I
,
EDWARD A. MASON and JOSEPH T. VANDERSLICE Institute of Molecular Physics, University of Maryland, College Park, Md.
Calculation of Virial and Joule-Thomson Coefficients at Extremely High Temperatures The methods illustrated can be used to calculate the equation of state and thermodynamic properties of gases at temperatures so high that direct measurements are almost impossible. The methods depend on the existence of precise intermolecular force information valid at close distances of approach of two molecules. Most of this information has been obtained from scattering measurements performed at room temperature, which are the most reliable source of information. Molecular quantum mechanics probably will soon be sufficiently developed to supply useful information on short-range intermolecular forces
Q
4
*
UANTITATIVE information on the equation of state and the thermodynamic properties of gases a t elevated temperatures is necessary in problems dealing with such phenomena as combustion, detonation, and high speed gas dynamics. Direct experimental measurements a t such temperatures cannot usually be made, and extrapolation from measurements a t low temperatures may yield erroneous results. However, reliable results can be calculated if precise intermolecular force information is available. Unfortunately, intermolecular forces obtained from analysis of low temperature measurements are generally not satisfactory for the calculation of high temperature properties, as different regions of the interaction curve are emphasized a t different temperatures (6). Accurate expressions for the repulsion part of the interaction curve are required because the high temperature properties are relatively insensitive to the attraction part (5,6). Such information can be obtained from accurate quantum-mechanical calculations, from measurements on the scattering of high velocity molecular beams in gases (4), or from high temperature measurements of some nonequilibrium property such as the viscosity. Repulsion energies are usually represented mathematically by an inverse
power of the separation distance, or by an exponential function of the separation. Although an inverse power is a satisfactory representation over a limited range of separations, theory indicates and experiment seems to confirm that the interaction is more satisfactorily represented by an exponential, especially at small separations (5). This article presents approximations for the second and third virial coefficients for cases where the interaction energy is represented by an exponential. Such expressions should be useful when the depth of the potential energy minimum, E , is small compared to kT, where k is Boltzmann's constant. The parameters in the exponential can be obtained from scattering measurements. By means of these parameters, the second and third virial coefficients have been calculated a t high temperatures for helium, neon, and argon, and compared with available experimental data. Aoy thermodynamic properties which depend on equation of state data can then be obtained from standard thermodynamic relations. As an illustration, the zero-pressure JouleThomson coefficient and its first pressure derivative have been calculated for helium and compared with experimentaI data. Second Virial Coefficient T o obtain the second virial coefficient,
B ( T ) , the following expression must be integrated : B(T) =
Unfortunately, even for extremely high temperatures, x is a t least of the order of magnitude of 100, so that the above series, although convergent, is useless for actual computation. However, an asymptotic series for F(x) can be obtained. The expression for the wellknown exponential integral function is (77)
L"
- dti =
El(x) =
(-1jn+1 + C ___ n n! xn
-1nyx
n=l
from which we obtain by comparison the relation,
where In y is Euler's constant, equal to 0.577216.. From Equation 4 we obtain by integration the relation
..
Equation 5 is an identity, as can be verified by substitution of the series for E l ( x ) . However, from Equation 5 an asymptotic expression can be obtained for F(x), which is very accurate when x is large. Substituting an asymptotic expansion (77) for El(%), El(x)= X
2 T ~ 0 J m (1 - exp[-
' P ( r ) / k r l ) r ~ d r (1)
where N Ois Avogadro's number, and q ( r ) is the potential energy of interaction of two molecules a t a separation distance of r . For the exponential potential,
a(r) = 'PO .xp(--r/p) (2) where qo and p are constants, the integrand of Equation 1 can be ekpanded in a power series and the expression integrated term by term. The restilt is
and carrying out repeated integration by parts, we obtain the asymptotic expansion for F(x) : F(x) =
1 e-" e-z (In yxj3 + - - 6 - + 6 x3 X4
-
,
,
,
The results can finally be written as:
+
B( T ) = bo [(In 6 -e --z
-
x3
3 6 -e+ -x X4
...I
(6)
where I
B ( T ) * 4irNop3F(x)
where y =
x = i,oo/kT
1.781072 . . .
