QUANTUM DEVIATIOXS FROM
THE
PRINCIPLE OF CORRESPONDING STATES
2567
Quantum Deviations from the Principle of Corresponding States
by R. M. Gibbons Air Products and Chemicalrr,Inc., Allentown, Pennsyhania 18105
(Received January $2, 1968)
An extension to the principle of corresponding states in terms of reduction by molecular constants is suggested which permits the calculation of consistent thermodynamic properties for simple molecules where quantum effects are important in the gas phase. The method correlates differences among the reduced properties of Ar, Ne, and 4He with the reduced de Broglie wavelength. The thermodynamic properties of Ne, nHz, and aHe are calculated from empirical equations for Ar, Ne, and 4He. The agreement with experimental data is good for the prediction of the properties of Ne and nHz but is less good for the properties of aHe. The method fails for the liquid and critical regions of the phase diagram.
to obtain the reduced properties are all listed in Table I. The two basic parameters for each substance (r and E The properties of light molecules deviate considerably were obtained from second virial coefficient data5 asfrom the classical principle of corresponding states a t suming a Lennard-Jones potential. The three reduclow temperatures and high densities. de Boer, et uL,l tion parameters €/a3, N u 3 , and elk are not mutually first suggested that these deviations were due to quanindependent, and the reduced PVT surface for a given tum effects and could be correlated with the reduced molecule does not appear to be particularly sensitive de Broglie wavelength; using this approach they were to the choice of c and u, so the errors associated with successful in the prediction of the critical properties using inexact constants should not be great. and vapor pressure properties of 3He. A similar apTo obtain the dependence of P* on A*, plots of P* proach has been used recently by Rodgers and Brickvs. f(A*) at several T* and V* values were made to find wedde2 to predict the vapor pressure of tritium. Ata form so that P* varies linearly with f (A"). The plot tention has also been given to the influence of quantum of P* vs. A*z in Figure 1 shows P* to vary linearly with effects on phase equilibria3 and critical-point b e h a ~ i o r . ~ A*2, so that approximately the equation It is proposed here to examine the importance of quan-
Introduction
tum effects in the gas phase at higher temperatures than have been considered hitherto, with particular attention to the prediction of the PVT surface of 3He. The formulation of the law of corresponding states by de Boer, et al., in terms of reduction of properties by molecular units, derived from the collision diameter u and the intermolecular potential constant E, is the most suitable for examining the influence of quantum effects. de Boer showed if molecules had the same form of pairwise additive intermolecular potential, that in the reduced equation of state the reduced pressure was a function of the reduced temperature, volume, and de Broglie wavelength, h / d z . The dependence on A*, the reduced de Broglie wavelength, was not given explicitly and the effect of different statistics was ignored. I n fact, de Boer, et al., obtained the dependence property on A* graphically, and similar methods were also used by Rodgers and Brickwedde.2 For example, to obtain the vapor pressure curve of 3He, de Boer, et uL,l plotted P* vs. A* at constant T* for Ar, Ne, Hz, and 4He; V* was not held constant. The formulation used here is slightly different in that T* and V* are held constant in determining the dependence of P* on A* and the dependence on A* is represented analytically instead of graphically. The method by which this was done is outlined in the following section. The Dependence of P* o n A*. The parameters used
P*
=
f,(V*, T*)
+ A*'fz(h*, T*)
(1)
represents the dependence of P* on A* for fixed T* and V*. f,(V*, T*) is a universal classical function and the term A*zf2(V*, T*) represents the total quantum effects. Rigorously, quantum effects can be represented by a series expansion ZA*nfR(V*, T*) wherever such series converge; here this series is approximated by A*2fz(V*,7'"). To obtain f, and f z at a given T* and V * , the actual pressures of Ar or S e and 4He are substituted into eq 1, which can than be solved for f, and fz. I t is possible that the differences in reduced pressures are due to other effects, such as nonadditivity of the intermolecular interactions or errors in the pressures of Ar and "e; for convenience, any differences in reduced pressures will be referred to as quantum effects, though it must be borne in mind that other causes can contribute too. Naturally, when quantum effects become
(1) J. D. de Boer, Physica, 14, 138 (1948); J. D. de Boer and R . J. Lunbeck, ibid., 14, 510, 520 (1948). (2) J. D. Rodgers and F. G. Brickwedde, J . Chem. Phys., 42, 2822 (1965). (3) R. H. Sherman and L. Hammel, Phys. Rev. Lett., 15, 9 (1965). (4) M . E. Fisher, ibid., 16, 11 (1966). (5) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids," John Wiley and Sons, Ino., New York, N. Y., 1964, pp 1110, 1212. Volume 73, n'umber 7 July 1968
2568
R. M. GIBBONS
I(
.7
P"
.2 T * = 1.350 V": 6.76
T* 5973 V*:6.76
[J
-
rc
Ne
I
I
2.5
H2
I 50
I
I
He4
7.5
A* Figure 1. Reduced pressure
v8.
