=AO*CXDT*A1»2»X»OXCT+A2*3*X**2*OXOT

RETURN

END

FUNCTION OP0<DI,T>

46
C CALCULATE OPOCO, LNITS ARE THE SAME AS IN THE FUNCTION HE2

DATA<A0 = 2.2«»ii»56>

OATA<Al=. 1757<»82)

OATA<A2*.00«»70035>

X«0I-SATL2CT)

DP0*A0+A1»2.»X*A2*3.*X**2

RETURN

END

FUNCTION SATL2CT)

C CALCULATE THE SATUARATEC LIQUID OENSITY IN HOLES/LITER FOR

C A TEMPERATURE IN KELVIN

X»2.172-T

0*-3. 310 07-.00 7*»2«i3913*X + .0 05916W3 7533*X*ALOG<XJ

D*EXP(DI*1000.

SATL2=0

If (T.LT.. 8) 0=36.27877

RETURN

END

FUNCTION 0SATL2CT)

C 1ST DERIVATIVE CF SATL2

X»2.172-T

OSATL2 = *.007<»2«i3i»913-. 05916^ 737533* <i.*ALOG<X>)

0SATL2=0SATL2*SATL2<T)

RETURN

END

FUNCTION D2SATL(T>

C 2N0 DERIVATIVE OF SATL2

X=2.172-T

0*SATL2(T)

02*.0 059l6<«737553*D/X+<.007<»2<t3*»913-.0 59i6<t7 37553*<ALOG(X)+i>)

1*DSATL2<T>

02SATL=D2

RETURN

ENO

FUNCTION CVVP(DtT)

C CALCULATES THE CONTRIBUTION OF THE VAPOR PRESSURE TERM TO CV

D0*SATL2(T»

CVVP=OVPHE2<T*.00 5)-OVPHE2(T-.0 05>

CVVP=CVVP*T*101.325*(l./D-l./D0)/.0i

RETURN

END

SUBROUTINE HE2EGCP»Q f T)

C CALCULATES THE F SUB S TERM IN THE EQUATION OF STATE

C UNITS ARE THE SAME AS IN THE FUNCTION HE2

DIMENSION G<23),A(23)

COMMON/DATA 2/G,R

1 Nsl

0=Q-36. 27877

T2=T*T

T3=T2*T

T*»*T3»T

T5=T«»*T

T6=T5*T

T7=T6*T

T8=T6*T2

T9=T8*T

T10*T8*T2

Tll=T10*T

02*0*0

03=02*0

0^=03*0

05 = 0*»*0

06=05*0

GO TO (10»20 v 30«<»0f50«60)*II

47
ENTRY PRESS2

ll'l

GO TO 1 10 A<l)=-0 2*T4/<*.

AC2»=-Q2*T5/5.

A(3)=-D2*T6/6.

AI<»)=-03*T3*2./3.

A(5)=-D3*T*»»2.M.

A(6)=-2.*03*T5/5.

A(7)s-2.*03*T6/6.

A(8»s-2.*D3*T7/7.

A(9>=-3.*0<»*T3/3.

A(10)s-3.*0«i*T<»/4.

A(ll»=-3.*0<»*T5/5.

A(12)=-3,»D*t»T6/6.

A(13>=-3.*0*»*T7/7.

A(JAI=-i».»D5*T3/3.

A(15J=-<».*05*T5/5.

A(16)=-«».*D5»T7/7.

A(17>=-<«.*05 # T9/9.

Afl8> = -«*.*D5»Tll/ll.

AC19)=-5.*06*T3/3.

A(20)=-5.*06»T5/5.

A(21)=-5.*06*T7/7.

A(22)=-5.*06»T9/9.

A(23)=-5.»D6»T11/11.

