Intermolecular Forces and the Properties of Gases. - The Journal of

INTERMOLECULAR FORCES AND THE PROPERTIES OF GASES‘ J. 0.HIRSCHFELDER AND W. E. ROSEVEARE Department of Chemistry, University of Wisconsin, iMadison, Wisconsin Recsived October 6, 1958

The equation of state for gases is intimately connected with the law of force between the individual molecules. We are interested in making this relationship explicit. From a set of accurate measurements of PVT or Joule-Thomson coefficients, we should obtain the energy of interaction, E(r), of a pair of molecules separated by a distance r. Or, conversely, if E(r) is calculated from the quantum-mechanical perturbation equations, we should derive the corresponding equation of state. Fowler (6), Lennard-Jones (14),and others have expressed the second virial coefficient in terms of E @ ) , so that the experimental data a t low pressures determine the interaction energy. However, the experimental difficulties in obtaining the imperfections of gases accurately a t low pressures restrict the use of this method of analysis to a comparatively few simple substances which have been studied with extraordinary care. For this reason, it is desirable to perfect an equation of state which is fairly accurate for high pressures and which still can be interpreted in terms of the intermolecular forces. I n this paper we consider first what information can be obtained from experimental data at low pressures and then that from data a t high pressures. I. INTERPRETATION O F LOW-PRESSURE DATA

The equation of state for gases a t reasonably low pressures can always be written in the form (6, 27):

where B’(T), C‘(T), D’(T) are functions of temperature but not of pressure. These are usually referred to as the second, third, fourth . , . . vinal coefficients, respectively. There is some ambiguity in the literature as to the definition of the virial coefficients, but in the course of this paper we shall always define them with reference to equation 1. The interpretation

* Presented a t the Symposium on Intermolecular Action, held a t Brown University, Providence, Rhode Island, December 27-29, 1938, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 15

16

J. 0. HIRSCHFELDER AND W. E. ROSEVEXRE

of these virials in terms of the law of force between a pair of molecules was carried out by Ursell (27), Fowler (6), and others so that we can write:

B’(T) = 2nN

r2 (1

- e = = ) dr

(2)

Here E(r) is the energy of interaction of a pair of molecules separated by a distance r , k is the Boltamann constant, and N is Avogadro’s number. The derivation of this formula requires that the collisions between the molecules be treated according to classical mechanics. We wish to consider equation 2 as an integral equation for the determination of E(r). If we knew B’(T) over a large temperature range, we should be able to determine E(r) with some precision. Lennard-Jones (14) made a real contribution to the study of gases when he showed how to perform the integration in equation 2 on the assumption that

=

- c/r6 + I/ra

(3)

This form for the potential is roughly what we would expect on the basis of quantum mechanics. Here ro is the separation for which the potential is zero, and Emis the energy required to separate the molecules from their equilibrium separation. Substituting the Lennard-Jones potential into equation 2 we obtain

B‘(T) = b 0 F . G )

(4)

where bo is the van der Waals b for molecules with a diameter ro (Le., bo = &rNri) and F, is a function of E,/kT, which Lennard-Jones has tabulated for different values of s. It has not been possible to determine s accurately from the experimental data, since the curves corresponding to different values of s have nearly the same shape. For many substances the indications seem to point to a value of s in the neighborhood of 12, and we shall use this in our analysis. Figure 1 shows B’(T) or FU plotted as a function of E,/kT. This form for the second vinal coefficient is valid for all substances to which this Lennard-Jones potential applies. It is interesting to examine the van der Waals constants which would be obtained for molecules obeying this inverse twelfth-power repulsion and inverse sixth-power attraction law. Expanding van der Waals’ equation in powers of l / V it is easy to show that B’(T) should equal b - a/RT. From this relationship a and b might be determined from the experimental second virial coefficient. Plotting B’(T) against 1/T (as in figure 1)’ a straight line would be drawn through the experimental points and the

INTERMOLECULAR FORCES AND PROPERTIES OF GASES

17

intercept for 1/T = 0 would be the value of b. From the curvature of

F I it~ follows that if the experimental data are given for high temperatures, the value of b obtained will be considerably smaller than bo, while lowtemperature observations will yield a much larger value for b. Table 1

Em/KT

FIG.1. Second virial coefficient as a function of temperature

shows the temperature variations of a and b obtained in this manner. Since these experimentally determined a and b change by almost a factor of 2 in going from the critical temperature to the low-pressure Boyle point (Le., the temperature at which B’(T) = 0), we should not think of them as constants.

