FREE-VOLUME MODELS FOR LIQUIDS TERRELL L. HILL Department of Chemistry, The Cniversity of Rochester, Rochester, S e w York Received April 8, 1947 I. Ih-TRODUCTION
It was found desirable, in another connection, to compare approximately, in the neighborhood of the critical point, the fundamental functions occurring in the van der Waals and Lennard-Jones and Devonshire models of classical liquids with the same functions obtained from experimental data. I t is believed that these results, although necessarily of only a semiquantitative nature, might be of interest, and they are presented here. A new approximate free-volume model for classical liquids is discussed briefly in an appendix. For a classical perfect gas
where u = T ’ / N and J ( T ) is the internal partition function. I n passing from a classical perfect gas and equation 1 to the free-volume model of a classical imperfect gas or liquid, v e may replace, in equation 1, the volume per molecule, v, by an effective free volume per molecule u, , and introduce a potential energy of interaction per molecule - x,evaluated when each molecule is in the center of its cell. I n the most general form of this model both v j and x would be functions of v and T . Actually, the dependence of x on T is not very strong. Seglecting this dependence we have, for an imperfect gas or liquid (classical), according to this model:
The theory of Lennard-Jones and Devonshire (LJD) (7) serves as an illustration of equation 2. We shall actually be concerned primarily with the very crude, but still helpful, simplest free-volume model in lyhich both 01 and x are taken as functions of v only. The physical picture associated with this latter more restricted model is that each molecule moves freely in a uniform potential field ( - x ( v ) ) over the free volume 5 o j . The remainder of the volume, V - Nu,, is excluded because the potential energy is infinite in this region. The excluded volume arises, of course, owing to the presence of the other molecules. I n the more general case (equation 2 ) , each molecule moves in a non-uniform (periodic in three dimensions, on the average) potential field over the entire volume N u = Ti. The molecules spend most but not all of their time near potential minima (at which the potential energy has the value - x ( v ) ) , following the Boltzmann distribution law. The temperature dependence introduced by the Boltemann distribution appears in v,(c,T). Models of this type define “effective” free volumes. 1219
1220
TEBRELL L. HILL
For future use, it will be convenient to defme several new quantities (following Fowler and Guggenheim (2) to some extent) : AJ.(v) = x(v)
We assume that the potential energy of interaction between two molecules, as a function of the distance r between them, is given by
Equation 4 defines e and r*. Also, z is the average number of nearest neighbors and y is a geometrical constant depending on the type of packing (y = d5 for close packing, and z = 12). A, $(v), f(v,T), and v* are now defined by equations 3 in terms of e, T * , y, z,x , and vi. The functionf(u,T) is the fraction of the actual volume that is effectively "free." From equation 2 and
we have
We expect the functions p and B to be universal functions, to a good approximation, for suitable gases such as nitrogen, argon, neon, etc. The question now arises as to the correct form of the functions p and B. Unfortunately there is no satisfactory theory to which to turn, though perhaps the most successful tractable effort is that of LJD. Writing x = v * / v , the LJD model leads to
FREE-VOLUME
1221
MODELS FOR LIQUIDS
where g, gz, and g, are functions of v/v* and A / k T defined by certain integrals. These equations give A / k T , S 9, v,/v* E 1.8, and P,v,/kT, S 0.64. If we compare these critical constants with experimental values for neon, nitrogen, and argon (see table l ) , the first is seen to be quite good (discussed further below) but one should have v,/v* = 2.76 and P,v,/kT, = 0.292. TABLE I t
35.3
~
I I
t See pp. 285 and 345 of
i 0.294 0.298 ~
reference 2; original references are given there.
It is of interest to discuss van der Waals’ equation also, from this point of view. We have : (P a/v?)(v - b ) = k T (12) Using (reference 8)
+
and
equation 12 can be rewritten in the form
with
and
The functions f and # are found from p and B by integrating (with respect to x, say), taking f = 1 and = 0 a t x = 0. We can also write f as f = ( v - b ) / v . Van der Waals’ equation thus follows from a free-volume model with p, as well aa B, taken as a function of v/v* only. These van der Waals functions (equations 16 and 17) are included in figures 1-4. In these figures, for van der Waals’ equation, we take 2 = 10.5 and find y from equation 20 (see Section 11),leading to K I= 2.59 and K z= 0.247.