2 SUM = DO 3 1=1,23

3 SUM=SUM+A<I)*G(I>'Q*»2/G2

psSUM

RETURN

ENTRY 0S0N2

11 = 2 GO TO 1 20 AC1>=T3*D

A(2) = D*T<»

A(3)=D*T5

A(<») = D2*T2

A(5)=02»T3

A<6) = 02»T*»

A(7)=D2*T5

A(8)=D2*T6

A(9)=D3*T2

A(10)=O3*T3

A(li>=0 3*T<*

A(12)=03*T5

A(13I=D3»T6

A(l«t) = 0U*T2

A<15>=D***Ti+

A(16)=D<»»T6

A(17)=D<»»T8

A(18)=O^*710

A<19)=05*T2

AC20)=D5*T*»

A(21I=D5*T6

A(22)=05*T8

A(23)=05*T10

21 SUM=0

00 22 1=1,23

22 SUM=SUM*A(I)*G(I)

P=SUM

RETURN

ENTRY 0SDT2

11 = 3 48
GO TO 1 30 AC1)=3»T2*D

AC2I**»»T3»0

AC3)=5»T*»»0

A<*»»s2»D2*T

A(5)«3»T2*02

A<6l**f*T3»02

A(7I*5*T*»*D2

A(8)=6»T5»02

A<9>*2»T»03

A(10)s3»T2»03

A(11)=«»»T3*D3

A(12)s5*T'»»03

A(13)=6»T5»03

A(HO=2*T*0<»

A(15I*<»*T3»0«»

A<16» = 6M5»0*»

A(17)=8*T7*0*»

A(18)=10»T9*D<»

A«19I=2»T»D5

A«20l*i»»T3»05

A(21M6*T5*05

A(22>=8*T7*05

A(23)=10*T9*O5

31 SUHsO

00 32 1*1,23

32 SUMsSUM*A(I)»6CI)

P*SUM»T

RETURN

ENTRY 0P0C2

GO TO 1

i*0 ACl)*-2.»Q*Ti»/<f.

A«2)s-2.*Q»T5/5.

A<3>= -2.»Q*T6/6.

AU)=-(Q*»2*2»Q»0)*2»T3/3.

A<5l=-(Q^2*2*Q*D)»2*T<i/*».

A(6)s-(Q»»2*2»Q»0)»2»T5/5.

A(7)=-(Q**2+2»Q»0)*2*T6/6.

A(8)=-(Q*»2*2»Q»D)*2*T7/7.

A(9»=-(Q»»2*2»0*2»Q»0 2)*3»T3/3.

A<10)*-<Q*»2»2»0*2»Q»02)»3»T<»M.

A<ll)=-(Q*»2*2»0*2*Q*02)*3*T5/5.

AC12)=-<Q**2»2*D+2»Q*02)»3*T6/6.

A(13»=-(Q*»2*2*0*2*Q*D2)*3*T7/7.

A(ima-(Q*»2*0 2»3+2»Q»D3)*«*»T3/3.

A<15)=-(Q**2*D2*3*2*Q»0 3)*^*T5/5.

A<16>*-CQ»*2»D2»3*2»Q»03)*^T7/7.

A(17|s-(Q»»2»02*3*2»Q»03>**»*T9/9.

A(18»=-«Q»»2»02»3*2»Q*D3)*«t*Tli/ll.

A(l9)=-(Q»*2»0 3*«»*2*0*0*»)*5*T3/3.

A(20)=-(Q»»2»D3«n»+2»Q»0M*5*T5/5.

AC21ls-(Q*»2»03*i»»2*Q»0'«»»5»T7/7.

A<22)s-<Q*»2*D3»*»*2*Q»0«»>»5*T9/9.

A(23>*-<Q*»2*03*«»»2»Q*D«t>*5*Tli/ll.

*»1 SUM=0

00 kZ 1=1,23

HZ SUM«SUM*A(I)»GCI)

P*SUH

RETURN

ENTRY 0PDT2

IIs5

GO TO 1 50 A<l)s-02»T3

49
A(2>=-D2*T«*

A(3)=-D2*T5

A<*»)*-2*03*T2

A<5)=-2*D3*T3

AC6I=-2*03»T«»

AC7>=-2»D3*T5

A(8I=-2»03»T6

A<9)=-3*0<4*T2

A<10)=-3»D'»*T3

A(ll)=-3»0i»»Ti»

A<12>=-3*D«»»T5

A<13>=-3*0VT6

A<K»I=-<,»D5»T2

A(15)=-*»*D5»Ti»