18

J. 0. HIRSCHFELDER AND W. E. ROSEVEARE

The Beattie-Bridgeman equation (1, 2, 3, 4) is perhaps the best empirical equation for representing the p , V , T for gases up to pressures of the order of 100 atm. Here:

pV2 = RT(1 - c / ( V T 3 ) ) ( V

+ Bo - bBo/V) - Ao(1 - a / V )

(5)

And, expanding the right-hand side in powers of V , it follows that

B'(T) = Bo - Ao/RT - c / T ~

(6) This expression for the second vinal coefficient agrees with the second vinal coefficient obtained from the Lennard-Jones potential over a wide temperature range, if c has the proper value with respect to BOand A o . Let us suppose for the time being that the molecules which we are considering actually obey the Lennard-Jones potential with the inverse twelfth-power repulsive and inverse sixth-power attractive energy and that our gas has been studied in a temperature range lying roughly between the critical TABLE 1 Temperature variations of a and b

0.893 0.709 0.563

0.448 0.355 0.282 0.224 0.178

1.616 1.272 1.072 0.943 0.861 0.801 0.754 0.716

i::

T,,critical temperature

3.397 3.141 2.934

i2.363 ::

T,, Boyle point (low pressure)

temperature and the low-pressure Boyle point. Then we expect equations 4 and 6 to agree a t the three temperatures corresponding to E,/kT = 0.7906, 0.4989, and 0.2924, from which it follows:

E,,, = 0.04127 X 10-16Ao/(RBo) ergs

(74

bo = 1.249Bo

(7b)

Xow, there is a third relationship between c and the constants A Oand BO which is fulfilled only i f the molecules actually satisfy the Lennard-Jones potential: c = 0.0236 A30/(R3B!)

(7c)

Equation 7c can therefore be used as a test for the validity of this potential energy function. Table 2 compares the values of E, and of bo calculated from equations 7a and 7b with the values given by Lennard-Jones (15). Here we use the values for the constants given by Beattie and Bridgeman

19


(4). In the last two columns we compare the values of c calculated by equation 7c with the values observed. The agreement isgood for neon, argon, nitrogen, and carbon monoxide and fairly good for hydrogen, oxygen, carbon dioxide, nitrous oxide, and methane. However, it is poor for helium, ethylene, ammonia, and diethyl ether. Most probably the constants for the helium mere fitted at a temperature very high compared to the critical point, and so our method of calculating the molecular constants for helium is not justifiable. Ammonia may be partially polymerized. The Lennard-Jones potential is quite satisfactory for the analysis of the experimental second virial coefficients, but not quite accurate enough to TABLE 2 Values of bo, E,, and c

I :;E\.

bo

CALCU-

GAS

I ~

bo

(CALCWLATED

FROM

E, :CALCULATED FROM BEATTIEBRIDGEMAN)

ergs

A . . . . . . . . . . . . . 49.10; 50.02

~

(c~cnLATED I

0.776 25.45 35.24 34.61

C

'

FROM ;(OBSERVED)' CQUATION

~0~x81

7c)

€9'@

I

~

literaper degrees mole

j 1

9661

AQREXYENT

!

