.I.A. Barker and D. Henderson1 C.S.I.R.O.
Chemical Research Laboratories Melbourne, Australia
I
I
The Equation of State of Simple Liquids
I t is almost one hundred years since van der Waals (I) made the first theoretical attempt to p r e dict the equation of state of a liquid. During the intervening years great progress has heenmade in the theory of solids and gases. However, until recently, little progress has been made in developing a theory even for simple liquids. By a simple liquid we mean one composed of molecules, such as those of the heavy inert gases, which are spherical, non-polar, and which have a mass large enough so that quantum effects may be neglected. This discussion will be limited to such liquids. There are two basic assumptions in the van der Waals theory. Firstly, it is assumed that two molecules attract each other according to some definite law as long as their separation exceeds a (the diameter of a mole cule) hut at the intermolecular distance a the attraction ceases and is replaced by a infinite repulsion. Secondly, it is assumed that (particularly at high densities) the structure of the liquid is determined primarily by the repulsive forces and that the effect of the attractive forces is to provide a uniform background potential in which the molecules move. With these assumptions the Helmholtz function, A, of a liquid of N molecules occupying a volume V and a t a temperature T will be that of a gas of hard spheres2 of diameter a at the same T and V except that it is lowered because of the background potential field. Thus where A. is the free energy of the hard-sphere gas,
and u(r) is the intermolecular potential. The factor of in eqn. (1) arises because the energy 4 is shared by two molecules and hence would be counted twice if this factor were not inserted. The functiona g(r) is the radial distribution.functia and is the average density of molecules at a distance r from another molecule located a t the origin divided by the overall density of molecules, p = N/V. Thus, the average number of molecules in a This research has been supported in part by grants from the National Research Council of Canada and the U S . Department of thehterior, Officeof Saline Water. Alfred P. SloanFoundation Fellow and Ian Potter Foundation Fellow. Permanent address: Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. On leave 1966-67. a A gas of hard spheres is one in which the molecules exert no forces on each other until they come into contact (i.e., 7 equals the diameter of the hard spheres) and then exert an infinite repulsion. a The d i a l distribution function is, for a given intermolecular potential, afunction of T, T, and p = N/V. However, we follow the usual practice of only explicitly writing down the r dependence.
2
/
Jovrnol of Chemicol Education
spherical shell of thickness dr surrounding a molecule a t the origin is zero for r < c and (N/V)g(r)4ar2 dr for r > a and eqn. (2) follows immediately. We have put a subscript zero on g(r) in (2) to emphasize that it is the radial distribution function of a hard-sphere gas and not that for the liquid itself. Before the consequences of eqn. (1) can be investigated it is clear that the properties of the hard-sphere gas must he well-understood. Van der Waals, of course, lacked this knowledge. Thus, he made the further assumption that the molecules are randomly distributed (i.e., go@)= 1). From this it follows that 4 is proportional to p : +,
2Na = --
(3)
v
If eqn. (3) is used the precise form of u(r) affects the value of a hut does not otherwise affect the results. Van der Waals approximated A. by assuming it to he the free energy of a perfect gas with Ti replaced by a smaller "free volume," V,, because the molecules themselves occupy a finite volume. Thus where v,=v-Nb A
=
h/(2mkT)'h
his Planck's constant, k is Boltzmann's constant, and m is the molecular mass. The parameter b is given by:
because, when two molecules collide, the center of mass of one of the molecules is excluded from a volume of 4aa3/3. This excluded volume is divided by two because it is shared by two molecules. Combining eqns. (1) and (3) to (6) and differentiating with respect to V yields the van der Waals equation of state:
(P + 7 '7)(V - N b )
=
NkT