AC17)=-<,*05»T8

A<18I=-*»*D5*T10

A(19)=-5»06»T2

A<20I=-5»D6»T4

A<21)=-5*D6»T6

A(22)=-5»D6*T8

A(23)=-5»06*T10

51 SUM=0

00 52 1=1,23

52 SUM=SUM*A(I)*G(I)*Q»»2/C2

P*SUM

RETURN

ENTRY PI02

11=6

60 TO 1 60 Mlls-0*T<t/*»

A(2)=-0»T5/5

At3)=-0»T6/6

A((»)=-02»T3/3

A(5)=-D2»T«»/i|

A(6)=-D2»T5/5

A(7)=-D2*T6/6

A(8)=-D2*T7/7

A(9>=-D3*T3/3

A(10l=-0 3*T*»/«»

A(ll)=-03*T5/5

A(12)=-03»T6/6

A(13)=-03*T7/7

ACl<»l=-0«»»T3/3

A«15l=-0<»*T5/5

A(16l=-0'»*T7/7

A(17)*-0*»*T9/9

AC18)=-0<,*T11/11

A<19»=-D5*T3/3

A(20»=-05*T5/5

A<21>=-D5*T7/7

AI22)=-D5»T9/9

A(23»=-05*T11/11

61 SUM=0

00 62 1=1,23

62 SUM=SUM+A(I)*G(I)

P*SUM

RETURN

ENO

FUNCTION SO(TT)

C CALCULATES THE PRESSURE VALUE FOR S,H, ANO CV

C UNITS ARE THE SAME AS IN FUNCTION HE2

DIMENSION S<<»<») ,C(i»<») ,H <<<) ,T<*»*»>

DATA(T=0,.l,.15,.2,.25,.3,.35,.<t,.**5,.5,.55,.6,.65,.7,.75,.8i

50
1.85,. 9,. 95, 1., 1.05, 1.1, i.15, 1.2,1. 25 , 1.3, 1.35,1. * ,i.*5, 1.5,

21.55, 1-6, 1-65, 1.7, 1.75, 1.8, 1.85,1. 9, 1.95, 2. ,2.05,2.1,2.15,2.172)

DATA(C=0,.20597E-*,.688 27E-*,.161*3E-3,.311**E-3,.5313*E-3,

1.13306E-3,. 001228*,. 017327,. 0237*9,. 032236,. 0**293,

2.1) 062 75*,. 091 73*,. 0138 2,. 21 058,. 031759,. 0*7598 ..07 00 05, . 105 0*,

3. 1*65,. 19 816,. 260 7*,. 33523, . *2128,. 526*2, .65752 ,. 80923,

*.97 015, 1.133*, 1.3169, 1.55 06, 1.8*9*, 2. 1868, 2.5*6*,

52.9278,3. 3332, 3.861* ,*. 5 113, 5. 19, 5. 90 09, 6. 6*2 8,

68.298*66,1*.)

OATA<S=0,.68889E-5,.23112E-*,.5*338E-*,.105 3*E-3,.18 0*lE-3,

1.28386E-3,.*198 3E-3,.5925*E-3,.8 07 09E-3,.107i*E-2,.0 01*00 5,

2. 018 226,. 23865,. 0316*9, . 0*2735 ,. 058517, . 08 08<»3,. 01123,

3.915562,. 021*57,. C29233,. 0392 75, .0516 89 ,. 066 910,. 08 5256, . 10675,

*.13301, .16*81,. 20 076,. 2*020,. 28*31,. 33558,. 3955*,. *650 2,

5.5*155,. 62676,. 721 89,. 826 13,. 95 09 8, 1.09 2 0,1. 2 39*, l.*137

679,1.559)

DATACH=0,.*17*E-5,.6261E-5,.1181E-*,.2337E-*,.**12E-*,.77 87E-*,

1.1290E-3,.2025E-3,.30*6E-3,.**36E-3,.63 31E-3,.8 97*E-3,.1278E-2,

2.18**E-2,.2705E-2,.*0 8E-2,.5966E-2,.8878E-2,.01311,.01916,

3. 0275*,. 03883, .053**,. 721,. 095*3 ,.12*,. 16 02,. 2 05*,. 258*, .31 86,

*• 3882, .*7 15, .5 721, .6917, .82 76, .98 32, 1.161, 1.36 3, 1.608, 1.891,

52.197,2.525,2.916)

S0=ATKINTITT,S,T,**,6,NES,. 01)**. 026

RETURN

ENTRY CO

S0*ATKINT(TT,C,T,**,6,NES,. 01)»*.00 26

RETURN

ENTRY HO

S9=ATKINT(TT,H,T,**,6,NES,. 01)**. 0026

RETURN

END

FUNCTION FIN0D2(PI,T)

C CALCULATES THE DENSITY IN MOLES/LITER FOR AN INPUT

C OF PRESSURE IN ATMOSPHERES ANO TEMPERATURE IN KELVIN.