~

5.19 X 10-1 4.881Xlo-' 16.51 16.50 13.24 13.40 13.40 13.36 4.74 16.22 24.03 24.03 20.49

He . . . . . . . . . . . l7.49! 21.41 CzHi.. . . . . . . . . 151.8 I YHs. . . . . . . . . . 42.65; ( C Z H S ), ~. , ~. .. 567.6

i

E, (CALCULATED FROM LENNARD-

litespm degrees

I

mde

1.010')

4.246 16.97 19.70 0.950

1

2.19

fit the Joule-Thomson coefficients of Roebuck and his coworkers (21, 22, 23, 24). The Joule-Thomson coefficients can be obtained with great accuracy, since they involve only a measurement of temperature and pressure. The second virial coefficients are accurate to several significant figures less than the pressure-volume data froin which they are obtained. It was shown in a previous communication (10) that the Joule-Thomson coefficient extrapolated to zero pressure, P O , is simply related to the second virial coefficient :

20

J. 0. HIRBCHFELDER AND W. E. ROBEVEARE

where C: is the specific heat at constant volume extrapolated to zero pressure. c", can be calculated very accurately for many substances from I

I

I

I

t I ,!

I

a 1

,

I.I

e

4, N.

O.!

.

'%-.-- e-,

I 3

-100

I 0

I 100

%-L*

200

3

T 'C.

FIQ.2. Nitrogen: Joule-Thomson coefficient extrapolated to zero pressure. 0 , experimental observations by Roebuck and Osterberg; 0 , calculated LennardJones potential, 8 = 12; X, calculated Lennard-Jonea potential, 8 9. an analysis of the spectroscopic energy levels. Substituting equation 4 into equation 8 we obtain the relation:


21

Here the G,(E,/kT) are functions of temperature which have been tabulated for different values of s. The determination of the two constants bo and E, is a simple job in curve fitting. For the case of nitrogen, we used the spectroscopically determined specific heat values of Trautz and Ader (26) in equation 9. Figure 2 shows the agreement which is obtained between the calculated Joule-Thomson coefficients and the values measured by Roebuck and Osterberg (23). The best values for bo and E , were as follows:

I

bo cc.

s = 9 s = 12

73.3 64.9

E, ergs

9.95

x 10-15

13.24

These are the same as the values obtained by Lennard-Jones from his analysis of the second virial coefficients (14). It will be noticed that the calculated and observed values agree well for the high-temperature region, but the calculated values are too small a t lower temperatures. If the two constants were chosen so that the fit was good for both the high and the low temperatures, it would be unsatisfactory in the intermediate region. We believe that a more accurate potential curve, perhaps one in which the repulsion vanes exponentially with the separation, would give a better fit. Gaseous mixtures are the simplest example of chemical solutions, and they should be useful in studying the laws of mixing and the interactions between unlike molecules. We can apply the same method of analysis to these mixtures that we used for the pure components. The laws of mixing a t low pressures are well known (6). If we let NI and N Z be the mole fractions of components 1 and 2,

and B’(T)mix. =

+

N : B ’ ( T ) ~ 2N1N2B’(T)12

+N~B’(T)~

(11)

where B’(T)12is the second virial coefficient concerned with the collision of unlike molecules. It differs from the coefficient for pure components (2), only in having E ( T ) ~the z , energy between unlike molecules, in place of E ( r ) . Substituting equations 10 and 11 into equation 8 for the JouleThomson coefficient:

22

J. 0. HIRBCHFELDER AND W. E . ROSEVEARE

Here

And, using the Lennard-Jones type of potential, The analysis of the law of force between two unlike molecules then consists in solving equation 12 for A B for many different values of the temperature. These experimental values of AIZserve to determine the (bo)= and (E,,JI2 of equation 14. For each mixture we obtain a different value for Alz at a particular temperature. The consistency of the experimental data is then demonstrated by the agreement or disagreement of these values obtained from different mixtures. The law of mixtures has been derived from rigorous statistical mechanical considerations, so that any disagreement must be due either to experimental errors or to factors which have not been taken into account in this treatment. We examined the experimental Joule-Thomson coefficients for mixtures of helium and nitrogen measured by Roebuck and Osterberg (24). The spread in the experimental Alz is large, and it is difficult to make an accurate analysis. We obtained the best agreement with the experimental data when the collision diameter for the unlike molecules, (ro)HeN2, was taken to be the arithmetical mean of the collision diameters for the pure components, i.e. ,

and the coefficient of van der Waals attraction for the unlike molecules, cHe-N2, was taken to be the geometrical mean of the coefficients for the pure components12Le., cHe-~*=

