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The chemical potential of a liquid on the liquid-vapour coexistence line and the lattice-gas model
Volume
125. number
2
CHEMICAL
PHYSICS
LETTERS
28 March
1986
THE CHEMICAL POTENTIAL OF A LIQUID ON THE LIQUID-VAPOUR COEXISTENCE LINE AND THE LAITICE-GAS MODEL
J.G. POWLES Physics Depariment, University of Kent, Canterbury CT2 7NZ, UK Received
22 November
1985; in final form 10 January
1986
The lattice-gas model predicts that a certain chemical potential is constant along the liquid-vapour coexistence tine. It is shown that this is true to a few percent for two simulated atomic liquids and for several real liquids (argon, nitrogen and hydrogen chloride). More complex liquids (water) show substantial deviations.
model) and in fact J = f E, so that,
1. Introduction Lee and Yang [I]
showed that Ising model for ferromagnetism could be reinterpreted as a model for a liquid, the so-called lattice-gas model. This model reproduces quite well the properties of a typical simple liquid, but attention has been directed mostly to the behaviour of the coexisting liquid and vapour near the critical point. Although conceptually attractive the model has been little used for the analysis of the properties of real, or simulated, liquids. Indeed the standard books on liquids either do not mention the model at all [2-61 or only very briefly [7-91. However, the model and the translation from Ising to lattice-gas is discussed nicely and in some detail in refs. [IO-121. It is an exact result of this model that the chemical potential, ~1,is given by, /A*=-2qJ-H,
(1)
where (in notation common to the Ising formulation) 4 is the maximum number of nearest neighbours and J is an interaction constant (positive for a ferromagnet). H is proportional to the applied magnetic field. Moreover for the coexisting liquid and vapour we have to put H = 0. We use the subscript, I, to distinguish this chemical potential from others introduced below. We now introduce the interaction energy, e, between nearest neighbour atoms (for other than nearest neighbours the interaction is zero in this 0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CCI~ex=-~4e.
(2)
It is a remarkable fact, not previously stated explicitly to my iknowledge, that this ccfor the latticegas model is a constant, independent of temperature, along the liquid-gas coexistence line. In this paper we discuss the validity of relation (2) for some simulated and some real liquids. However before doing this we have to note that /.+ is not the conventional chemical potential as used in the statistical mechanics of liquids. As explained below, there is a contribution, i kT911, which is sometimes called the kinetic term, which is omitted. It involves the particle mass, m, and Planck’s constant, h (see eq. (5)) which are not’ explicitly involved in this static classical model. This is of no consequence in all pre/vious analyses of the lattice-gas model since these have, as far as I can discover, all been concerned with the behaviour on isotherms. Of course when moving along the coexistence line the temperature is the independent variable and usually i kTBI, is a large contribution to ~1. Before proceeding further it is convenient, especially for the simulation results, to use dimensionless units. We reduce with the energy parameter, E, and a length parameter, o, the latter being, in the simplest model, a molecular diameter and sets the length scale of the “lattice”. Thus c(* = p/e, T* = kT/e and p* = pus. Henceforward all quantities will be in these re113
Volume 125, number 2
CHEMICAL PHYSICS LETTERS
duced dimensionless units, but for simplicity we omit the asterisk. It is useful to divide p into two parts, P = Pr + #d ,
(3)
where pti is the value for the ideal gas at the same temperature and density, p, and, @=Tlnp-iTlnT+;TSX,
(4)
where 5X= ln(h2/2nm ea2)
(5)
for atoms, with straightforward generalisation to molecules. pT is the “residual” contribution due to the interactions between the particles. For argon (m = 39.9 amu, e/k = 119.8 K and u = 3.405 A), 81 is -5.2 1, so that the 91 term in (4) is substantial when T is of order one, as stated above. As discussed in section 2, p* is the quantity usually determined in a simulation [ 131. In fact in ref. [ 131 we introduced the quantity, pl , where, /.ll =/.l-ST%,
(6)
so that it is in fact, /.Q which is the quantity, pI, discussed in the Ising model. For liquids we use the notation, pl , henceforward. The result for the latticegas model is therefore that, independent of temperature (N.B. reduced units), Pl coex
Z-f&
(7)
In particular for a simple-cubic lattice (4 = 6), ccl mx=-3
(7a)
and, for a body-centered cubic lattice (Q = 8), Pl mex=-4, (7b) etc. (7) is the relation which we now test for simulated and real liquids.