P*PI

iriPI.LT..00 001)P=. 00 001

0*SATL2(T>

D=D*(*3.-D)*P/25.

10=1

DO 10 1=1,25

PP=PHE2<0,T)

IPCABSCPP-P).LE..00001»P)GO TO 11

OP=0P0HE2(D,T)

10 D=D-(PP-P)/OP

FINDD2=0

RETURN

11 FIN002=0

RETURN

END

FUNCTION PMELT2CT)

C INPUT IN KELVIN, OUTPUT IN ATMOSPHERES (FROM GRILLY, 1972)

IF(T.LT.1.*6*)G0 TO 1 PWELT2=31.168-17. 122*T*9. 29 2*T**2

RETURN

1 PNELT2=2*.996*.0799 01**22»T-.6729*2 7939*T**2*1.87 853 695*T»*3

1-2. 3265 09762*T**** 1.0611 363 53*T**5

RETURN

END

FUNCTION FIND TD(CO)

C CALCULATE THE TEMFERATURE IN K CN THE LAMBDA LINE FOR

C AN INPUT OF DENSITY IN GM/CC (KIERSTEND 1967)

D=DO

T=2.172

DO 10 1=1,20

51
OC*DENLAM(T)

IFCA8S(DD-DC).LE..00000i*DD)GO TO 11

10 TeT-<DENLAM<T)-DD)/(OELAMP(T) > GO TO 12

11 FINO TO=T

RETURN

12 FIND TO=0

RETURN

END

FUNCTION OENLAM(T)

C CALCULATE THE DENSITY IN GM/CC ON THE LANBOA LINE FOR A C TEMPERATURE IN K (KIERSTENO 1967)

D0 = 0.1'»8<tl388

Dl=-0. 150735

D2=-0. 3298225

D3=-0. 53031333

D«»=-0. 383035

05=-0. 00226388

06 = 36. 73*»8

X=T-2.172

DENLAM=00+01*X+02*X**2+03*X**3*01»*X***»*C5*EXP(06*X)

RETURN

ENO

FUNCTION DELAMPCT)

C CALCULATE THE DERIVATIVE OF DENLAM WITH RESPECT TO TEMPERATURE

O0 = 0.1<*8«»1388

Dl=-0. 150735

D2=-0. 3298225

O3=-0. 53031333

D<*=-0. 383035

05=-0. 00226388

06 = 36. 73t»8

X=T-2.172

DEL AMP=D1 + 2*D2*X + 3*03* X**2*tt*Oi»*X**3 + D5*C6*EXP(06*X>

RETURN

ENO

FUNCTION FIND TP(PP)

C CALCULATE THE TEMFERATURE ON THE LAMBOA LINE FOR A GIVEN

C PRESSURE IN ATM (KIERSTENO 19671

P=PP

T=2.172

00 10 1=1,20

BC=PRSLAMCT)

IF (ABS(PP-BC) .LE.. 000001*PP) GO TO 11

10 T=T-(PRSLAM(T)-PP)/(PRLAMF(T) ) GO TO 12

11 FIND TP=T

RETURN

12 FIND TP=0

RETURN

END

FUNCTION PRSLAM(T)

C CALCULATE THE PRESSURE IN ATMOSPHERES FOR AN INPUT OF

C TEMPERATURE IN KELVIN (KIERSTENO 1967)

80 = 0.<»29007^9

Bl=-95.0719

B2=-86.i»17

B3=-103.3^1

B«*=-77. 52175

B5=-0. 37827065

B6=^2.2507

X=T-2.172

PfSLAM = B0 + Bl*X + B2*X**2+e3*X**3+B<**X**<4 + B5*EXP(B6*X)

