J. Phys. Chem. 1995,99, 12355-12362
12355
Statistical Mechanical Theory of the Partial and Other Derivatives of Thermodynamic Quantities of Liquid Mixtures Masaharu Ohbat and Hiroyasu Nomura* Department of Chemical Engineering, School of Engineering, Nagoya Universiv, Chikusa-ku, Nagoya-shi, 464 Japan Received: February 3, 1995; In Final Form: June 1, 1 9 9 9
The independent variables used in ordinary experimental studies are p (pressure), T (temperature), and N; (number of molecules of the ith species), while the variables in integral equation theory are V (volume), T, and e (number density of the ith species). The formula for the transformation between these different sets of independent variables has been obtained. In this formula, the isothermal compressibility, the expansibility, and the partial molar volume play an important role. The expressions of these quantities in terms of the derivatives have been obtained on the basis of either the radial distribution functions and their T or Kirkwood-Buff theory or the virial equation of state. The p, T,or Ni derivatives of the radial distribution functions have been represented in terms of the radial distribution functions and their T or ei derivatives. Then, the expressions of the p , T,or N,derivatives of the enthalpy have been obtained in terms of the radial distribution functions and their p , T, or N; derivatives. The calculation has been carried out for the case of the infinitely dilute solution of two-component Lennard-Jones liquids applied with the PY-approximation.
ei
1. Introduction In the integral equation theory of solutions, the thermodynamic state is specified by the set of independent variables V, T, and e, (the number density of the ith species). On the other hand, experimental thermodynamic quantities are usually described as functions of the independent variables p , T, and N,. To discriminate the phenomenon which takes place under the condition of constant volume from the one under the condition of constant pressure is very important. For example, we have shown that the behavior of the partial molar volume of the solute arising from the conformational change of the solvent molecules is quite different depending on whether the behavior is observed under the condition of constant volume or constant pressure.' We have also obtained the similar conclusion for variation of the heat of vaporization of the tetraatomic molecular liquid accompanied with the conformational change of molecules.2 Thus, it is highly essential in solution chemistry to obtain the statistical mechanical expressions of the partial molar thermodynamic quantities and other thermodynamic derivatives in terms of the independent variables p , T, and N, in order to shed light on the solute-solvent interactions at the molecular level. Recently, some authors have reported the theoretical investigation of the derivatives of thermodynamic quantities. Patey et al.3 have obtained an integral equation of the number density derivatives of radial distribution functions and calculated the solvation energy for Lennard-Jones mixtures and ionic solutions under the condition of constant volume. Karplus et aL4 have also studied solutions of argon or monoatomic ions and a waterlike solvent. Using the RISM-HNC theory, they obtained an expression for the solvation energy under the condition of constant volume and for the chemical potential in terms of sitesite radial distribution functions and their number density derivatives. In these works, although these have a theoretical aspect, the independent variables used are still e, and T, not p , T, and N,. In the work of Karplus et aL5 the solvation enthalpy
at constant pressure has been proposed. However, their formulas are expressed in terms of the direct correlation functions, their number density derivatives, and temperature derivatives at constant volume. Hence, their formulas are complicated and the physical meaning is difficult to grasp. Obviously, their formulas are incomplete with regard to expressing them in terms of the independent variables p , T, and Ni. If the thermodynamic quantities are expressed in terms of the radial distribution functions and theirp, T, or Ni derivatives, the formulas will provide clear physical insight. The first purpose of this work is to establish a way to calculate the p , T, or N; derivatives of radial distribution functions. The second is to obtain the formulas of the p , T, or N; derivatives of enthalpy expressed in terms of radial distribution functions and their p , T, or N; derivatives. These are discussed in section 3. In section 2, we summarize the result of integral equation theory applied for the temperature and number density derivatives of pair correlation functions. In the final section, calculated results are illustrated for the case of infinitely dilute solutions of a twocomponent Lennard-Jones liquid.