2. Simulated liquids 2.1. LJsf3
We have already calculated ccl coex (we call it just pl henceforward), for several temperatures in ref. [ 131 for the LJsf3 potential (the LennardJones 12-6 potential (@= 4(r-12 - r-6)) cut off, shifted and twisted 114
28 March 1986
CHEMICAL PHYSICS LETTERS
Volume 125, number 2
-rT-T77l Tt -4.
where L is the side of the simulation basis cell). However, quite reliable corrections can be made to allow for this [ 1.51.The simulation results for the pressure, P, for the W fluid have been conflated by Nicolas et al. [ 141 and P@, T) is given in parametric form, with 33 constants. The quantity, r_cT, can be calculated from the relation [ 131
Tc
LJ
h_,
28 March 1986
P
CL’=T
s (P/p’T-l)(l/p’)dp’+P/p-T,
(8)
0
0.6
0.8
1.0
1.2
1.4
Fig. l. The modified chemical potential (see eq. (6)) on the liquid-vapour coexistence line as a function of temperature (both in reduced units) for two simulated liquids, one with a cut-off, shifted and twisted Lennard-Jones 12-6 potential [ 141 LJsf3, data from ref. [ 131 extended, and a second for the complete LennardJones 12-6 potential, LJ. The left-hand bracket indicates the triple point temperature, when known. The right-hand brackets indicate the critical temperature. Note the supressed zero and the large scale. The point at the far right for LJsf3 is for the fluid at the critical density.
so that both the potential and the force are zero for T> 3 [ 141. Quoting from ref. [ 131, with the addition of two extra temperatures, we have the values given in table 1.These values are plotted on fig. L Note the suppressed zero and large scale for ccl. A value for, T = 1.28 for p = pc = 0.30, above the critical temperature, T, = 1.14, is also given. The chemical potential on the coexistence line is indeed remarkably constant, to f 3%, over the whole temperature range from the triple point (which is not known but probably, q w 0.63) to the critical point. It actually has a shallow minimum. Moreover, /+ x-
3.
‘This suggests (cf. (7a)) that the lattice-gas, simple-cubic, model corresponds closely to the L&f3 model liquid, at least for this property. The extent to which it does in other respects is worthy of investigation and will be discussed elsewhere. 2.2. LJ The Lennard-Jones 12.-6 model with the complete potential cannot, strictly speaking, be simulated since the potential must be cut-off somewhere (r,
and hence p1 from (3) (4) and (6). On the coexistence line, T, P and ccl are the same for the coexisting liquid and vapour phases and so the coexistence value of ~1~ (and P,, pp and p,) can be determined. The values of P, and the p are given in ref. [ 161,but the actual values of ccl were not given. We have calculated them, via (8) and they are given in table 1 and fig. L Again the chemical potential, PI, has a shallow minimum within the coexistence and only varies by about f 3% from the triplet point (Tt = 0.68 [17]) to the critical point although the range of temperature is large (T,/T, w 2). In this case, ccl m-4, which is the value (cf. (7b)) for the body-centred cubic lattice gas. It would seem that the value of the chemical potential, but not the constancy, is sensitive to the form of the potential.
3. Real liquids We now test the validity of eq. (7) for a variety of real liquids using experimental measurements to calculate ccl. The chemical potential (strictly @) can be calculated from the power series in the density (e.g. eq. (5.11) of ref. [ 131) but this series is slowly convergent and normally only the second virial, B2(T), is well known, and only rough values of B3(T). However, as’ I have pointed out elsewhere, the power series in the pressure (eq. (14) of ref. [IS]), rather than the density, is much more rapidly convergent because P already implicitly contains higher-order virial coefficients. Moreover, as shown in ref. [ 181 (appendix B) it is usually a good approximation to neglect B3 with respect to 822, etc. (remember, B2 is actually 4103, 115
CHEMICAL PHYSICS LETTERS
Volume 125, number 2
etc.), In the following formulae we mean 82 per particle. In this way we find, after some manipulation, p1 =-;T
In. T+ T ln Pv + B2Pv-a (Bl-
t ($B; - B2B3 t $B,>P;/?