reduced de Broglie wavelength a t fixed reduced temperature and pressure.
gross, this simple representation will no longer be adequate to describe the dependence of P* on A* but the method should be of much greater accuracy than the classical form of the law of corresponding states by itself. The enthalpy and entropy can be derived from the equation of state by standard thermodynamic arguments.6 This implies the same dependence on A*2 for H and S as has been shown by the pressure. It remains then to generate consistent P V T , SVT, and HVT surfaces of the substances used to determine The Journal of Physical Chemistry
s He3
f, and fz. I n this way, a consistent set of values for the thermodynamic properties can be obtained for other similar substances. The modified Strobridge equations for 4He,7Ar,* and Ne9 are suitable. These equations (6) J. G. Hust and A. L. Gosman, Advan. Cryog. Eng., 9, 227 (1964). (7) D. B. Mann, National Bureau of Standards Technical Note 154,
U. S. Government Printing Office, Washington, D. C., 1962. (8) A. L. Gosman, J. G. Hust, and R. D. McCarty, National Bureau of Standards Report 8293, U. 8. Government Printing Office, Washington, D. C., 1964.
QUANTUM DEVIATIONS FROM
THE
+20
h
+i 4.12
-4
2569
PRINCIPLE OF CORRESPONDING STATES
0
100'K
A
70° K TO0 K (NBS Carrelotior
w A 35OK
0 45OK 50°K
05S0 K 0
Q 2 0 0 ° K
A
I SO
'
0.0
I
I
100
150
p Alm
Figure 2. (a) The percentage error in the computed pressure us. the experimental pressure a t 70 and 100"; (b) the difference between the computed pressure and the pressure obtained from the McCarty-Stewart equation a t 200, 55, 50, and 45°K.
Table I : Values of Molecular and Reference-State Constants a. € i k , 'K
Ar Ne 4He 8He Hz
119.8 35.6 10.22 10.22 37.0
u,
Molecular Constants
B
3.405 2.749 2.556 2.556 15.11961 b. I
Ne 'He
Hz *He
Nus, oo/mol
A*1
413.54 233.40 83.3795 83.3795 200.807
23.778 12.5127 10.05796 10.05796 15.1196
0.034969 0.349281 6.9696 9.025 2.9929
Reference-State Constants To OK
Ar
dua, atm
87.28 27.09 4.2144 20.39 3 1905 I
so,
HQ,
J/g mol dag
J/g mol
129.826 68.6195 37.0714 80.1069 34.0131
9530.76 1863 85 146.3425 1486.50 66.3771
have restricted ranges but are suitable in most cases. The Ar equation has the smallest temperature range being limited to T* < 2.5, but both the Ne and 4He equations have ranges extending to T* > 9.0. All these equations are inaccurate in the critical region. The equations for Ar7 and 4He8have been used in conjunction with eq 1 and the reduction parameters of Table I to calculate thermodynamically consistent properties of Ne and Hz. Similarly, calculations using the equations for Ne9 and 4Heshave been made to obtain thermodynamically consistent properties for 3He. These calculations will now be discussed.
I
(9) R. D. McCarty and R. B. Stewart, "Advances in Thermophysical Properties at Extreme Temperature and Pressure," American Society of Mechanical Engineers, MoGraw-Hill Book Co., Inc., New York, N. Y., 1966, p 84.
Volume 72?Number 7 July 1968
2 570
T h e Pyoperties of Neon. There is only a very limited amount of PVT data available for Ne with which comparisons can be made. The data of RIichels, et ~ 1 . at room temperature and above are the main source, together with the recent data of Sonntag and Sullivan" in the region 70-120°K. In addition, there is the equation of McCarty and Stewartl9which is more comprehensive but is not based upon experimental data at temperatures below 55"K, since none were available when the correlation was developed. Neon is normally thought of as a gas which obeys the classical version of corresponding states closely. However, Figure 1 shows that for V* = 2.411 and T* = 1.545, quantum effectscan be as large as 10%. At low densities the influence of quantum behavior rapidly becomes very small especially at high temperatures. Figure 2a shows an error plot for the values of P predicted by our correlation compared with the experimental results of Sonntag, et al.," at 70 and 100°K. The average error in the predicted values of our correlation was lo0 atm. Figure 2b shows plots of differences between the values predicted b y the RlcCarty-Stewart equation and the present correlation. At 200'K the average difference is