2. The Integral Equation for the Temperature and Number Density Derivatives of the Radial Distribution Functions In this section, we summarize integral equation theory for the number density derivatives of the pair correlation functions proposed by Garisto et aL3 and Yu and Karplus4 and for the temperature derivatives by Yu et aL5 For simplicity, we consider a s-component monoatomic molecular fluid. This theory can easily be extended to polyatomic molecular liquid mixtures and to the higher order derivatives of the pair correlation functions. The theory described in this section is an extension of the theory of Gan and Eu6 to multicomponent systems. The Omstein-Zemike equation for the multicomponent system is expressed in Fourier transform space as
* To whom correspondence should be addressed.
' Present
address: Kawaijuku Educational Institution. Chikusa-ku, Nagoya-shi, 464 Japan. Abstract published in Advance ACS Abstracts, July 15, 1995. @
0022-365419512099- 12355$09.00/0
h=e+i!@h where
h
(1)
and E are matrices whose elements are the Fourier
0 1995 American Chemical Society
12356 J. Phys. Chem., Vol. 99, No. 32, 1995 transforms of the total correlation function, &ij(k),and the direct Correlation function, Z&), respectively, between the ith and the jth species. The ij element of the matrix e is eidij, where dij is the Kronecker delta. &(k) and Z.u(k) are functions of the thermodynamic variables T and @i (i = 1, ..., s). Usually, the term “temperature T’means the absolute temperature. However, in this paper the absolute temperature multiplied by the Boltzmann constant is simply called the “temperature T’.By differentiating both sides of eq 1 with respect to T, we have
where hTeand fTeare the matrices whose elements are hjTe(k) and Ef,(k), respectively. Hereafter,yb denotes the a derivatives off at constant b, that is, (@&)b. The subscript @ means that the differentiation is carried out under the condition that the number densities of all species are held constant. In a similar way, by differentiating both sides of eq 1 with respect to e k , we have
Here, the subscript e’ means that the differentiation is carried out under the condition that all number densities except for @ k are held constant. Equations 2 and 3 are the Omstein-Zemike equations of the temperature and number density derivatives for pair correlation functions, respectively. The closure relation for the derivatives of the pair correlation functions is obtained by differentiating the closure relation for the pair correlation functions such as the PY approximation. The PY approximation is expressed as
Here, yij(r)= hij(r)- cij(r),Jj(r) is the Mayerffunction defined byfij(r) = exp(-uU(r)/T) - 1, and u ~ ( ris) the pair potential between two molecules of the ith and jth species. By differentiating both sides of eq 4 with respect to T and @ k , we have
Ohba and Nomura where F V ~ ,=, F/V (if F is the extensive variable) or = 0 (if F is the intensive variable). From the equation of state V = V(T,p,”, ,...As), we have
where a, is the expansibility, KT the isothermal compressibility, and vi the partial molecular volume of the ith species. From the definition of the number density of the kth species, @ k = Nk/v, the small variation of @k is
(9) After eliminating dV and d @ k from eqs 7-9, we have
d~ = (FTV,@ - a,Y) d T + KTYdp + i= 1
where
and X = F (if F is the extensive variable) or = 0 (if F is the intensive variable). From eq 10, we obtain the relations for transforming the independent variables of the derivatives of thermodynamic quantities from V , T, and (i = 1, ..., s) to p , T, and Ni (i = 1, ..., s)
F’,.~ = F ~ , -, apY
and and
respectively, wherefijr(r) = u ~ ( rexp{ ) - u ~ ( r ) / n / PEquations . 5 and 6 are the PY approximations for the T and @ k derivatives of the pair correlation functions, respectively. In a similar manner, we can obtain the closure approximations for the derivatives of the pair correlation functions in the HNC and other approximations. 3. Transformation of the Independent Variables of the Derivatives of Thermodynamic Quantities and Their Correlation Function Expressions