B3>PzlT
+ ... .
(9)
Note: This method avoids the explicit evaluation of
the integral in (8). In this equation Pv is the vapour pressure, which is usually available, as is B2. In the following we use the recommended values of B2 in ref. [ 191, unless otherwise stated, and when+ is avaliable we have confirmed that the approximation to ignore B,, in (9) for n > 2 is adequate. Although we are dealing with real liquids it is still convenient to express the results in reduced units and we use the values of E and u given in ref. [20] and, when these are lacking, those in ref. [21]. They were obtained by fitting the equation of state and/or the viscosity of the dilute gas. The values used are given in table 1. We have calculated p1 for argon (the prototype atomic liquid), nitrogen (the prototype non-polar diatomic liquid), carbon tetrachloride (roughly spherical molecules), hydrogen chloride (a strongly polar liquid) and finally, for water (the most important liquid). The vapour pressures are conviently tabulated in ref. [8] (where they are called p,). The values of ccl are given in table 1 and are plotted on fig. 2. For argon pl has a shallow minimum and the values
+rYFFRn --e---c -3
0.6
-0.6
T1.0
1.2
1.4
Fig. 2. As fig. 1 but for a selection of real liquids using reduction parameters (E and u) as given in table 1. Note the shifted scale of p1 with respect to fig. L Note also that the result for argon is very similar to that for LJ in fig. L
116
28 March 1986
are quite close to those for LJ liquid. This is consisten with the well known fact that the thermodynamic, am indeed other, properties of liquid argon are well reproduced by assuming that the effective pair interatomic interaction is of LJ form [ 15,221. Since the other “rare-gas” liquids are known to obey the law of corresponding states (except for helium at low temperatures and to a lesser extend neon) there is no need to test them in the present respect. Nitrogen has a shallow minimum in PI at about the same value as for LJ consistent with the fact that its thermodynamics is broadly of LJ form [ 231. No doubt similar results would be obtained for oxygen. The rather polar liquid hydrogen chloride (B2 from ref. [24]) also has a shallow minimum in p1 at a value comparable with that for the rather short range potential, LJsf3, in spite of the long range of the multipolar interactions for this molecule, which has large dipole and quadrupole moments. For carbon tetrachloride values for B2 are only quoted for a very restricted range of temperature (320 to 420 K) far from Tt (= 25OK) and Tc (= 556 K). However, for T < 320 K for this material even the B2 terms in (9) are quite small (the vapour is almost a perfect gas) so that a quite reliable estimate of pl can be obtained. The results for the low temperatures are given in table 1 and fig. 2. (The critical point is T = 1.34.) Thus in this case /.I~is far from constant and has a large (negative) value. The reason for this is not clear but it is worth noting that carbon tetrachloride, although globally non-polar, has strong bond dipole moments and for the present purpose may be more like water (see below). For water ample data are available for P, and for 4. The results are given in table 1 and fig. 2. In this case /.fl shows a minimum near T, and rises sharply as Tt it approached. Also the absolute value of cc1 is unusually large even at the minimum. Were it not for the behaviour of carbon tetrachloride it would be tempting to attribute the behaviour of water at low temperatures to the well known anomalies in liquid water in the low temperature region. Finally we note that the results as presented are dependent on the choice of the values of the parameters (E, a) and the way they are used in the lattice-gas model. Thus the choice of the pair-interaction energy to be the E in the effective LennardJones potential and the scale of the lattice to be the corresponding u
Volume 125, number 2
CHEMICAL PHYSICS LETTERS
warrants closer investigation, However, the choice is not critical. Thus it may be shown that the constancy of /.fl coex is not very sensitive to (e, u) although its actual value is rather more so. Furthermore, when there are two sets of (E, a) available, say from refs. [202 11, the BI term in eq. (6) only changes by a few tenths of a unit, which hardly changes the results in fig. 2.
4. Conclusion We have shown that the constancy of the modified chemical potential for the coexisting liquid predicted by the lattice-gas model is quite well obeyed for many real and simulated liquids, especially for the simplest liquids. For liquids when the molecules have more complex interaction potentials there are appreciable deviations. However, in all cases it has a minimum in the coexistence range. Our results suggest that a more detailed comparison of the lattice-gas models with realistic liquids, not only for the coexistence properties, would be of interest. Moreover the present analysis directs attention to the possibility that other theories of liquids may imply that the modified chemical potential is almost constant along the coexistence line. In this respect the generalised van der Waals and the penetrable-sphere models seem good candidates (e.g. see eqs. (5.9) and (5.11) of ref.