(CH~CN~)'= "

(0.1522 X 14.0)"2 X 10"'

= 1.46 X

ergs-cm.E

Figure 3 compares the calculated and observed Joule-Thomson coefficients for the four mixtures studied. It is only in the case of mixture No. 1 that the discrepancy is large. An error of 1per cent in the quantitative analysis of the mixture would be sufficient to account for the difference. I n any event, (ro)HeNI is determined within 2 per cent and CH-N, within 10 per J. Corner, an associate of Lennard-Jones, has pointed out to us in a private communication that the Slater-Kirkwood formulation of the coefficient of van der Waals attraction betweenunlike moleculeswould leadto a vslue of CNr-He within 1 per cent of the geometrical mean. For other mixtures he might not expect this law to hold so accurately. f

23


cent for the potential between the unlike molecules. This type of interaction between unlike molecules has been utilized extensively by Hildebrand (8) in his work on the solubilities of non-polar substances. We attempted to make a similar analysis of mixtures of argon and helium, which have also been studied by Roebuck and Osterberg (work unpublished). In this case, the A n obtained from Werent mixtures had systematic variations, so that no definite conclusions could be reached. The discrepancy might be attributed to an unmixing of the gases in passing through the porous plug, although this explanat,ion seems very unlikely.

-

a

*005-

- 0 @ 5-100

-50

50

100

150

200

250

T 'C.

FIQ.3. Helium-nitrogen mixtures: Joule-Thomson coeliicient extrapolated to zero pressure. -, experiments of Roebuck and Osterberg; X, calculated values, Curve I, mixture No. 1, 75.5 per cent helium; curve 11, mixture No. 2, 61.0 per cent helium; curve 111, mixture No. 3, 33.2 per cent helium; curve IV, mixture KO.4, 16.6 per cent helium. II. A LIbfITINQ EQUATION OF STATE FOR HIGH TEMPERATURES

As the temperature becomes high compared to the critical temperature we might suppose that the equation of state for a gas would approach some simple limiting form. Experimentally, it is found that a t sufficiently high temperatures the internal energy of a gas a t constant temperature is a linear function of the density. Michels, Bijl, and Michels (19) have found that this is true for carbon dioxide up to pressures of 3OOO atm., where the Beattie-Bridgeman equation no longer applies (20). The black lines in figure 4 show the results of measurements of Michels, Bijl, and Michels. The circles are points calculated from the Beattie-Bridge-

24

J. 0. HIRSCHFELDER AND W. E. ROSEVEARE

DENSITY, 9 ,IN AMAGAT UNITS FIG.4. Carbon dioxide: internal energy as 8 function of density

man equation for a temperature of 15OoC. For this equation of state, the internal energy, U , has the form:

vi=

Uo(T) -

(A0

+ 3Rc/TZ)/V - 1/2(-aAo + 3RCBll/P)/V* + RcbBo/(TZV3) (15)

At high temperatures it approaches the form:

U = Uo(T) - Ao(1

- 1/2a/V)/V

(16)

25


This expression would give the best agreement with the experimental data a t the high temperatures if a were taken to be 0. The internal energy cannot keep on decreasing with increasing density, but perhaps the correction factor should be introduced as an exponential with volume, as has been pointed out by Kincaid and Eyring (13). Our present difficulty is that we do not know how’to interpret the Beattie-Bridgeman constants, a and b, in terms of the intermolecular forces. However, if we take the internal energy to be a linear function of the density, we are led to a simple equation of state which has a simple interpretationon the basis of the Menke probability functions (18).