Van der Waals, Of course, did not Obtain (8) in this manner. However, the method we have used is equivalent to his but leads more easilyto generalizations of (8). ~f the valuesof a and b given by (2), (3), and (7) are used, eqn. yields which are in poor agreementwith experimental data. The situation may be improved somewhat by regarding a and b as parameters to be varied so as to fit *me experimental data. However, case the results are only fair, even in All this is, of course, well-known. What is not wellknown is the unsatisfactory results obtained from the
J. A. Barker and D. Henderson1
C.S.I.R.O. Chemical Research Laboratories Melbourne, Australia
I
1
The Equation of State of Simple Liquids
I
It is almost one hundred years since van der Waals (1) made the first theoretical attempt to predict the equation of state of a liquid. During the intervening years great progress has been made in the theory of solids and gases. However, until recently, little progress has been made in developing a theory even for simple liquids. By a simple liquid we mean one composed of molecules, such as those of the heavy inert gases, which are spherical, non-polar, and which have a mass large enough so that quantum effects may be neglected. This discussion will be limited to such liquids. There are two basic assumptions in the van der Waals theory. Firstly, it is assumed that two molecules attract each other according to some definite law as long as their separation exceeds a (the diameter of a molecule) but a t the intermolecular distance a the attraction ceases and is replaced by a infinite repulsion. Secondly, it is assumed that (particularly at high densities) the structure of the liquid is determined primarily by the repulsive forces and that the effect of the attractive forces is to provide a uniform background potential in which the molecules move. With these assumptions the Helmholtz function, A, of a liquid of N molecules occupying a volume V and a t a temperature T will be that of a gas of hard spheres2 of diameter a at the same T and V except that it is lowered because of the background potential field. Thus A - Aa
=
'ISNQ
(1)
where A. is the free energy of the hard-sphere gas, @ =
[" u(r) N go.o(r)4mzdr
spherical shell of thickness dr surrounding a molecule a t the origin is zero for r < a and (N/V)g(r)4ar2dr for r > a and eqn. (2) follows immediately. We have put a subscript zero on g(r) in (2) to emphasize that it is the radial distribution function of a hard-sphere gas and not that for the liquid itself. Before the consequences of eqn. (1) can be investigated it is clear that the properties of the hard-sphere gas must be well-understood. Van der Waals, of course, lacked this knowledge. Thus, he made the further assumption that the molecules are randomly distributed (i.e., go(r) = 1). From this it follows that -$ is proportional to p: @
2Na = --
(3)
v
If eqn. (3) is used the precise form of u(r) affects the value of a but does not otherwise affect the results. Van der Waals approximated A. by assuming it to he the free energy of a perfect gas with V replaced by a smaller "free volume," V,, because the molecules themselves occupy a finite volume. Thus Ao/NkT = 3 i n A
- 1 -lnV, - InN
(4)
where V, = V h
=
- Nb
h/(2rmkT)'h
h is Planck's constant, k is Boltzmann's constant, and m is the molecular mass. The parameter b is given by:
(2)
J o
and u(r) is the intermolecular potential. The factor of in eqn. (1) arises because the energy 6 is shared by two molecules and hence would be counted twice if this factor were not inserted. The function3 g(r) is the radial distribution function and is the average density of molecules a t a distance r from another molecule located at the origin divided by the overall density of molecules, p = N / V . Thus, the average number of molecules in a This research has been supported in part by grants from the National Research Council of Canads and the U.S. Department of theInterior, Officeof Saline Water. Alfred P. Sloan FoundationFellow and IanPotter Foundation Fellow. Permanent address: Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. On leave 196647. gas of hard spheres is one in which the molecules exert no farces on each other until they come into contact (i.e., equals the diameter of the hard spheres) and then exert an infinite repulsion. 8 The radial distribution function is, for s given intermolecular potential, B function of r, T, and p = N/V. However, we follow the usual practice of only explicitly writing down the T dependence.