[lOI>. Again, our results may be useful in technical applications. For instance, if the vapour pressure is known at one temperature its value at another temperature can be predicted - effectively by use of eq. (9) and its inverse, say. This may be more effective than currently used empirical relations such as the Antoine equation [ 251.
Acknowledgement The author thanks the Science and Engineering Research Council for encouragement and J.S. Rowlinson (Oxford) and S. Murad (Illinois) for helpful suggestions.
28 March 1986
References
ill T.D. Lee and C.N. Yang, Phys. Rev. 87 (1952) 410. I21 P.A. Egelstaff, An introduction to the liquid state (Academic Press, New York, 1967). (31 H.L. Frisch and Z.W. Salzburg, Simple dense fluids (Academic Press, New York, 1968). [41 C.A. Croxton, Liquid state physics (Cambridge Univ. Press, Cambridge, 1974). 151 R.O. Watts and I.J. McGee, Liquid state chemical physics (Wiley, New York, 1976). 161 J.N. Murrell and E.A. Boucher, Properties of liquids and solutions (Wiley, New York, 1982). [71 W.W. Wood, In: Physics of simple liquids, eds. H.N.V. Temperley, J.S. Rowlinson and G.S. Rushbrookie (North-Holland, Amsterdam, 1968). [81 J.S. Rowlinson, Liquids and liquid mixtures, 2nd Ed. (Butterworths, London, 1969); J.S. Rowlinson and F.L. Swinton, Liquids and liquid mixtures, 3rd Ed. (Butterworths, London, 1982). PI J.-P. Hansen and I.R. McDonald, Theory of simple liquids (Academic Press, New York, 1976). [lOI J.S. Rowlinson and B. Widom, Molecular theory of capillarity (Clarendon Press, Oxford, 1982). illI R. Brout, Phase transitions (Benjamin, New York, 1965). [121 R.K. Pathria, Statistical mechanics (Pergamon, Oxford, 1972). 1131 J.G. Powles, W.A.B. Evans and N. Quirke, Mol. Phys. 46 (1982) 1347;Sl (1984) 1511. U41 J.J. Nicolas, K.E. Cubbins, W.B. Streett and D.J. Tildesley, Mol. Phys. 34 (1979) 1429. 1151 L. Verlet, Phys. Rev. 159 (1967) 98. [I61 J.G. Powles, Physica 126A (1984) 289. [I71 J.-P. Hansen and L. Verlet, Phys. Rev. 184 (1969) 151. 1181 J.G. Powles, Mol. Phys. 41 (1980) 715. [W J.H. Dymond and E.B. Smith, The virial coefficients of gases (Oxford Univ. Press, Oxford, 1969,198O). WI J.O. Hirschfelder, CF. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). WI F.M. Mourits and F.H.A. Rummens, Can. J. Chem. 55 (1977) 3007. WI I.R. McDonald and K. Singer, J. Chem. Phys. 47 (1967) 4766; 50 (1969) 2308. (231 I.R. McDonald and K. Singer. Mol. Phys. 23 (1972) 29. v41 B. Schramm and U. Leuchs, Ber. Bunsenges. PhysIk. Chem. 83 (1979) 847. WI J.M. Prausnitz, C.A. Eckert, R.V. Orye and J.P. O’Connell, Computer calculations for multicomponents vapour-liquid equilibria (Prentice-Hall, Englewood Cliffs, 1967).