3

500-

I

1

I

0.2

0.4

06

DENSITY

G/CC

FIG.5. Steam: internal energy as a function of density

For pressures in the neighborhood of 100 atm. the Beattie-Bridgeman equation is excellent. Up to pressures of this magnitude a t high temperatures, this equation gives the internal energy to be very nearly a linear function of the density. In figure 5 we have plotted the measurements of Keenan and Keyes (12) for the internal energy of steam as a function of density for various temperatures. At the high temperatures we obtain accurate straight lines. For temperatures near the critical point there are deviations. The experimental values of Roebuck for air (21) provide another example where the internal energy is a linear function of the

(

density. This relationship requires that aa):”,e

a constant independent

26

J. 0. HIRSCHFELDER AND W . E. ROSEVEARE

of pressure for each temperature. Table 3 shows that this is true within the irregular variations of the data. At low temperatures the linearity of the internal energy with density still holds for all dilute gases. Hildebrand (8) has shown that it is also true for the so-called “normal liquids.” We do not know whether the internal energy of these liquids lies along the same straight line as the internal energy of the gas. In figure 6 we have plotted the internal energy of n-butane in the temperature range lying between the critical temperature and the ordinary boiling point (from the work of Sage, Webster, and Lacey (25)). The fact that the internal energy of the liquid has a different slope from that of the gas in the cases of water and of butane may be due to hindered rotation in the condensed phase. Many liquids must be studied in this manner before any definite conclusions can be reached.

TABLE 3 values of

(5)

jor air

V T

P

200%.

atm.

1 20

60 100 140 180

1767 1685 1692 1712 1753

1618 1673 1693

1606 1510 1601 1499

1498 1529 1544 1574

1429 1400 1380 1392

1603 1576

The deviations in the critical region can be explained on the basis of fluctuations. I n the vicinity of the critical point it follows from statistical mechanics that there is almost the same probability that in a small unit of volume a t any instant there will be n molecules more than the average as that there will be n molecules less than the average. But the dense regions give disproportionately large negative contributions to the internal energy. We know from opalescence and related effects that the fluctuations in the critical region are large. Perhaps the most pertinent evidence is obtained through the specific heat atconstant volume. Gibbs has shown that quite generally,

where U i s the instantaneous value of the internal energy and the bars represent the time averages (6). Thus C. is a direct measure of the fluctuations in the internal energy. Michels, Bijl, and Michels (19) find

27


that the specific heat a t constant volume of carbon dioxide has a sharp maximum a t the critical point. Quantitative estimates of the effect of these fluctuations are of the right order of magnitude to explain the devia-

\

(

I

I

IA

tions from the linearity of the internal energy with density. The constants in the Beattie-Bridgeman equation are chosen so that the equation of state fit,s the experimental data in this region in spite of the fluctuations. Such a choice of constants is certainly desirable from a practical standpoint,

28

J. 0. HIRSCHFELDER AND W. E. ROSEYEARE

but it makes it more difficult to interpret these constants in terms of the molecular forces, than if bhey were chosen to fit a temperature range where fluctuations are unimportant. Suppose that the internal energy, U , is a linear function of the density, i.e.,

For the time being, B’(T) could be a general function of temperature, but we will soon identify it with the second virial coefficient defined in equation l. From thermodynamics we know that

where A is the maximum work function. Combining equations 17 and 18 and integrating with respect to the temperature, keeping the volume constant, we obtain the relation for A :

The last term is the constant of integration. Now, using this expression for A in the thermodynamical equation of state,

For large specific volumes the equation of state must reduce to the virial equation 1 and this gives a positive proof that B’T in equations 17 to 20 is actually the second virial coefficient. The linearity of the internal energy with density is equivalent to the temperature independence of the third, fourth, etc. virial coefficients. The form for f(V) can be obtained explicitly on the assumption that it is the same as for rigid non-attracting spheres. At very high temperatures where E, < < kT, the molecules approach this ideal and equation 20 should still apply. Under these conditions Happel (7) has evaluated the third and fourth virial coefficients in terms of the space in which the hard spheres overlapped. He obtained C‘ = 0.625b2and D’ = 0.2869b3,where b is the van der Waals constant corresponding to a rigid collision diameter, T . Hirschfelder, Stevenson, and Eyring (IO) showed that asymptotically, as the molecules become very close packed, the van der Waals or the Happel equation becomes : (p

+ a / V z ) ( V - 0.7163b”3Vz’3)= RT

(21)