This expression gives B( T ) to a computational accuracy of better than 0.1% even for very high temperatures where x is as
' ~
VOL. 5 0 , NO. 7
JULY 1 9 5 8
1033
Figure 1. leff. Helium. za
Cenfer. Neon.
20\
There i s essential agreement f o r helium and neon, but argon shows a l a r g e temperature g a p
Right. Argon. 1
c H o i b o r n and O t t o a Michels ond W o u t e r i S c h n e l d e i ond Coworkers Colculoted
1
Second virial coefficients a t high temperatures
,
i
I
2
°
?
I
1
C
I
22t
-
'O IO L
.
3 Holborn ond O t t o
Nicholson and - Calculated
Schnelder _n
--1
m 2
201 25
small as 3. At low temperatures, \vhere x is large, the "convergence" is even better, but the expression finally becomes inaccurate because of the neglect of molecular attraction. As an example. let us consider the second virial coefficient of helium a t verv high temperatures. T h e interaction energy between two helium atoms, V(T) =
(6.18 X 10-lo) exp[-4.55 r ( A , ) ] ergs, 1.3 A. < r < 2.3 A . (7)
has been obtained by -4mdur and Harkness ( 7 ) from their results on the scattering of fast helium atoms in room temperature helium gas. For this potential. the first term of Equation 6 would be adequate u p to temperatures as high as 105 O K., except for the fact that a t such temperatures excitation and ionization would vitiate the calculations. T h e results are shown graphically in Figure 1 as B'11 us. log T, xvhich should be nearly a straight line for an exponential repulsion potential. Included are some experimentally determined values of the second virial coefficient of helium (8, 9, 7 7 : 72, 76, 20, 22), nhere B is expressed in units of cubic centimeters per mole. At low temperatures, the experimental values curve downward, showing the effect of molecular attraction, but the agreement a t high temperatures is within about 4%. Such agreement seems good in view of the experimental errors involved and the fact that the calculated curve is based primarily on scattering experiments and not on compressibility measurements. An important question involves the size of the temperature range over which the foregoing calculations are valid. The second virial coefficient is negative at low temperatures because of attraction forces, becomes positive as the temperature is raised, passes through a maximum, and falls off a t very high temperatures. T h e repulsion forces are predominant at temperatures somewhat beyond the maximum, and a t still higher temperatures are completely dominant (6). T h e d a x i m u m occurs a t about T = 20 e / k . We expect that our expression for the second virial coefficient is about 5% too high, because of neglect of molecular attraction, when the temperature is as low as three times thatat the maximum, or 60 e / k . For the upper temperature
1034
30
3
c-
- Coiculoted
L 35 l o g T('K1
Holborn a n d O t t o Wholley, L u p i e n ond
Schne der
40
I I 6 -~
limit it seems reasonable to take this temperatwe as the one at which k T is equal to the maximum interaction energy knoivn from the scattering measurements. For the cases considered here the upper limit is about 15,000 OK. I n all figures the calculated curves are not extended beyond the regions in which they are expected to he reasonably valid. I n the case of helium, where the attraction forces are unusually weak, the experimental results overlapped the calculated curve. Let us now consider neon, for LLhich the attraction forces are stronger than for helium, but are still Iveak. The repulsion can be represented by the expression, exp[ -5.10 r ( A ) ] ( 7 ) = (1.39 X ergs. 1.8 A < r < 2.5 A . (8) Equation 8 was obtained by fitting the potential function from the scattering measurements ( 3 ) with an exponential, and requiring further that the exponential join on to the potential function calculated by Mason and Rice (74) from crystal data and gaseous viscosit)and compressibility measurements. The resulting high temperature second virial coefficient is shown in Figure 1 (center) Lvith the experimental data ( 7 7 , 72, 78). There is a gap in temperature between the experimental values and the calculated curve, but it is not too large to conceal the fact that there is essential agreement. \Ve finally consider a case where the molecular attraction is strong: argon. By a curve-fitting procedure as was used for neon, we obtain the potential energy function ppir) = (5.17 X exp[-4.47 r ( A . ) ] ergs, 2.2 A. < r < 3.4 A. (9) based on scattering measurements (2) and the calculations of Mason and Rice (74). The second virial coefficient calculated from Equation 9 is shown in Figure 1 (right) with the experimental data ( 9 , 70, 7 7 , 75, 27). There is a very large temperature gap between the experimental values and the calculated curve. This gap, as well as the smaller one for neon, is not caused by ignorance of the molecular interactions which predominate in this temperature range, but rather by the method of calculation,
INDUSTRIAL A N D ENGINEERING CHEMISTRY
25
1
30
35
40
loo T [ O K )
Tvhich ignores the molecular attractions. The gaps can be covered, but different calculation methods are required.