117
125. number
2
CHEMICAL
PHYSICS
LETTERS
28 March
1986
THE CHEMICAL POTENTIAL OF A LIQUID ON THE LIQUID-VAPOUR COEXISTENCE LINE AND THE LAITICE-GAS MODEL
J.G. POWLES Physics Depariment, University of Kent, Canterbury CT2 7NZ, UK Received
22 November
1985; in final form 10 January
1986
The lattice-gas model predicts that a certain chemical potential is constant along the liquid-vapour coexistence tine. It is shown that this is true to a few percent for two simulated atomic liquids and for several real liquids (argon, nitrogen and hydrogen chloride). More complex liquids (water) show substantial deviations.
model) and in fact J = f E, so that,
1. Introduction Lee and Yang [I]
showed that Ising model for ferromagnetism could be reinterpreted as a model for a liquid, the so-called lattice-gas model. This model reproduces quite well the properties of a typical simple liquid, but attention has been directed mostly to the behaviour of the coexisting liquid and vapour near the critical point. Although conceptually attractive the model has been little used for the analysis of the properties of real, or simulated, liquids. Indeed the standard books on liquids either do not mention the model at all [2-61 or only very briefly [7-91. However, the model and the translation from Ising to lattice-gas is discussed nicely and in some detail in refs. [IO-121. It is an exact result of this model that the chemical potential, ~1,is given by, /A*=-2qJ-H,
(1)
where (in notation common to the Ising formulation) 4 is the maximum number of nearest neighbours and J is an interaction constant (positive for a ferromagnet). H is proportional to the applied magnetic field. Moreover for the coexisting liquid and vapour we have to put H = 0. We use the subscript, I, to distinguish this chemical potential from others introduced below. We now introduce the interaction energy, e, between nearest neighbour atoms (for other than nearest neighbours the interaction is zero in this 0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CCI~ex=-~4e.
(2)
It is a remarkable fact, not previously stated explicitly to my iknowledge, that this ccfor the latticegas model is a constant, independent of temperature, along the liquid-gas coexistence line. In this paper we discuss the validity of relation (2) for some simulated and some real liquids. However before doing this we have to note that /.+ is not the conventional chemical potential as used in the statistical mechanics of liquids. As explained below, there is a contribution, i kT911, which is sometimes called the kinetic term, which is omitted. It involves the particle mass, m, and Planck’s constant, h (see eq. (5)) which are not’ explicitly involved in this static classical model. This is of no consequence in all pre/vious analyses of the lattice-gas model since these have, as far as I can discover, all been concerned with the behaviour on isotherms. Of course when moving along the coexistence line the temperature is the independent variable and usually i kTBI, is a large contribution to ~1. Before proceeding further it is convenient, especially for the simulation results, to use dimensionless units. We reduce with the energy parameter, E, and a length parameter, o, the latter being, in the simplest model, a molecular diameter and sets the length scale of the “lattice”. Thus c(* = p/e, T* = kT/e and p* = pus. Henceforward all quantities will be in these re113
Volume 125, number 2
CHEMICAL PHYSICS LETTERS
duced dimensionless units, but for simplicity we omit the asterisk. It is useful to divide p into two parts, P = Pr + #d ,
(3)
where pti is the value for the ideal gas at the same temperature and density, p, and, @=Tlnp-iTlnT+;TSX,
(4)
where 5X= ln(h2/2nm ea2)
(5)
for atoms, with straightforward generalisation to molecules. pT is the “residual” contribution due to the interactions between the particles. For argon (m = 39.9 amu, e/k = 119.8 K and u = 3.405 A), 81 is -5.2 1, so that the 91 term in (4) is substantial when T is of order one, as stated above. As discussed in section 2, p* is the quantity usually determined in a simulation [ 131. In fact in ref. [ 131 we introduced the quantity, pl , where, /.ll =/.l-ST%,
(6)
so that it is in fact, /.Q which is the quantity, pI, discussed in the Ising model. For liquids we use the notation, pl , henceforward. The result for the latticegas model is therefore that, independent of temperature (N.B. reduced units), Pl coex
Z-f&
(7)
In particular for a simple-cubic lattice (4 = 6), ccl mx=-3
(7a)
and, for a body-centered cubic lattice (Q = 8), Pl mex=-4, (7b) etc. (7) is the relation which we now test for simulated and real liquids.