29

Here the constant 0.7163 is a geometrical factor made up of pure numbers such as T,42,etc., calculated on the assumption that the molecules are packed into a body-centered lattice; face-centered packing would have led to a value of 0.6962 instead. The requirement that Happel’s equation should agree with equation 21 throughout the liquid range of densities serves to evaluate the fifth virial coefficient. Comparing this vinal equation with equation 20 we obtain the equation: p =

RT - (1 + B’(T)/V + 0.625bZ/V2 + 0.2869b3/V3+ 0.1928b4/V4) (22) V

This relation follows as a direct consequence of the linearity of the internal energy with density, and the assumption that a t sufficiently high temperatures the molecules behave like rigid non-attracting spheres. This equation of state differs from that of Hirschfelder, Stevenson, and Eyring (10) only in the form of the second virial coefficient. The Joule-Thomson coefficient as a function of pressure provides a very sensitive test for the validity of equation 22. At zero pressure po is a function only of the second virial coefficient; a t high pressure the Joule-Thomson coefficient depends on all of the virial coefficients in an important way. From simple thermodynamical considerations:

And from equation 22 it follows that:

(aa,v), = - V [ P + RTB’(T) -F - R T - d dV

(-)I

f(V) V2

-I

(26)

Here, of course, we have abbreviated b2 ’ 0 = 0.625 + 0.2869b3/V3+ 0.1928b4/V4 V2

(27)

We can compare the results of this equation with the experimental JouleThomson coefficients for argon obtained by Roebuck and Osterberg (22). In a previous communication, we found a potential energy function, E(r), from which we calculated po in agreement with the experimental values.

30

J. 0 . HIRBCHFELDER AND W. E. ROSEVEARE

This potential energy also served to evaluate B’(T). Therefore when we consider the Joule-Thomson coefficient a t high pressures, the only constant which we can adjust is the value of van der Waals b. Figure 7 compares the experimental Joule-Thomson coefficients of Roebuck and Osterberg with the values which we calculate on the assumption first that b = 40

P IN ATM.

FIQ.7. Argon: Joule-Thomson coefficient. 0,calculated, b

= 40 cc.; X, calcu-

lated, b = 35 cc.; 0 , observed by Roebuck and Oeterberg.

cc. and then that b = 35 cc. For b = 40 cc. the agreement of the calculated with the experimental values is surprisingly good. However, it is clear that a slightly smaller value of b would give better agreement at the higher temperatures and a slightly larger value for b is indicated for the lower temperatures. Van der Waals’ b for argon, calculated in the manner

INTERMOLECULAR FORCES AND PROPERTIES OF QABES

31

of the first section from low-pressure measurements, is 37 cc. for 300OC. and 47 cc. for 0°C. From the Cailletet and Mathias relationship considering the liquid to be close packed, b = 41.2 cc. The critical constants give 6 = 32.3. Thus the value of b = 40 cc., which the experimental Joule-Thomson coefficients indicate, is in accord with the value of b determined by the usual methods. The van der Waals b which we have used corresponds to the molecules having collision diameters 7 per cent smaller than the diameter at which their energy of repulsion exactly compensates the energy of attraction. We conclude from the above analysis: (I) The assumption that the third and higher virial coefficients are constants provides a good first approximation. (2) The values of the third and higher vinal coefficients can be closely linked with the collision diameters or van der Waals b, and the values thus obtained are in close agreement with the results of other experiments. Up to this point, our approach to the equation of state has been from the standpoint of pure thermodynamics. Now let us examine the significance of the linearity of the internal energy with density from the standpoint of the individual molecular interactions. There are two methods of doing this, which have become almost standard practice in statistical mechanics. First, we might think of all of the molecules except one being held in their mean positions and study the effective, or the free, volume in which this one molecule can move. The second method also fixes our attention on one molecule, but here we are interested in the instantaneous distribution of molecules in the vicinity of our chosen one. This distribution leads to the well-known Menke probability function (B), which may be evaluated from x-ray diffraction patterns. The free volume, V,, may be defined by means of the partition function:

F N = [V,(2~kT/hz)aizJ]Nexp(-U I R T )

