Solution Phase Chemical Potentials

In the thermodynamic treatment of condensed phase solutions, two limiting forms for the chemical potential are often used. One focuses on the limit of a dilute solution of one component (solute) in another component (solvent); the other addresses the limit of nearly pure solvent. The first leads to Henry's Law for solubilities and the second to Raoult's Law for the vapor pressure of solvent above a dilute solution. Later we will present the microscopic form of these laws from the standpoint of the potential distribution theorem, but here we give the thermodynamic treatment. We assume a uniform condensed phase system.

The Henry's Law limit chemical potential can be written as

$$\mu_i = \mu_i^{\circ} + k_BT \ln (\gamma_i x_i)$$ (1.54)

where $\mu_i^{\circ}$ is a reference chemical potential for an infinitely dilute solution. The constant $\gamma_i$ is termed the activity coefficient, and $x_i$ is the mole fraction of component $i$ in the solution. The activity coefficient goes to one as the solute concentration goes to zero. We assume that the solution is very dilute, so $\gamma_i = 1$. If we now place the solution phase in equilibrium with an ideal gas of the solute, and use Eq. 1.51, we find

$$\mu_i = \mu_i^{\circ} + k_BT \ln x_i = k_BT \ln (P_i\Lambda_i^3/k_BT)$$ (1.55)

$P_i$ is the gas phase pressure. Solving this equation for $P_i$, we obtain

$$P_i = \frac{k_BT}{\Lambda_i^3}e^{\mu_i^{\circ}/k_BT} x_i$$ (1.56)

or

$$K_i x_i = P_i$$ (1.57)

where

$$K_i = \frac{k_BT}{\Lambda_i^3}e^{\mu_i^{\circ}/k_BT}$$ (1.58)

is the Henry's Law constant.

At the other extreme of considering the chemical potential of a component which is nearly pure, we can write the chemical potential as

$$\mu_i = \mu_i^{*} + k_BT \ln (\gamma_i x_i)$$ (1.59)

with $\mu_i^{*}$ a reference chemical potential for a pure component $i$. For this case, the activity coefficient approaches one as the mole fraction of $i$ approaches one. Using the same argument as above for the Henry's Law case, we find

$$P_i^{\circ} x_i = P_i$$ (1.60)

where

$$P_i^{\circ} = \frac{k_BT}{\Lambda_i^3}e^{\mu_i^{*}/k_BT}$$ (1.61)

is the pure $i$ vapor pressure.