T. L. Beck
Department of Chemistry
University of Cincinnati
Cincinnati, OH 45221-0172

The Widom formula and mixture chemical potentials

These notes connect the statistical mechanical derivation of the chemical potential derived by Widom to standard bulk expressions. The purpose is to clarify the notions of standard state chemical potentials and activity coefficients in molecular language. This is a helpful way to understand the traditional textbook expressions for the chemical potential. It also gives transparent views on the origins of non-unity activity coefficients and limiting behavior of solutions. Finally, the Widom test particle approach provides a fruitful starting point for theoretical developments relevant to solvation phenomena.

Contents

1  Introduction

The chemical potential is a key thermodynamic variable which controls the transport and equilibrium distribution of matter. It is crucial for understanding phase equilibria, solvation phenomena, solute partitioning between phases, equilibrium adsorption, reversible binding, and chemical reaction equilibria.

In 1963 Widom proposed a direct molecular-level route to the chemical potential starting from the canonical ensemble expression for the Helmholtz free energy:

The classical expression for the partition function¹ is

where N is the number of particles, h is Planck's constant, k is Boltzmann's constant, m is the mass, and U is the total potential energy. The discrete approximation to the chemical potential is then

By substituting Eq. 2 into Eq. 3 and rearranging terms, we get

Here DU is the total change in the system energy upon adding another particle to the (N-1) other particles. Notice that there is no assumption of pairwise additivity in this expression. Also, the volume V has been arbitrarily inserted and can take on any value. The typical assumption would be the total system volume. By rearranging further, we can obtain the following formula:

The density of particles in the volume V is r, L is the deBroglie wavelength, the integration in the final term is a uniform integration over the system volume, and the averaging is a thermal average over all (N-1) particles signified by the ° subscript. The factor inside the averaging is the Boltzmann factor for the hypothetical insertion of the Nth `test particle' into the system.

For a mixture, one follows the same procedure and obtains the formula:

The insertion is then into a solution of (N-1) solvent particles whatever they may be. This expression is easily extended to quantum mechanics in the path integral formulation (Beck and Marchioro, 1993). The solvent averaging can be classical or quantum. The inserted particle is a cyclic polymer which interacts with the solvent but not with itself. The quantum formula appears quite similar to the classical one:

The inner averaging is over the Gaussian quantum fluctuations in the path integral expression for the solute.

The second term on the rhs of Eq. 6 is the excess chemical potential:

If the insertion probability is very low, it is clear the excess chemical potential becomes large and positive.

Eq. 5 can be solved for the density as follows:

where z_i is the absolute activity for species i. This provides the starting point for theoretical developments such as cumulant and cluster expansions (Pratt and Rempe, 1999).

2  Local nature of the Widom formula

Let us define a new partition function Z¢_N(V, T) in which the Nth particle (the test particle) is restricted to trial insertions in a small portion of a much larger system. This could correspond, for example, to sampling in two different phases at equilibrium or two regions of space accessible to a solute such as in the bulk liquid and bound to a protein. Then the integration in the second term on the rhs side of Eq. 6 only covers the zone of interest (denoted by the prime):

This expression is perfectly well-defined and could be viewed as the result of a strong external perturbing potential which didn't allow insertion outside the region of interest. The sampling volume V¢ can be shrunk to a volume of any size; the choice depends on the problem. For example, one might want to know the excess chemical potential for a solute near a pocket of an enzyme. If the excess chemical potential was desired as a function of the distance from an interface, one would take flat slabs in the z direction as the sample volumes. For the case of solvation of a single object (molecule or ion) in bulk solvent, the sampling volume V¢ can be essentially zero due to the invariance with translation in the uniform fluid; in this sense we can speak of the excess chemical potential or particle density at a `point'. In addition, the region of the solvent which must `distort' to accomodate the solute at this point is surely local. Hence, only local information is required to obtain an accurate excess chemical potential. This physical picture provides the basis for cluster expansions originating from Eq. 9. (Lawrence does this make any sense in relation to your comments in your paper on cluster expansions? This is my crude interpretation anyway.) In a sense, the Widom expression is `doubly local' in that one can compute the excess chemical potential for any desired finite volume region, and only local solvent fluctuations contribute to the insertion probabilities at a given point in space.

The chemical potential itself is constant at equilibrium so the local densities and excess chemical potentials adjust themselves to maintain that constancy. This leads directly to the formula for the solute partition coefficient between two phases in a large system:

The two sampling volumes for the calculation of the excess chemical potentials are the two bulk phases, omitting details of the interfacial region.

3  Bulk chemical potentials in the Widom language

This section derives the appropriate formulas for chemical potentials in mixtures that are standard in physical chemistry texts. The purpose is to understand exactly what are the standard state chemical potentials and the activity coefficients on the molecular scale. Also, the Henry's and Raoult's Law constants are derived. We will assume here a single homogeneous phase composed of two components [labeled by 1 (solvent) and 2 (solute)].

3.1  Chemical potential of a solute in a solvent

The traditional expression for the chemical potential of a solute in a solvent is

The reference chemical potential m₂^° is for the infinite-dilution limit, x₂ is the mole fraction, and g₂ is the activity coefficient. What is the connection of this formula to the Widom expression?

We go back to Eq. 6 and rearrange to obtain

Now, we multiply and divide by an integral over the insertion probability for a solute at infinite dilution:

Then the chemical potential can be written:

where the first term is the infinite-dilution chemical potential and the activity coefficient is

A couple of points are clear from these expressions. First, the infinite dilution chemical potential m₂^° involves only solute-solvent interactions, since the insertion probability is computed with pure solvent comprising the (N-1) particle bath. Second, the activity coefficient is a measure of deviations from the infinite-dilution limit: if finite solute concentration leads overall to more favorable insertion probabilities (due to solute-solute attractions and/or solvent reorganization due to the solutes), the activity coefficient will be less than one (and vice versa for net repulsive interactions). Also, as the solution becomes very dilute, g₂ ® 1 as it should.

A similar path leads to the chemical potential of the solvent species:

where

The superscript p denotes the pure solvent reference state, and the reference chemical potential is typically denoted by m₁^*; m₁^* involves computation of the insertion probability for a solvent test particle into the pure solvent. Then, as x₁ ® 1, g₁ ® 1.

3.2  Henry's and Raoult's Laws

It is now easy to derive the limiting laws for solutions from the above expressions. For the infinite-dilution limit solute case in equilibrium with an ideal gas, the chemical potential is

The Henry's Law expression relating the mole fraction of the volatile solute to the vapor pressure is

where K₂ is Henry's constant. Solving Eq. 19 for P₂, we get

Therefore, Henry's Law constant is

Computation of the infinite-dilution chemical potential via the Widom route thus yields Henry's Law constant.

In the same way, Raoult's law is derived as

where the pure solvent vapor pressure is

The vapor pressure of the pure solvent is determined by the pure solvent chemical potential.

4  Theoretical approximations originating from the Widom formula

Discussion of ways to proceed from Widom formula.

Footnotes:

¹ Here assumed for a single component with no internal structure; the extension to multiple particles and internal modes is straightforward