2
/
Journal of Chemical Education
because, when two molecules collide, the center of mass of one of the molecules is excluded from a volume of 4av3/3. This excluded volume is divided by two hecause it is shared by two molecules. Combining eqns. (1) and (3) to (6) and differentiating with respect to V yields the van der Waals equation of state: (V - Nb) (p + , 7)
=
NkT
(8)
Van der waals, of course, did not obtain (8) in this manner. However, the method we have used is equivalent to his but leads more easily to generalizations of (8). 1fthe yaluesof a and b given by (2), (3), and (7) are used, eqn. (8) yields results which are in poor agreement with experimental data. The situation may he improved somewhat by regarding a and b as parameters to he varied SO as to fit some experimental data. However, are only fair. even in this case the All this is, of course, well-known. What is not wellknown is the unsatisfactory results obtained from the
The differentiations in (18) are lengthy and involved and are omitted. However, the final result is remarkably simple (8). It is
~ the free energy, radial diswhere Ao, go and ( b p / b ~ )are tribution function, and compressibility of a system of hard spheres of diameter d. The first-order terms in (19) are exact whereas the second-order term in y2is approximate. The y2 term is much smaller than the y term so that this does not greatly affect our results. The ay and a2 terms have been omitted. They should he very small for a potential which has the physically necessary property of rising steeply in the repulsive region. For a = y = 1eqn. (19) is the free energy for the potential U(T). We still have the parameters a and d a t our disposal. The most reasonable choice of o is the value of r for which u(r) = 0. We choose for d the value given by:
This choice makes d dependent on T but not p and has the effect of making the term of order or zero. For go and (bp/i)p), one could use results from machine calculations. However, since very good analytic expressions are obtained from the PY theory it is more convenient to use these expressions. Thus,
Figure 1. Equation of state for the *quare-well potential. The curves ore isothermr colcvlated from eqnr (211 and (221. The points given by the open circle ore molecular dynamics valuer colevlated by Alder and Wainwright (3) ot r / k T = 0 while the four other liner are molecular dynamics valuer calculated by Alder I101 a t e/kT = 0.331, 0.478 -0.59 and -0.87.
chine calculations. The parameter Vo= No2/ -\/Z is the volume of the system when close-packed. The agreement is excellent. Our calculations are also in good agreement with Rotenherg's results. I n Table 1 we compare the critical constants obtained from our perturbation theory with those obtained by Alder. Here again the agreement is good. The most pleasing aspect of our calculations is that our perturbation series converges very rapidly. Even a t the lowest temperatures the term of order y2 is a t least an order of magnitude smaller than the term of Table I.
where ?I = apd3/6. The equation of state, the entropy, and the internal energy can he obtained by numerically differentiating (21).
Critical Constants for the Square-well Potential (X = 1.5)
Alder (10)
Present Cdo.
Potentials with a Hard Core
For molecules with a hard core u(r) = m for r < a and thus f(r) = -1 for r < o. From this and (20) it follows that d = a. Hence, to first-order, (21) is identical with (1) and (2). Our perturhation theory is thus a generalization of the van der Waals theory. We have calculated (8) the equation of state for the squarewell potential:
This is the most interesting potential with a hard core since "quasi-experimental" Monte-Carlo and molecular dynamics machine calculations for h = 1.5 have recently been made by Rotenberg (9) and Alder (lo), respectively. I n Figure 1we compare the results of our calculations of the equation of state with Alder's ma4 / Journal of Chemical Educafion
order y. This coupled with the good agreement of our results with the machine calculations strongly suggests that the higher-order terms can be neglected (except perhaps in the region of the critical point). Potentials with a Soft Core
Our perturhation theory represents an advance over the van der Waals theory not only because t,he term of order yZis included but also because it can he applied to potentials with a soft core. We have recently calculated the propert,ies of a liquid whose molecules interact according to the 6:12 potential:
Table 2.
Values of d/rr for the
6:12 Potential
AT/.
d/a
Recently, Monte Carlo machine calculations have been made by Wood and Parker ( 1 1 ) and by McDonald and Singer (12) and molecular dynamics machine calcula tions have been made by Verlet (18). Except near the critical point, the results of these machine calculations are in good agreement with the experimental properties of argon (14-20). This indicates that although (23) is not a good representation of the true pair potential it is a good efeetive pair potential at high densities. I n Table 2 we have listed values of d at a few temperatures. I n Figure 2 we compare our values of the e q u a
Figure 3. Reduced volume at saturated vapor pressures for tho 6:12 potentid. The curve giver the rervltr of eqnr. (21) ond (221 and the points ond are machine calculation vduer and ore token from given by (12) and 1131, respectively. The pointr given by 0,0, and 0 ore experimental value. and ore taken fmm 1151, (161, and 120). rerpectivaly.
e
4
I n Figure 3 we have plotted our calculations of the volume of liquid argon a t saturated vapor pressures. The results of our perturbation calculations are in good agreement with the machine calculations and experimental results. I n Table 3 we have listed our values of the critical constants together with Verlet's molecular dynamics estimates and the experimental values for Table 3.