2. Simulated liquids 2.1. LJsf3
We have already calculated ccl coex (we call it just pl henceforward), for several temperatures in ref. [ 131 for the LJsf3 potential (the LennardJones 12-6 potential (@= 4(r-12 - r-6)) cut off, shifted and twisted 114
28 March 1986
CHEMICAL PHYSICS LETTERS
Volume 125, number 2
-rT-T77l Tt -4.
where L is the side of the simulation basis cell). However, quite reliable corrections can be made to allow for this [ 1.51.The simulation results for the pressure, P, for the W fluid have been conflated by Nicolas et al. [ 141 and P@, T) is given in parametric form, with 33 constants. The quantity, r_cT, can be calculated from the relation [ 131
Tc
LJ
h_,
28 March 1986
P
CL’=T
s (P/p’T-l)(l/p’)dp’+P/p-T,
(8)
0
0.6
0.8
1.0
1.2
1.4
Fig. l. The modified chemical potential (see eq. (6)) on the liquid-vapour coexistence line as a function of temperature (both in reduced units) for two simulated liquids, one with a cut-off, shifted and twisted Lennard-Jones 12-6 potential [ 141 LJsf3, data from ref. [ 131 extended, and a second for the complete LennardJones 12-6 potential, LJ. The left-hand bracket indicates the triple point temperature, when known. The right-hand brackets indicate the critical temperature. Note the supressed zero and the large scale. The point at the far right for LJsf3 is for the fluid at the critical density.
so that both the potential and the force are zero for T> 3 [ 141. Quoting from ref. [ 131, with the addition of two extra temperatures, we have the values given in table 1.These values are plotted on fig. L Note the suppressed zero and large scale for ccl. A value for, T = 1.28 for p = pc = 0.30, above the critical temperature, T, = 1.14, is also given. The chemical potential on the coexistence line is indeed remarkably constant, to f 3%, over the whole temperature range from the triple point (which is not known but probably, q w 0.63) to the critical point. It actually has a shallow minimum. Moreover, /+ x-
3.
‘This suggests (cf. (7a)) that the lattice-gas, simple-cubic, model corresponds closely to the L&f3 model liquid, at least for this property. The extent to which it does in other respects is worthy of investigation and will be discussed elsewhere. 2.2. LJ The Lennard-Jones 12.-6 model with the complete potential cannot, strictly speaking, be simulated since the potential must be cut-off somewhere (r,
and hence p1 from (3) (4) and (6). On the coexistence line, T, P and ccl are the same for the coexisting liquid and vapour phases and so the coexistence value of ~1~ (and P,, pp and p,) can be determined. The values of P, and the p are given in ref. [ 161,but the actual values of ccl were not given. We have calculated them, via (8) and they are given in table 1 and fig. L Again the chemical potential, PI, has a shallow minimum within the coexistence and only varies by about f 3% from the triplet point (Tt = 0.68 [17]) to the critical point although the range of temperature is large (T,/T, w 2). In this case, ccl m-4, which is the value (cf. (7b)) for the body-centred cubic lattice gas. It would seem that the value of the chemical potential, but not the constancy, is sensitive to the form of the potential.
3. Real liquids We now test the validity of eq. (7) for a variety of real liquids using experimental measurements to calculate ccl. The chemical potential (strictly @) can be calculated from the power series in the density (e.g. eq. (5.11) of ref. [ 131) but this series is slowly convergent and normally only the second virial, B2(T), is well known, and only rough values of B3(T). However, as’ I have pointed out elsewhere, the power series in the pressure (eq. (14) of ref. [IS]), rather than the density, is much more rapidly convergent because P already implicitly contains higher-order virial coefficients. Moreover, as shown in ref. [ 181 (appendix B) it is usually a good approximation to neglect B3 with respect to 822, etc. (remember, B2 is actually 4103, 115
CHEMICAL PHYSICS LETTERS
Volume 125, number 2
etc.), In the following formulae we mean 82 per particle. In this way we find, after some manipulation, p1 =-;T
In. T+ T ln Pv + B2Pv-a (Bl-
t ($B; - B2B3 t $B,>P;/?
B3>PzlT
+ ... .