RETURN

52
ENO

FUNCTION PRLAMP (T)

C CALCULATE THE DERIVATIVE OF PRSLAM

B0*0.<»28007*»9

Bl»-95.0719

B2*-86.«»17

B3s-103.3«il

B<»*-77.52175

B5*-0. 37827065

B6*<i2.2507

X**T-2.172

PR.AMP=Bl + 2*B2*X+3*B3*X»*2*-it*8*»*X**3*B5*E6*EXP(B6*X)

RETURN

ENO

FUNCTION POO(OI)

C CALCULATE THE PRESSURE IN ATM AT T=0 FOR AN INPUT OF DENSITY

C IN MOLES/LITER

XtOI-36. 27877

P0 0*2.281877372»X*. 16 82 886 *X**2+.0 052778 8<»96 8*X»*3

RETURN

ENO

FUNCTION PO0H<DI)

C CALCULATE THE CONTRIBUTION OF POO TO ENTHALPY

0ATA<A=36.27877), (8=2.28187 73 72), (C=. 16 82 8 86 ) , (F=. 05277 88*»968)

PO0HsB*ALOG(OI)+B*A/OI*C*OI-2*C»A»ALOG(DI>-C*A**2/OI+F*OI**2/2.

1-3»F*A»DI+3*F*A»*2*AL0G(DI)+F*A»»3/DI

RETURN

ENO

FUNCTION OP00<DI>

C CALCULATE THE DERIVATIVE OF P0 WITH RESPECT TO DENSITY

X*DI-36. 27877

DP 00*2. 28187 7372*. 16820 886*2. *X + . 005 277 88 »»968*3.*X**2

RETURN

ENO

FUNCTION ATKINT(X,YMAT,XMAT,NELMTS,NMAX,NESSY,ACRCY)

DIMENSION YNAT(99«), XMAT (999) , A (21,20)

C FIRST TWO SUCCESSIVE VALUES OF THE XMATRIX THAT STRADDLE THE

C VALUE X MILL BE SOUGHT

JJ1=NELMTS-1

DC 20 1*1, JJ1

0IF1*X-XMAT<I>

DIF2*XMAT(I*1>-X

IFC0IF1)16,15,16

15 ATKINT=YMAT(I)

NESSY =NMAX

RETURN

16 IFCDIF2) 18,17.18

17 ATKINT=YMATCI*1)

NESSY =NMAX

RETURN

18 RATIOaDIFl/ 0IF2

IFIRATIO) 20,20,19

19 I«IO»I

GO TO 32

20 CONTINUE

NESSY*!

ATKINT=0.0

RETURN

32 CONTINUE

C NOTE THAT RATIO IS POSITIVE IF THE TWO POINTS STRADDLE X C REGARDLESS WHICH IS LARGER

JJJalMIO

JUPxIMIO

JONsIMIO

53
IFf JJJ+NMAX-NELMTS+1) 98,98, 10 2 98 00 201 J=1,NMAX

JJJ=IMID*J-1

A(1,J)=XMAT<JJJ)

201 A(2,J)=YMAT(JJJ)

GO TO 203

102 00 41 J=1,NMAX

JJ=J/2

J0E=J-2»JJ

C JOE IS IF J IS EVEN ANO 1 IF J IS 000

IF<J-1)33,40,33

33 IF(JON-l) 34,36,3**

34 IF(JUP-NELMTS)35,37,35

35 IFCJOE) 37,36,37

36 JUP=JUP+1

JJJ=JUP

GO TO 40

37 JDN=J0N-1

JJJsJON

GO TO 40

«»0 A(1,J)=XMATIJJJ)

A<2,J)=YMAT(JJJ)

41 CONTINUE

203 NNN=NMAX*1

DO 6 J=3,NNN

L=J-1

DO 5 K=L,NMAX

C J IS THE COLUMN NUMBER

C K IS THE ROM NUMBER

0A(J,K)=(A<J-i,K)-A<J-l,J-2) )* (X-AU,J-2))/(A<l,K)-A<l,J-2))

1 + AU-l,J-2)

IF<K-L)3,2,3

2 IF<ABS(<A<J,L»-A<J-1,L-1))/A(J,L))-ACRCY/10 0.0) 7,7,3

3 CONTINUE

5 CONTINUE

6 CONTINUE

NESSY=0

ATKINT=A(NNN,NMAX)