(28)

where J is the partition function for the internal degrees of freedom of the molecule, rn is the mass of the molecule, and h is Planck's constant. Other definitions of free volume have been proposed, but this one has the advantage that the free volume appears in the partition function as the effective volume in which the molecules move, it may be given a pictorial interpretation when the molecules are rigid, and it is simply related to the entropy. Eyring and his coworkers have developed the free volume description for the equation of state of liquids (5,9, 13), and more recently Lennard-Jones and Devonshire (15, 16, 17) have started to use it in their description of dense gases. In terms of the free volume, the thermodynamical equation of state is p =

- ($)T

+ RT ($log V,)T

32

J. 0. HIRSCHFELDER AND W. E . ROSEVEARE

Substituting our expression for U in terms of B’(T) it follows that

Integrating this expression with respect to volume and requiring that the free volume approach the true volume as the density becomes small, we obtain:

V,

= V exp

[If$

dV

- v d-!-T (TB’(T))]

The second term in the exponential corresponds to an increase in free volume with increasing temperature when the system is confined to a fixed amount of space. This shows that the model corresponding to the linearity of the internal energy with density is considerably more general than a gas of rigid spheres. This temperature dependence was missing in the free volume for gases proposed earlier by Hirschfelder, Stevenson, and Eyring (10). Another virtue of our free volume is that it leads to an increase in C, with pressure, as shown by equation 24. Eyring and his coworkers (5,9, 13) introduce a similar term in the specific heat, considering a temperature variation of b and supposing van der Waals’ a to be independent of temperature. The linearity of the internal energy with density is described most easily in terms of the Menke probability function. This function, W(r), is defined as the number of molecules which lie within a unit of volume R distance T from a particular molecule divided by the average number of molecules lying in an arbitrary unit of volume. In terms of the probability function, the internal energy is

U

=

’$*1-W(r)E(r)r2dr + Uo(T)

If U is a linear function of 1/V, it follows that W ( r )must be a function of r independent of volume, Le., the same function for a dilute as for a dense gas. For the dilute gas,

W(r) = exp( - E ( r ) / k T ) For liquids, the probability function has been determined from x-ray diffraction data for mercury (18), carbon tetrachloride (18), gallium (18), water (11), and for a few comparatively complicated substances, but as far as we were able to ascertain, it has never been determined for helium, hydrogen, neon, nitrogen, or argon, which are the substances for which we know E(T). Figure 8 shows W ( T )for a dilute gas and its likely appearance for the case of a “normal” liquid or a dense gas. In the examples where the linear relationship holds, we should expect that the first maximum in W(T)is the same for the dilute gas as for the liquid or dense gas, since this


33

part of the probability function depends on the actual collisions between a pair of molecules and is not greatly affected by the presence of neighbors. The two curves should differ in the relatively unimportant second and third maxima. From this standpoint, it follows that W(r) cannot be exactly independent of volume, but this assumption may represent a good first approximation. The deviations of the internal energy of gases and liquids from the linearity with density might be used to study the structure of these substances. As long as a change in the density produces only a change in the frequency but not a change in the character of molecular collisions, W(r)is independent of volume. However, aa the system becomes more and more dense the molecules will become geometrically constrained to move in a more or less regular lattice. Under these conditions W ( r )

DENSE GAS OR

_ - --

I.0 W(r)

0

0

r,

r-’

Fro. 8. W ( r ) for a dilute gas and its likely appearance for the case of a “normal” liquid or a dense gas

will vary with the density of the system in something like the following manner : (33) W ( r ) = g(r)/V - h(r)/V“ where g(r) and h(r) are independent of volume, and have maxima corresponding to the lattice points. For these dense systems it is clear that the lattice structure will set in, and the deviations of the internal energy from the linearity will occur in the neighborhood of the density of the crystal. It is in this region that we should like the correction factor for the internal energy (corresponding to a/2V in the Beattie-Bridgeman equation) to be large. SUMMARY

1. Any good equation of state for gases may be used to estimate the energy of interaction and collision diameter of molecules. The BeattieBridgeman equation is excellent for this purpose.