Critical Constants for the 6: 12 Potential
Present Calo.
Verlet (15)
Exot (16)
argon (16). The agreement is reasonably satisfactory. Our results are appreciably closer to the molecular dynamics results than to the experimental results. Our perturbation theory gives good results for the entropy of a liquid as well as the equation of state. This can be seen from Figure 4 where we have plotted
Figure 2. Equation of slate for the 6: 12 potential. The curves are isotherms calculated from eqnr I211 and 123) ond are labelled with the appropriate volve of T*. The pointr given by 0, and 0 ore machine colcvlation valuer and are taken fram (1 I), 112). and 1131, re~pestirely. The pointr given by O were calculated udng the soefficienh in o Rve-term density expansion (21) and the points given by X, ond ore experimentd valuer and are token fram 1171, 1181, and (191, respectively
+,
+
e,
tion of state with the results of the machine calculations, with results, valid a t low densities, based on the calculation (21) of the first five coefficients in an expansion of the equation of state in powers of the density, and with the experimental results for argon. I n reducing the experimental data we used the parameters e l k = 119.B°K and o = 3.405&. derived by Michels (14) from the second virial coefficientof argon and defined the following reduced quantities: T*
=
AT/.,
V*
=
V/Nm3,
p* =
I/#*,
p*
=
pva/e (24)
%* Figure 4. Reduced vapor pressures for the 6 ' 1 2 potential. p he curve giver the results of eqnr (21) and (221 ond points are experimental valuer (16).
Volume 45, Number
I, January 1968
/
5
the vapor pressure as a function of the temperature. Again the agreement with experimental results is good. As in the application to the square-well potential, the convergence of the expansion is very rapid. There is no indication (except presumably near the critical point) that the higher-order terms make an appreciable contribution.
nineteenth century he was able to give an essentially correct physical description of the liquid state and that much of the recent progress has been to put his ideas on a rigorous mathematical basis.
Summary
The authors are grateful to Drs. Alder, Rotenberg, and Verlet for sending them the results of the machine calculations.
In this note we have described a systematic perturbation theory of liquids which yields results which are in good agreement with both the machine calculation results and with the properties of liquid argon. In fact our results are better than those of any other non-empirical theory presently available. Much, of course, remains to be done. At preserit we are calculating the term of order r2in the free energy by Monte-Carlo methods. From our preliminary results, it appears that our approximate expression is somewhat small at high densities and that if the exact results are used the agreement with experiment and with machine calculations will be improved. In addition, we are applying this approach to the theory of mixtures. This may very well be the most exciting application of our perturbation theory. As we have pointed out, the essence of the van der Waals theory is that a liquid can be considered, to a good approximation, to be a hard-sphere gas moving in a uniform background potential provided by the attractive forces and that the structure and entropy of the liquid are unaffected by the attractive forces. I n this paper we have shown that this is a good first approxima, tion and have shown how this idea can be incorporated into a systematic theory of the equilibrium properties of liquids. Recent work by Alder (%%) suggests that a similar approach leads to a satisfactory theory of transport properties. I n a recent review of a book on the liquid state a reviewer stated: "There is a saying that the nineteenth century was the era of the gaseous state, the twentieth century of the solid state, and that perhaps by the twenty-first century we may understand something about liquids." This is certainly a widely held view but, in view of our results, is unduly pessimistic. It would appear that the era of the liquid state spans both the nineteenth and twentieth centuries. It is a tribute to the insight of van der Waals that in the
Acknowledgments
Literature Cited (1) VAN DER WAALS,J. D., Thesis, Leiden (1873). ( 2 ) WOOD,W. W., PARKER,F. R., A N D J.