RETURN

7 NESSY=J-1

ATKINT=A(J,L)

RETURN

END

FUNCTION VPNHE2CTT)

C GIVES A VAPOR PRESSURE FOR HELIUM IN ATMOSPHERES GIVEN A C TEMPERATURE IN KELVIN. THE FUNCTION REPRODUCES THE 1968 HELIUM

C TEMPERATURE SCALE TO .0001 KELVIN

DIMENSION CC12) ,D<14)

OATA(C=-3. 93946352 87, 141. 2 7<+97598, -1640. 7 741565, 11974. 557102,

1-55283.30 9818,166219.565 04,-325 212.8284 0,398 843.22750,

2-277718.06992,83395.2 041831

DATA<D=-49.5105««0 356,651.92 36417,-37 07.5«i3085 6,1288 0.673491,

1 -300 48. 5«»5554, 49532. 26 7436, -59337. 55 8548,52311. 296025,

2-3 3950.233134,160 28.674 003,-5 354.103896 7,1199.0 301906,

3 -161. <t6362959, 9.8811553386)

TteTT P«0

IF(T-2. 1720)10, 10,1

1 TfeT-DELT(T)

DO 5 1=1,10

5 Pi=P*C<I)*T»M2-I)

VP =EXP(P)/.76E+6

VPNHE2=VP

RETURN

54
10 CONTINUE

T=T-OELT<T>

IF(T.LT..8)G0 TO 20

00 15 1 = 1, 1<» 15 P=P*D<I)*T**t2-I)

VP =EXP(P)/.76E+6

VPNHE2=VP

RETURN

20 VPNHE2=VPNL0H(TT)

RETURN

ENO

FUNCTION DELT(TT)

T=TT

DELT=.001*.002*T

RETURN

ENO

FUNCTION VPTHEZ(PP)

C SOLVES THE VAPOR PRESSURE EQUATION FOR TEMPERATURE GIVEN A PRESSURE

C THE FLUID IS HELIUM ANO THE UNITS ARE ATMOSPHERES AND KELVINS

P*PP

IF(P.LT..8A2105)GO TO 12

T*5.0

PCAL=VPNHE2(T)

GO TO 13

12 PCAL=.0A9737

IF«ABS(P-PCAL)-.0 00 01*PP) 11,11,1

1 T=2.1720

13 DO 10 1=1,26

DP=0VPHE2(T1

DEL = (PCAL-P)/OP

T=T-OEL

PCAL=VPNHE2(T)

IF(ABS(P-PCAL)-.0 00 01»P)il,ll,2

2 IF (A BS (DEL)-. 000000 1*T) 11,11,10

10 CONTINUE

PRINT 100,

T

11 VPTHE2=T

RETURN

100 FORMAK* TEMPERATURE ITTERATION FAILED AT T=*,E1W.7>

END

FUNCTION CVPHE2(TT)

C GIVES THE DERIVATIVE OF THE VAPOR PRESSURE FOR HELIUM GIVEN A C TEMPERATURE IN KELVIN9 PRESSURE IS IN ATMOSPHERES

OIMENSION C(12) ,0(1*1)

OAT A <C=-3.939<»6 352 87, 1«»1. 27^975 98,-16*»0. 7 7A15 65, 1197^.557 102,

1-5528 3.3 9818, 166 219. 565 0A, -325212. 828 k 0,398 8^3. 2 275 0,

2-277718. 0699 2, 8 3395. 2 0A 183)

DATA<0=-A9.5105A0 356,651.92 36A17,-37 07.5A3 085 6,1288 0.673A91,

1 -3 0A8.5i»555'»,t»9532.2e7*»36,-59337.55 85«t8,52 311.2960 25,

2-3 395 0.23 31 3<4, 160 28 .67A 003, -53 5A.1 038967 , 1199 . 030 1906,

3 - 161. A6362959, 9. 8811553386)

P=0

If (TT-2.1720) 10,10,1

1 T=TT-OELT(TT)

DO 5 1=1,10

5 P = P«-C(I)*T*M1-I)*(2-I)

0VPHE2=P*VPNHE2(TT)