Third Virial Coefficient The fundamental expression for the third virial coefficient is much more complicated (73) than Equation 1, and does not yield to the method by which Equation 6 for the second virial coefficient was derived. However, a reasonable estimate can be obtained by analogy xvith the inverse power repulsion potential, q ( r ) = K,'P. For such a potential it has been found (73) that the ratio of the third virial coefficient, C( 7') to the square of the second virial coefficient depends onl>- weakly on the steepness of the repulsion, lvhich is measured by the parameter n. The ratio C,:Bz varies only from 0.372 to 0.625 as n varies from 6 to infinity. As the steepness of an exponential is intermediate between these values, it is reasonable to expect that the ratio CiB2 for an exponential lies betTveen 0.372 and 0.623. Taking the ratio to be 0.5, we obtain the expression ~
C( T ) m 0.5 bo2 (In
yx)6
(10)
This expression is actually more useful than it might seem at first, because experimental measurements of third virial coefficients are themselves rather uncertain. From Equation 10 third virial coefficients have been calculated for helium, neon, and argon, which are compared in Figure 2 Tvith available experimental values ( 7 7 , 72, 75, 76, 78, 27). The range of validity of the calculations is estimated in the same way as for the second virial coefficients-i.e., from about 15,000° K. down to three times the temperature of the maximum. in the C us. T curve, which occurs a t about ?' = 1.5 e l k . I n all cases the agreement \vith experiment is excellent, and for helium and neon Equation 10 appears to be valid even below room temperature. This is merely a n illustration of the fact that the third virial coefficient is much less influenced by molecular attraction forces than is the second virial coefficient.
Figure 2.
Third virial coefficient of helium (left), neon (center), and argon (right) In all cases agreement with experiment is excellent v -
f 2
3 5i
1
__--
oe
i
o Holborn and O t t o h Michals and Woutera
9 - 30-
.
&
0
- calculated
C
0
% ' %4=
-~
Holborn a n d O f t a M LhaIs o n d Coworkers
Wholley. L u p e n and S c h n e d e r
- CaICulot$d
:\
I
3
i I
I
0 201.
I
u 25-
101 IO
20
-
i_______
4.0
30 log T ( O K 1
Calculated
log T
10
20
30 log
(Cgo)-l[T(dB/dT)
=
-
(B/cPO)[l
- B]
-k 3(bo/B)1'3
(11)
'
(I2)
w
Values of for helium calculated from Equation 12 are compared in Figure 3 with the experimental results (79). The agreement is not SO good as for the virial coefficients, but is reasonable considering the fact that p0 depends on the derivative of B, which puts a considerable strain on a potential energy function derived from scattering measurements. The first pressure derivative of the Joule-Thomson coefficient can be written
(7) as (dp/dfl)" = ( RTC,O)
d2B
-1
2BCg0) - 2C
50
40
-I
, is the zero-pressure molar where CO heat capacity a t constant pressure. Substitution of Equation 6 into Equation l l leads to the following expression for po for an exponential repulsion:
@'
_--_A
--
The deviations of the thermodynamic properties of gases from the ideal gas values can be written in terms of the virial coefficients (7), so that the preceding calculations can easily be extended to include many thermodynamic functions. This extension may be illustrated by calculating the zero-pressure JouleThomson coefficient and also the zeropressure limit of its first derivative with respect to pressure. T h e zero-pressure Joule-Thomson coefficient, po, is related to the second virial coefficient as follows (7) : =
__-
('K)
T (OK1
Joule-Thomson Coefficient
*
40
30
20
+ T dC fi]
means that ( d p / d f ) o is less than about deg./atm.2, in agreement with the
(dp/@)O = ( p 0 B / R T )[1 - 3(R/CP")X (bo/B)'I3 - 6(R/Cp0)(b~/B)2/3 . . . ] (14)
+
For helium Equation 14 predicts a value of (dp/dp)O a t 600" K. of only -1.5 X deg./atm.2 Roebuck and Osterberg (79) were unable to observe any variation of p with pressure, which
I -
f
E 4.