+
1222
TERRELL L. HILL
Our object is to compare, in the neighborhood of the witical point, the functions p, B , f , and IC. of the LJD and van der Waals models with corresponding functions calculated from experimental P V T data (International Critical Tables, for example). Unfortunately, the experimental data for argon and nitrogen do not appear to be adequate to study the temperature dependence (of p and f ) satisfactorily, and, for neon, the data seem inadequate even for the volume dependence. In any case the available data restrict one to temperatures slightly above the critical temperature. In view of the above restrictions, it mill be necessary to make the approximation, in examining the experimental data, of neglecting the temperature dependence of p andf. That is, we assume that the experimental PVT data follow
= rp
);(
+ & B );(
(in other words, we “force” the experimental data to obey the rather restricted equation 18). This is, of course, a fairly crude assumption, but it should lead to results of semiquantitative significance. The procedure is then as follows: Suppose Pv/kT is available for approximately the same range in v/v*, for a gas, a t two temperatures, TI and 2’1, iyhich do not differ greatly. We may then interpolate (numerically or graphically) ( P v / k T ) ]and (PujkT)?for desirable values of v/v*. From equation 18, we have
from which we calculate p and B for each value of u/v*. In this may, if sufficient data are available, we can obtain the approximate volume dependence of rp and B, near the critical temperature. 11. CALCULATIONS FOR ARGON h N D NITROGEX
It is first necessary to make some preliminary calculations concerning y, v*, etc. For close packing of either type, z = 12 and y = 4%Actually, z is probably somewhat less than 12. The only experimental data available are for argon (1, lo), leading to z = 10.2-10.9. We adopt z = 10.5 here as an average value. In order to estimate z for neon and nitrogen we assume that h / k T , = ze/kT, has the same value for argon, neon, and nitrogen. For argon (table l ) , taking z = 10.5 and B = 16.5 X ergs per molecule, we find A / k T , = 8.31, and we adopt this as an approximate universal value for suitable classical gases. It may be compared with h / k T , 9 in the LJD theory. Using A / k T , = 8.31, z is given for neon (the degeneracyjn neon should be negligible a t its critical temperature) ana nitrogen in table 1, and also for hydrogen and helium assuming for the moment that they too are classical. The values of z thus obtained for
FREE-VOLUME MODELS FOR LIQUIDS
1223
hydrogen and helium (in parentheses in the table) are approximate upper limits, since degeneracy aids condensation and hence, with degeneracy, other things being equal, the same critical temperature can obtain with a smaller number of nearest neighbors. The value of z for hydrogen, taking degeneracy into account, is estimated roughly in Appendix I1 as approximately 8.8. It may be pointed out in this connection that the relatively large discrepancies between T, (calcd.)' and T , (obsd.) in table 5* of Fowler and Guggenheim (from LJD) for hydrogen and helium are actually of the wrong sign to be explained by degeneracy. This is because z is assumed the same for all gases listed, including hydrogen and helium. From these considerations (table l ) , one therefore expects that z should be relatively small for hydrogen and especially for helium. There is some tentative experimental evidence for this ( z = 6 ) in the case of helium (6). In the absence of other information we make the rather reasonable assumption that the values of z in table 1 result from the occurrence of occasional random vacancies in a close-packed lattice. This allows us to estimate y from z by y = 2/2 2/12 (20) The remainder of the values in table 1 for neon, nitrogen, and argon may then be calculated using T * ~ (see footnote to table 1) and the experimental critical constants. All of the non-experimental values in table 1 should be considered approximate.