34

J. 0. HIRSCHFELDER AND W. E. ROBEVEARE

2. The Joule-Thomson coefficients extrapolated to zero pressure lead to an alternative method of determining these molecular constants which has some advantages. 3. Experimentally it is found that a t high temperatures the internal energy of a gas at constant temperature is a linear function of the density. This leads to a simple equation of state which holds up to pressures of the order of 3000 atm. For such high pressures the Beattie-Bridgeman equation does not hold when the usual values of the constants are employed. 4. The linearity of the internal energy with density at constant pressure is explained in terms of the Menke probability function. Deviations from this law in the neighborhood of the critical temperature are due primarily to fluctuations.

We should like to thank Professor J. R. Roebuck for permitting us to study his unpublished data, and one of us (J. 0. H.) would like to express his appreciation to the Wisconsin Alumni Research Foundation for financial support throughout the course of this work. REFERENCES (1) BEATTIE, J. A., AND BRIDGEMAN, 0. C.: J. Am. Chem. SOC.49,1665 (1927). (2) BEATTIE, J. A., AND BRIDGEMAN, 0. C.: J. Am. Chem. SOC.50,3133 (1928). (3) BEATTIE, J. A., AND BRIDGEMAN, 0. C.: J. Am. Chem. SOC.SO, 3153 (1938). (4) BEATTIE, J. A., AND BRIDGEMAN, 0. C.: Z. Physik 62, 95 (1930). (5) EYRING, H., AND HIRSCHFELDER, J.: J. Phys. Chem. 41,249 (1937). (6) FOWLER, R. H.: Statistical Mechanics, 2nd edition. University Press, Cambridge (1936). (7) HAPPEL, H.: Ann. Physik SO, 246 (1906). (8) HILDEBRAND, JOEL H. : Solubility of Non-electrolytes, American Chemical Society Monograph. Reinhold Publishing Corporation, New York (1936). (9) HIRSCHFELDER, J., EWELL,R. B., AND ROEBUCK, J. R.: J. Chem. Phys. 8, 205 (1938). (10) HIRSCHFELDER, J., STEVENSON, D., AND EYRING,H . : J. Chem. Phys. 6, 896 (1937). (11) KATZOFF, S.:J. Chem. Phys. 2, 841 (1934). (12) KEENAN,J. H., AND KEYES,F . G.:Thermodynamical Properties of Steam. John Wiley and Sons, Inc., New York (1936). (13) KINCAID, J., AND EYRING, H.: J. Chem. Phys. 6, 587 (1937). (14) LENNARD-JONES, J. E . : Interatomic Forces, Chap. X in Statistical Mechanics (reference 6). (15) LENNARD-JONES, J. E.:Physica 4, 10, 941 (1937). (16) LENNARD-JONES, J. E., AND DEVONSHIRE, A. F.: Proc. Roy. SOC.(London) Al63, 53 (1937). (17) LENNARD-JONES, J. E., AND DEVONSHIRE, A. F.: Proc. Roy. SOC.(London) A166, 1 (1938). (18)MENKE,H.: Physik. Z. 39\593 (1932). (19) MICHELS,A., BIJL, A., AND MICHELS,C.: Proc. Roy. SOC.(London) M W , 376 (1937).


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MICHELS, A., AND MICHELS, C . : Proc. Roy. SOC.(London) Al60, 348 (1937). ROEBUCK, J. R . : Proc. Am. Acad. Arts Sci. 64, 287 (1930). ROEBUCK, J. R., . ~ N DOSTERBERG, H.: Phys. Rev. 46, 785 (1934). ROEBUCK, J. R., AND OSTERBIRQ, H.: Phys. Rev. 48,450 (1935). J. R . , AND OSTERBERG, H.: J . Am. Chem. Soc. 60,341 (1938). ROEBUCK, SAGE,B. H., WEBSTER, D. C., AND LACEY,W. E.:Ind. Eng. Chem. 29, 1188 (1937). (26) TRAUTZ, M., AND ADEII, H.: Z. Physik 89, 1 (1934). (27) URSELL,H. D.: Proc. Cambridge Phil. Soc. 23, 685 (1927).

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Intermolecular Forces and the Properties of Gases. - The Journal of

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