RETURN

10 IF(TT.LT..8)G0 TO 20

T=TT-DELT(TT)

00 15 1=1, Ik

15 P=P+0(I)*T**(1-I)*(2-I)

0VPHE2=P*VPNHE2(TT)

RETURN

55
20 DVPHE2*DL0*HTT)

RETURN

END

FUNCTION VPNLOW(T)

R*. 08205616

H*H0!T)/«».0026

S*S0(T)M.0026

GL*H-T*S

0*. 00 01

IMT.LT..7ID*.0000001

IP<T.LT..5)0*.lE-8

IF(T.LT..35)0*1.E-12

IFCT.LT..3)D*1.E-13

IFCT.LT.. 25)0*1.E-16

IFCT.LT.. 2)0*1. E-30

IF<T.LT..15)D=.lE-3<»

00 10 1*1,20

GV*36.8 05«i68*5.1930<»3*(T-<».22-T*ALOGCT/<«.22))-T*9.37855

l+«5.31*i79»T»R»AL0G<R»T»0)

IF<ABS<GL-GV).LE..001MA8S(GL)))GO TO 11

P*D»R»T

0GV=25.31«i69»R*T/D

10 0*D-IGV-GL)/OGV

PRINT 100

100 FORMAT*' NO CONVERGENCE**

11 VPNLOW*0*R*T

RETURN

END

FUNCTION DLOH(T)

TUP=T*.05*T

TON*T-.05»T

PUP=VPNLOHCTUP)

PDN*VPNLOW<TDN)

OtOH* (PUP-PCN) / (TUP-TON)

RETURN

EHO

FUNCTION ENTR2(D,T)

DIMENSION A(20) *GC20)

C0MMON/DATA2/G,A»F,N,I0

10*1

CALL 0SQN2<S1,D,T)

0D*SATL2(T)

CALL 0SDN2<SO,DD.T)

S*S0m+<OVPHE2<T)»(l./0-l./DD)-MSl-SO> )* 101. 325

EHTR2=S

RETURN

END

FUNCTION CPHE2C0»T)

C CALCULATE THE SPECIFIC HEAT CAPACITY AT CONSTANT PRESSURE

C UNITS ARE THE SAME AS IN FUNCTION HE2

CPHE2*CVHE2!0,T)+101.325*(OPTHE2(O f T)**2/DPCHE2(C,T) /D**2)

RETURN

EWO

FUNCTION WHE2<0,T)

C CALCULATE THE VELOCITY OF SOUND FOR AN INPUT OF DENSITY IN

C MOLES/LITER

HHE2*C (CPHE2(0»T)/CVHE2(DtT))*DPDHE2(D,T)*25311.) **,5

RETURN

END

56
HBS-1UA (REV. 0-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-1029

4. TITLE AND SUBTITLE

THE THERMODYNAMIC PROPERTIES OF HELIUM II FROM K TO THE

LAMBDA TRANSITIONS

5. Publication Date December 1980

7. AUTHOR(S)

Robert D. McCarty

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

8. Performing Organ. Report No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP) 13. Type of Report & Period Covered

W. Sponsoring Aj«cy Coda — -

15. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

if. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) The equation of state of He-II is modeled by an equation of state explicit in

pressure as a function of density and temperature. The equation of state is divided

into three regions of temperature to .8 K; .8 to 1.2 K and 1.2 to the lambda

temperature for which similar functional forms are used with different adjustable

parameters. The combined functions are valid over the entire PT range of the

super fluid and may be used for all classical thermodynamic properties. Comparisons

between calculated and experimental data are presented. A computer program for

calculation of thermodynamic properties (PVT, isochoric heat capacity, isobaric heat

capacity, internal energy, enthalpy, entropy, and velocity of sound) is included.

17. KEY WORDS (aix to twelve entries; alphabetical order; capitalize only the drat letter of the firat key word unleaa a proper name;

separated by semicolons)

Computer program; equation of state; deviation plots; helium II; mathematical model;

8uperfluid; thermodynamic properties.

18. AVAILABILITY \X} Unlimited

For Official Distribution. Do Not Release to NTIS

UJ Order From Sup. of Doc, U.S. Government Printing Office, Washington, DC

20402

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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(THIS REPORT)

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