3.1. General Expressions for the Transformation of the Independent Variables V , T,and pi (i = 1, s) top, T,and Ni (i = 1, s) for the Derivatives of Thermodynamic Quantities. Let us consider an arbitrary extensive or intensive variable F. If independent variables are V, T, and e; (i = 1, ..., s), then the small variation of F is
...,
...,
3.2. Pair Correlation Function Expressions for the Expansibility, the Isothermal Compressibility, and the Partial Molecular Volume. In eqs 12-14, a,, KT, and v k , play a special role in the transformation of the independent variables. Thus, in order to calculate the T, p , and Nk derivatives of some thermodynamic properties based only on the integral equation theory, we must have the expressions of a,, KT, and v k in terms of pair correlation functions and their derivatives given by the theory described in section 2. There are two routes for calculating these three quantities. One is based on the virial equation of state, and the other is based on the compressibility equation of state. The virial equation of state is
where u( is the abbreviation of dug(r)/dr. Hereafter, gc(r), gi,Pkr,,,!r), etc. are shortened as gij, g$’T.@’, etc. whenever confusion does not occur. By differentiating both sides of this equation with e, (the total number density), @k and T, we have
J. Phys. Chem., Vol. 99, No. 32, 1995 12357
Partial Derivatives of Thermodynamic Quantities 1
s
s
S
Thus, from eqs 24 and 25, we have
and
1
1
s
and
n u s , we can calculate KT, vk, and a, using the pair correlation functions and their derivatives. AII alternative route for calculating KT, v k , and a, is the way starting from the Kirkwood-Buff theory and PY approximation. According to the Kirkwood-Buff theory,' KT and v k can be expressed in terms of the direct correlation functions as follows; S
;=I
Here, we use the relation qai
0
-5
z -10
.._.
-....
-15
0
0.1
0.2
0.3
b......
0.4
0.5
?a 60 Temperature dependence of
H':
p,x.
system 2, la =0.45
-20;
I
2
4
6
8
1
0
T Figure 14. Partial molecular enthalpy of the solute: (0).H>T.,,.N,; (A) U2 of eq 39; (0),-U1 of eq 39; ( x ) U3 of eq 39; (M), pV,.
arising from the change of solvent structure around the solute molecule, is small. This, however, does not mean that U3 is negligible for understanding the enthalpic behavior of solutions. The excess partial molecular enthalpy HbE and the terms contributing to it are listed in Table 2. As shown in this table, U3 plays an important role in determining the value of H b E ,
12362 J. Phys. Chem., Vol. 99, No. 32, 1995 though the contribution of U3 to H N b ~ . p . ~isa small. It is also seen in this table that all terms contributing to HbE are equivalently impgrtant and thus if we neglect one or two terms, such as U3 or pvb, on the basis of their smallness compared with H N b ~ . p . ~ we a , have an incorrect conclusion with respect to HbE. U3 of system 2 is positive whereas that of system 3 is negative. At first sight, this fact seems to contradict the result described in section 4.2, that is, the solute molecule with a strong interaction is the structure maker and that with a weak interaction is the structure breaker. However, this is not true. The packing fraction of the system calculated is so high that the average distance between solvent molecules becomes smaller than the minimum distance of the interaction potential with the
Ohba and Nomura compression of solvent around the solute molecule. Thus, U3 of system 2 takes a positive value.
References and Notes (1) Ohba, M.; Kawaizumi, F.; Nomura, H. J . Phys. Chem. 1992, 96, 5129. (2) Ohba, M.; Nomura, H. J. Phys. Chem. 1991, 95, 1399. ( 3 ) Garisto, F.; Kusalik, P. G.;Patey, G. N. J. Chem. Phys. 1983, 79, 6294. (4) Yu, H.; Karplus, M. J. Chem. Phys. 1988, 89, 2366. (5) Yu,H.; Roux, B.; Karplus, M. J. Chem. Phys. 1990, 92, 5020. (6) Gan, H. H.; Eu, B. C. J. Chem. Phys. 1992, 96, 558. (7) Kirkwood, J. G.; Buff, F. P. J . Chem. Phys. 1951, 19, 774. (8) Baxter, R. J. J . Chem. Phys. 1967,47, 4855. JP950349M