$. lntermediafe Regions of Temperature I n the case of the second virial coeficients of neon and argon, there is a region of temperature where no values are given. For neon this range is from looo" to 2ooo" K., and for argon it runs from 1000' to 7000' K. I n these regions the repulsion energy is the dominant factor in determining the second virial coefficient, but the attraction energy-is not small enough to be methods of calculating B ( T ) in these regions can be developed. One method might be to consider the attraction forces as perturbations and carry out a perturbation procedure such as used by Zwanzig (23) for the equation of state in a somewhat different form, An even simpler method is to Some standard form of Potential energy function, such as the exp-six potential ( I d ) ,such that it reproduces the Potential curves from the scattering measurements and also reproduces approximately the position and depth of the Potential energy minimum as determined from analysis of low temperature Properties. Such Potentials can be used to cover the intermediate temperature range with reasonable accuracy. This method has been extensively used by Amdur and Mason in a series of calculations which have not yet been published.
(13)
where R is the gas constant per mole. Substitution of Equations 6, 10, and 12 into Equation 13 yields the following approximate expression :
70
literature Cited (1) Amdur, I., Harkness, A. L., J . Chem. Phys. 22, 664 (1954). (2) Amdur, I., Mason, E. A., Zbzd., 22, 670 (1954). (3) Zbid.,23,415 (1955). (4) Ibid., 25, 624, 632 (1956); and vre-
vious papers.
(5) Cottrell, T. L., Discussions Furadav SOC. 22, 10 (1956). (6) Hirschfelder, J. O., Curtiss, C. F.,
Bird, R. B., "Molecular Theory of Gases and Liquids," p. 164, Wiley, New York, 1954. * (7) Zbid., pp. 230-2.
50-
J 11
'
' -
1
*
Roebuck and Osferberg
- Calculated 10
eo
40
30 log 5
~i~~~~ 3.
(OK)
joule-~horn-
son coefficient of helium at high ternperatures Agreement is not so good as far virial coefficients
(8) Holborn, L., Otto, J., 2. Phystk. 10, 367 (1922). (9) Zbzd., 23,77 (1924). (lo) 30, 320 (11) Zbzd., 33, 1 (1925). (12) Zbzd., 38,359 (1926). (13) Kihara, T., Reus. Modevn Phys. 25, 831 (1953).
J.
(I4) Mason, E. 843 A,, RLcet w. Phys. 22, (1954). (15) Mlchels, A., Wljker, H., Wijker, H., P h y m a 15, 627 (1949). (16) Michels, A., Wouters, H., Zbzd., 8, 923 (1941). (17) Natl. Bur. Standards, "Tables of Functions and of Zeros of Func-
tions," Applied Mathematics Series 37, 57-8 (1954). (18) Nicholson, G. A., Schneider, W. G., Can. J . Chem. 33, 589 (1955). (19) Roebuck, J. R., Osterberg, H., Phys. Rev. 43, 60 (1933). (20) Schneider, W. G., Duffie, J. A. H., J . Chem. Phys. 17, 751 (1949). (22) Whalley, E., Lupien, Y., Schneider, W. G., Can. J . Chem. 31, 722 (19 53). (22) Yntema, J. L., Schneider, W. G., J. Chem. Phys. 18, 641 (1950). (23) Zwanzig, R. W., Zbtd., 22, 1420 (1954).
RECEIVED for review December 3, 1957 ACCEPTEDJanuary 15, 1958 Division of Industrial and Engineering Chemistry, ACS, Christmas Symposium, Cleveland, Ohio, January 1958. Research supported in part by the U. S. Air Force through the Air Force Office of Scientific Research, Air Research and Development Command, under Contract No. AF 18(600)-1562, and in part by the National Advisory Committee for Aeronautics. Reproduction in whole or in part is permitted for any purpose of the United States Government. VOL. 5 0 , NO. 7
JULY 1958
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