Nitrogen The only suitable data are for TI = 128.64 (six experimental values of Pv/kT in the approximate range V/V*= 1.8 to 6.0) and T1 = 12G.D (five experimental values in this range). As T I - T2 is quite small in this case, numerical interpolation of (Pv/kT)I and (Pzl/kT), was used (Lagrange's formula) rather than graphical interpolation. B(v/v*) and p(v/v*) were then calculated from table 1 and equations 19. These results are given in table 2 and figures 1 and 2. We also give, in table 2, the critical isotherm ( A / k T = 8.31) calculated from B , p, and equation 18. In order to calculatef(v/v*) and #(v/u*)we must extrapolated lnf/dx and dfi/dx to x = 0. We estimate (d In f/dx),,o and (d#/dx),,o from second virial coefficients, as follows: Assume that, for small x (large v), we can m i t e
where B(T) is the second virial coefficient. Hence
1 2
T, (calcd.) is given incorrectly for helium; it should be 8.O"K. Page 345 of reference 2.
1224
TERRELL L. HILL
TABLE 2 Nitrogen
-- - dU*
GI, (3
-E
- - __ 1.847 2.308 2.770 3.232 3,694 4.155 4.617 5.079 5.540 6.002
Pv
I
*
PX
9 E kTr - --
0.2464 0.2193 0.2268 2.092 0.2860 0.2645 0.1803 1.754 0.3330 0.3125 0.1716 1.729 0.3812 0.3608 0.1709 1.772 0.4270 0.4074 0.1643 1.765 0.4688 0.4509 0.1500 1.690 0.5061 0.4903 0.1327 1.586 0.5393 0.5251 0.1191 1.508 0.5690 0.5555 0.1138 1.496 0.5956 0.5819 0.1142 1,525 __ -
0.207 0.256
0.303 0.352 0.400 0.444
0.483 0.518 0.550 0.576
0.112 0,111 0.109 0.109 0.108 0.107 0.105 0.102
1.786 2.083 2.500 3.125 4.167 6.250
0.099 0.096
8 3
2
3
2
4
v /v"
FIQ. 1
6
a
8
0.3316 0.2985 0,2652 0.2269 0.1796 0.1295
-
4
x
0.2218 0.2587 0.2979 0.3510 0.4365 0,5434
IO
1225
FREE-VOLUME MODELS FOR LIQUIDS
That is, we assume that B(T) is a linear function of 1/T between TI and Ta For nitrogen, B( T) is available for TI and T2,and hence c1 and q may be calculated easily on this assumption. One finds c1 = -5.042 X lo4 A.a-deg. and c2 = 229.6 A.* It may be remarked that the limiting values of the derivatives obtained by equations 23 seem quite reasonable in actually performing the and f may now extrapolations. The same is true for 'argon (and hydrogen).
+
0 X
b
0
0
FIG. 2
be calculated by numerical integration, and these functions are also given in table 2, and in figures 3 and 4. Argon The temperatures T I = 157.24 and T2 = 152.86 were used with careful largescale graphical interpolation of PvIkT. Values of the second virial coefficient are not available for exactly these temperatures, but a tangent line was used, drawn through the smooth curve of B(T) versus 1 / T a t (TI T2)/2 = 155.05
+
1226
TERRELL L. HILL
(the smooth curve had to be extrapolated slightly to this temperature). There is obtained c1 = -0.361 x lo4 A.3-deg. and c2 = 102.8 ‘A.3 We give in table 3 and figures 1-4 the functions B , 9,f, and +, and the calculated critical isotherm in table 3. C
.8
LJD
,
.b
f .4
2.
4
- 6 v/v
e
IO
FIG. 3 IIX. DISCUSSION
The functions p, B , j , and $for argon and nitrogen, and for the van der Waals and LJD models, may be compared in figures 1-4. The curves given for the LJD model are for *\/kT = 9 (this is the critical temperature in this model-p depends on temperature here). It will be observed that neither the van der Waals nor the LJD functions resemble the “experimental” functions very closely, though, rather surprisingly, the van der Waals functions actually seem to be superior in some respects, Apparently the fact that we have, for small 5 , Pv/kT = 1 O(z) van der Waals (and actual gases) = 1 O(9) LJD is of importance even near the critical point.
