The chemical potential of a solvent

We can anticipate that the chemical potential of a species ought to increase with concentration because the higher its concentration, the greater its chemical punch . In the following, we use J to denote a substance in general, A to denote a solvent, and B to denote a solute. 

The key to linking the properties of a solution to those of a gas and setting up an expression for the chemical potential of a solute is the work done by the French chemist Fran( ois Raoult (1830-1901), who spent most of his fife mccisuring the vapor pressures of solutions. He measured the partial vapor pressure, pj, of each component in the mixture, the partial pressure of the vapor of each component in 

The partial vapor pressure of a substance in a liquid mixture is proportional to its mole fraction in the mixture and its vapor pressure when pure p, = Xjpf 

When the mole fraction of water in an aqueous solution is 0.90, then, provided Raoult s law is obeyed, the partial vapor pressure of the water in the solution is 90 per cent that of pure water. This conclusion is approximately true whatever the identity of the solute and the solvent (Fig. 3.22). 

The molecular origin of Raoult s law is the effect of the solute on the entropy of the solution. The entropy of the solvent arises from the random locations and the thermal motion of its molecules. The vapor pressure then represents the tendency of the system and its surroundings to reach a higher entropy. When a solute is present, the molecules in the solution are more dispersed than in the pure solvent, so we cannot be sure that a molecule chosen at random will be a solvent molecule (Fig. 3.23). Because the entropy of the solution is higher than that of the pure solvent, the solution has a lower tendency to acquire an even higher entropy by the solvent vaporizing. In other words, the vapor pressure of the solvent in the solution is lower than that of the pure solvent. 

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The fact that the chemical potential of a solvent in a solution (composed of solute + solvent) is less than that of the chemical potential of the pure solvent gives rise to a number of properties ... 

The concept of a chemical potential is germane to a discussion of water activity (aw), which is technologically defined as the ratio of the equilibrium water vapor pressure over a solution or dispersion (p0) and the water vapor pressure over pure water (jb ). Also by definition, the chemical potential of a solvent ( jl0) or a solute (p ) is the rate of change in energy of either with a change only in the molal content of that component in solution. 

Whenever a solution is separated from a solvent by a membrane that is permeable only to solvent molecules (referred to as a semi-permeable membrane), there is a passage of solvent across the membrane into the solution. This is the phenomenon of osmosis. If the solution is totally confined by a semipermeable membrane and immersed in the solvent, then a pressure differential develops across the membrane, which is referred to as the osmotic pressure. Solvent passes through the membrane because of the inequality of the chemical potentials on either side of the membrane. Since the chemical potential of a solvent molecule in solution is less than that in pure solvent, solvent will spontaneously enter the solution until this inequality is removed. The equation which relates the osmotic pressure of the solution. If, to the solution concentration... 

It must be remembered that irrespective of the composition of the solvation sphere, the chemical potential of a solvent molecule in this sphere, is the same as the bulk chemical potential, since the system is in equilibrium. However, following the commencement of the dissociative act in an octahedral complex, equilibrium may not exist at a particular site and entry may depend on the immediate availability of a potential ligand rather than on the statistical population of the solvation sphere or the bulk solvent composition. 

Chemical potential is a function of concentration, temperature, and pressure. Under isothermal conditions it is solely determined by concentration and pressure. The increase of solute concentration decreases the chemical potential of a solvent, which can also be expressed by its activity according to the following relationship ... 

Osmotic pressure is related to molar mass of the solute the smaller the mass, the higher will be the pressure at the same concentration. Electrolytes show higher osmotic pressure than nonelectrolytes because each ion affects the chemical potential of a solvent. 

In the region occupied by the polymer chain, solvent molecules are mixed. Let A/ro be the chemical potential of the solvent molecule measured from the value in the pure solvent. From the thermodynamic condition A/xo = (9AF/dNo)n = —(< / )(9 AF/d4>)=0 that the chemical potential of a solvent molecule inside the region occupied by the polymer should be equal to that in the outside region, we can derive Maxwell s rule of equal area for the osmotic pressure in the form... 

The theoretical importance of Raoult s law is that, because it relates vapor pressure to composition and we know how to relate pressure to chemical potential, we can use the law to relate chemical potential to the composition of a solution. As we show in the following Justification, the chemical potential of a solvent A present in solution at a mole fraction is... 

Figure 3.27 shows the variation of the chemical potential of the solvent predicted by this expression. Note that the chemical potential has its pure value at Xa = 1 (when only A is present). The essential feature of eqn 3.12 is that because Xa < 1 implies that In Xa < 0, the chemical potential of a solvent is lower in a solution than when it is pure. Provided the solution is almost ideal, a solvent in which a solute is present has less chemical punch (including a lower ability to generate a vapor pressure) than when it is pure. 

This snbchapter presents the results of theoretical and experimental studies on the influence of solvent properties on the nature of the swelling kinetics for elastomeric polymer networks at finite strains of the polymer matrix. Emphasis is focused on the study of the relationship of the asymptotic properties of the swelling kinetic curves with the thermodynamic quality of a solvent. The chemical potential of a solvent in the sample under swelling is given by the following equation ... 

Equation [6.6.1] describes the dependence of the chemical potential of a solvent absorbed by the sample under symmetrical biaxial stretching and compressiom Equating it... 

The exact definition of the theta state is given by chemical thermodynamics. The chemical potential of a solvent 1, Afi i, can be split into an ideal term and an excess term ... 

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