(9)
Note: This method avoids the explicit evaluation of
the integral in (8). In this equation Pv is the vapour pressure, which is usually available, as is B2. In the following we use the recommended values of B2 in ref. [ 191, unless otherwise stated, and when+ is avaliable we have confirmed that the approximation to ignore B,, in (9) for n > 2 is adequate. Although we are dealing with real liquids it is still convenient to express the results in reduced units and we use the values of E and u given in ref. [20] and, when these are lacking, those in ref. [21]. They were obtained by fitting the equation of state and/or the viscosity of the dilute gas. The values used are given in table 1. We have calculated p1 for argon (the prototype atomic liquid), nitrogen (the prototype non-polar diatomic liquid), carbon tetrachloride (roughly spherical molecules), hydrogen chloride (a strongly polar liquid) and finally, for water (the most important liquid). The vapour pressures are conviently tabulated in ref. [8] (where they are called p,). The values of ccl are given in table 1 and are plotted on fig. 2. For argon pl has a shallow minimum and the values
+rYFFRn --e---c -3
0.6
-0.6
T1.0
1.2
1.4
Fig. 2. As fig. 1 but for a selection of real liquids using reduction parameters (E and u) as given in table 1. Note the shifted scale of p1 with respect to fig. L Note also that the result for argon is very similar to that for LJ in fig. L
116
28 March 1986
are quite close to those for LJ liquid. This is consisten with the well known fact that the thermodynamic, am indeed other, properties of liquid argon are well reproduced by assuming that the effective pair interatomic interaction is of LJ form [ 15,221. Since the other “rare-gas” liquids are known to obey the law of corresponding states (except for helium at low temperatures and to a lesser extend neon) there is no need to test them in the present respect. Nitrogen has a shallow minimum in PI at about the same value as for LJ consistent with the fact that its thermodynamics is broadly of LJ form [ 231. No doubt similar results would be obtained for oxygen. The rather polar liquid hydrogen chloride (B2 from ref. [24]) also has a shallow minimum in p1 at a value comparable with that for the rather short range potential, LJsf3, in spite of the long range of the multipolar interactions for this molecule, which has large dipole and quadrupole moments. For carbon tetrachloride values for B2 are only quoted for a very restricted range of temperature (320 to 420 K) far from Tt (= 25OK) and Tc (= 556 K). However, for T < 320 K for this material even the B2 terms in (9) are quite small (the vapour is almost a perfect gas) so that a quite reliable estimate of pl can be obtained. The results for the low temperatures are given in table 1 and fig. 2. (The critical point is T = 1.34.) Thus in this case /.I~is far from constant and has a large (negative) value. The reason for this is not clear but it is worth noting that carbon tetrachloride, although globally non-polar, has strong bond dipole moments and for the present purpose may be more like water (see below). For water ample data are available for P, and for 4. The results are given in table 1 and fig. 2. In this case /.fl shows a minimum near T, and rises sharply as Tt it approached. Also the absolute value of cc1 is unusually large even at the minimum. Were it not for the behaviour of carbon tetrachloride it would be tempting to attribute the behaviour of water at low temperatures to the well known anomalies in liquid water in the low temperature region. Finally we note that the results as presented are dependent on the choice of the values of the parameters (E, a) and the way they are used in the lattice-gas model. Thus the choice of the pair-interaction energy to be the E in the effective LennardJones potential and the scale of the lattice to be the corresponding u
Volume 125, number 2
CHEMICAL PHYSICS LETTERS
warrants closer investigation, However, the choice is not critical. Thus it may be shown that the constancy of /.fl coex is not very sensitive to (e, u) although its actual value is rather more so. Furthermore, when there are two sets of (E, a) available, say from refs. [202 11, the BI term in eq. (6) only changes by a few tenths of a unit, which hardly changes the results in fig. 2.
4. Conclusion We have shown that the constancy of the modified chemical potential for the coexisting liquid predicted by the lattice-gas model is quite well obeyed for many real and simulated liquids, especially for the simplest liquids. For liquids when the molecules have more complex interaction potentials there are appreciable deviations. However, in all cases it has a minimum in the coexistence range. Our results suggest that a more detailed comparison of the lattice-gas models with realistic liquids, not only for the coexistence properties, would be of interest. Moreover the present analysis directs attention to the possibility that other theories of liquids may imply that the modified chemical potential is almost constant along the coexistence line. In this respect the generalised van der Waals and the penetrable-sphere models seem good candidates (e.g. see eqs. (5.9) and (5.11) of ref.
[lOI>. Again, our results may be useful in technical applications. For instance, if the vapour pressure is known at one temperature its value at another temperature can be predicted - effectively by use of eq. (9) and its inverse, say. This may be more effective than currently used empirical relations such as the Antoine equation [ 251.
Acknowledgement The author thanks the Science and Engineering Research Council for encouragement and J.S. Rowlinson (Oxford) and S. Murad (Illinois) for helpful suggestions.
28 March 1986
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