+ +
I227
FREE-VOLUME MODELS FOR LIQUIDS
A
.3
Y .2
A 0
.1
X
X
2
4.
v/v”
8
6
10
FIG.4
TABLE 3 Argon
0 3033
0.3244 0.3485 0 372 3.693 4.103 4.616 5.276
6.156 7.386 9.233 12.311
1
0.4436 0.4781 0.5178 0 4886 0.5630 0.53i1 0.6124 0.5914 0.6660 0.6493 I 0.7248 0.7110 ’ 0.7864 0.7761 1
0.1708 0.1693 0.1640 0.1557
1.703 1.714 1.895 1.655
0.1277 0.1133 0.0918 0.0730 0.0603 0.0451
1.567 1.537 1.467 1.345 1.249 1.207 1.146
0.285 0.307 0 332 0 361 ~
0.433 C.476
0.526 0.582 ~
0.643 0.706 0.772
0 108 0 108 0 108 0 108 0.107 0.105 0.103 0.loo 0.095 0,087 0.076 0.063
1
*
0.4098
0.2319
0.4497
0.2090
0.jOli
0.1830
0.5652
0.1547
0.6417
0.1242
0.7311
0.0917
0.8103
0.0626
1228
TERRELL L. HILL
The discrepancies between nitrogen and argon are rather small in figures 1 and 2, but they are of course considerably magnified in figures 3 and 4 because of the integrations involved. Although it has been emphasized that the above comparison between experimental and theoretica1,curves is only semiquantitative, it is felt that the general form of the “experimental” functions ‘p, B,f, and 4 exhibited in figures 1-4 should be of some value as a test in connection with future work on free-volume models of liquids. It will be noticed, in figure 3,that a t the critical point f is of the order of 0.30.5. Earlier work on liquids has given much smaller values (3, 4) :f = 0.0030.01. However, the actual discrepancy is not nearly so great as this, for these latter values refer to considerably smaller volumes and temperatures. Let us estimate roughly the magnitude of our f’s for argon and nitrogen a t their respective boiling points. We may correct for the volume effect by extrapolating TABLE 4
LJD funelions for A / k T = 6 PO -
alu’
1.195 1.291 1.414 1.581 1.826 2.236 2.575 3.162 4.472
kT
0.7 0.6 0.5 0.4 0.3 0.2 0.15 0.1 0.05
0.00262 0.00395 0.00659 0.0115 0.0213 0.0425 0.0615 0.0930 0,1470
0.00182 0.00380 0.00865 0.0222 0.0667 0.257 0.583 1.693 7.825
0.000333 0.000662 0.00140 0.00325 0.00&13 0.0255 0.0478 0.1021 0.2648
2.838 2.577 2.133 1.854 1.670 1.528 1.431 1.413 1.342
PP’ kT
2.375 1.996 1.508 1.173 0.915 0.683 0.556 0.447 0.300
f , in figure 3, to f = 0 a t v/v* = 0. We read from these extrapolated curves f = 0.21 and 0.14 a t the boiling-point values of u/v*, 1.04 and 1.06. For the temperature dependence we have to assume that it will be about the same as indicated (for g) for the LJD theory (table 7, page 348,of Fowler and Guggenheim (2)). This introduces in our case a factor of about 4.2,which gives f = 0.05 and 0.03. We are interested in temperatures below the boiling points for this comparison, which would give us still smaller values off than 0.05 and0.03. In view of the crudeness of the temperature correction, it is therefore hard to say whether there is in fact any actual discrepancy in order of magnitude. It may be noted in this connection that the LJD theory gives f = 21yg = 0.17 a t the critical point (compared to 0.3-0.5 here). APPENDIX I
In the course of the present work LJD functions were calculated for A / k T = 6. As these results may be of use to others, they are given in table 4.
1229
FREE-VOLUME MODELS FOR LIQUIDS APPENDIX I1
Hydrogen is probably slightly degenerate (Bose-Einstein) near its critical point. At the present time, no treatment of a degenerate imperfect gas or liquid is available, even for the simplest (hard sphere type) model, owing to mathematical difficulties. However, because the degeneracy in hydrogen a t the critical point is only slight, it would seem that one ought to be able to make the necessary small corrections with “order of magnitude” accuracy in this case (but not for helium) by adopting the simplest possible type of assumption: write the well-known (8) expression for the Helmholtz free energy of a perfect BoseEinstein gas
-A/NkT
=
rmkT In (2-- h?
)
3!2
pv f
1 -t
v
(-) h2
U0(T)= 1
< $(T)
312
2 . 6 1 2 ~ 2nmkT
where p is the degeneracy of the lowest energy level, and then formally replace u by u,(u) nrherever u appears and add a term x(u)/kT to both parts of the above equation for -A,INkT (this procedure being in formal analogy with the process: equation 1 + equation 2 for a classical liquid). The equation of state then follows immediately. Some calculations have been made on hydrogen, using the above procedure and experimental PVT data as for argon and nitrogen. d method of successive approximations must be used here to get the “experimental” functions q , B , f, and $. The fact that the experimental data refer to a mixture of ortho- and para-hydrogen was taken into account. It does not seem worthwhile to present these calculations in detail, because they probably have only order-of-magnitude significance. However, a few results might be mentioned: The observed critical temperature of hydrogen is about one-half of a degree lower than it would be if hydrogen were non-degenerate (Le., classical); z is about 8.8 (instead of the classical value 8.96) ; the functions p, B , f,and $ found for hydrogen agree as well with argon and nitrogen as the argon and nitrogen functions do with each other (figures 1-4). Because of the small amount of degeneracy, these hydrogen functions do not differ much from the results obtained in a classical calculation for hydrogen (as for argon and nitrogen). So the corrected functions for hydrogen should be fairly reliable. APPENDIX I11
We discuss here very briefly a new model of a classical liquid which might be of interest. It has about the same degree of crudeness as the harmonic oscillator
1230
TERRELL L. HILL
model of a liquid, but it is somewhat more general (it includes the harmonic oscillator model as a special case) and seems more realistic in a sense. It retains, on the one hand, the close connection between a crystal and the harmonic oscillator model of a liquid, and, on the other, it also shows clearly the transition from liquid to gas. In the harmonic oscillator model, each molecule is considered to be trapped in a cell and performs harmonic oscillations in three dimensions about the center of the cell. 9 s the distance r from the center of the cell increases, the potential energy becomes larger and larger. In the LJD model the potential energy also becomes infinite as r increases, but the form of the potential curve is more complicated than just parabolic (as it is taken to be in the harmonic oscillator model). In both of these models, because of this infinite potential energy property, it is necessary to introduce in a rather arbitrary fashion the possibility of a molecule moving from cell to cell (communal entropy) in the liquid state. The present model avoids this difficulty. Any given molecule in a liquid may be thought of roughly as being surrounded on the average by a number, z , of nearest neighbors in a close-packed lattice. If z is less than 12, say 10.5, we assume some of the lattice sites are vacant a t random. As a molecule moves outward from the center of its cell, the potential energy usually increases. In order for the molecule t o continue this outward motion and pass through its own coordinate shell into an adjacent cell it must in general pass over a potential barrier, but not an infinite one. If a molecule in a liquid is surrounded by nearest neighbor molecules in a face-centered cubic lattice, there are six most probable places, symmetrically distributed, through which the central molecule can pass in leaving the cell (these are the centers of the six squares formed by groups of four molecules in the coordinate shell). On the time average there is of course spherical symmetry, but for convenience we make use, as an approximation, of the simple cubic symmetry of the six passagemrays and write for the simplest possible potential function having the desired properties
That is, -x(v) is the potential energy a t the center of a cell (say x = y = z = O), -x*(v) is the potential energy a t the side of a cell (top of the potential barrier; say y = z = 0, x = d / 2), and we assume a cosine curve for intermediate positions. In the picture above, d , the wave length of the potential curve, is ( 4 3 ) X (distance between nearest neighbors). However, in the classical treatment used below, d does not appear in the final equations. The classical partition function for a single molecule in the volume V = l3 is
with H =
(d + P: + P:) + U l S , y, z ) 2m
FREE-VOLUME MODELS FOR LIQUIDS
1231
Hence
The value of the integral is &(u), where Io(u) is a modified Bessel function of the first kind and u = ( x - x . ) / 2 k T . The complete partition function for N molecules is Q = q N / N ! . We obtain, finally,
- A / N k T = 11 with
ur(v,T) = v exp = v(exp
As T .--f
m
or V-+O((x
or
V -+
m (x,
x, -+ 0 ) ,u -+ 0 and v j -+ v (perfect gas:.
As T -+ 0
- x.} + m ) , u - - t m and
since lim lo(u) = U-D
e" (27ru)1'2
In the usual harmonic oscillator model
where ~ ( u )is-the frequency of oscillation. The formal connection between the two models (for large u ) is therefore
In order to proceed further the functions x ( v ) and x.(v) must be specified. It is clear that a more general model of this type could be developed, replacing the arbitrarily assumed cosine form of the potential energy by a function obtained from more fundamental considerations. This would lead to a treatment similar to that of WD, except that infinite potential barriers would be replaced by finite ones. It will be noted that there is an obvious connection between the present model of a liquid and the treatment of the problems (5, 9) of ( 1 ) hindered rotation in molecules such as ethane and (2) the mobile-localized transition of adsorbed molecules.
1232
R. T. DAVIS, JR., T.
w.
DEWITT AND P. H. E~MMETT
SUMMARY
1. The fundamentat functions of the van der Waals and Lennard-Jones and Devonshire models of classical liquids are compared with corresponding approximate functions obtained from experimental PT’T data on nitrogen and argon. Seither of these models is very satisfactory from this point of view. The general form of the “experimental” functions should be useful as an approximate criterion of success of future free-volume models of liquids. 2. Related calculations on hydrogen are discussed very briefly. It is possible to take into account the slight degeneracy in hydrogen near its critical point in only a very approximate manner. 3. A new approximate model of a classical liquid is suggested which ret.ains, on the one hand, the close connection between a crystal and the harmonic oscillator model of a liquid, and which, on the other hand, shows clearly the transition from liquid to gas. REFERENCES (1) EISEBSTEIN, A . , ASO GINGRICH, N. S.: Phys. Rev. 62, 261 (1942). R . H., A N D GUGGENHEIM, E. A , : Statistical Thermodynamics. Cambridge (2) FOWLER, University Press, London (1939). (3) FRANK, H. S.: J . Cheni. Phys. 13, 478, 493, 507 (1945). S.:Theoretical Chemistry, pp. 46572. D . Van Kostrand Co., Inc., S e w (4) GLASETOXE, York (1944). (5) HILI,,T. I,.: J. Cheni. Phys. 14, 441 (1946). W . €I., A N D T A K O W SIi. , W.: Physica 6, 161, 270 (1938) (6) KEXSOM, J. E., ANI) DEVOSSHIRE, A . F . : Proc. Roy. SOC. (London) 163A, (7) LEKNARD-JONES, 63 (1937); 166A. 1 (1938). (8) MAYER,J. E . , ANI) MAYER, 31. G.: Statistical Mechanics. John Wiley and Sons, Inc., S e w York (1940). (9) PITZEH,Ii. S., ANI) GWIXN,W.D . : .J. Chem. Phys. 10,423 (1912). (10) I ~ I C E0. , li.: J. Cheni. Phys. 7 , 883 (1930).
ADSORPTION OF GASES O S SCRFACES OF POWDERS AS. METAL FOILS’ R. T . DAVIS, JR.,T. W. DEWITT,
AKD P. H. EMMETT Mellon Institute, Pittsburgh 1.9, Pennsylvania
Receiwd J u n e 11, 1947
During the past ten years many measurements have been made to test the validity for the determination of surface area of an improved method (9) and also a theoretical method (6) for interpreting multilayer adsorption isotherms taken near the boiling points of the gaseous adsorbates. The proposed methods 1 Presented at the Symposium on the Bdsorption of Gasos which was held under the auspices of the Division of Colloid Chemistry at the 110th Meeting of the American Chemical Society, Chicago, Illinois, September 11-12, 1946.
