Chemical Potential

Historical Context

The concept of chemical potential was introduced by J. Willard Gibbs in 1876 as the intensive variable conjugate to particle number. Gilbert Newton Lewis (1901-1907) developed the concepts of activity and fugacity to extend the ideal-solution formalism to real systems, providing the practical tools that physical chemists use daily.

Francois-Marie Raoult established his law for ideal solutions in 1887, while William Henry discovered his law for dilute solutions of gases in 1803. Together, these laws define the two limiting behaviors that bracket all real solution behavior.

Derivation 1: Chemical Potential

For a multi-component system, the Gibbs energy depends on T, P, and the amounts of each species. The fundamental relation becomes:

The chemical potential of species i is defined as:

Equivalently, $\mu_i$ can be expressed in terms of other potentials:

The chemical potential has a clear physical interpretation: it is the Gibbs energy per mole of species i when added to the mixture at constant T and P. Material flows spontaneously from regions of high $\mu$ to regions of low $\mu$, just as heat flows from high T to low T.

Derivation 2: Ideal Solutions and Raoult's Law

For an ideal gas at temperature T, the chemical potential depends on pressure:

For an ideal solution, we replace pressure with mole fraction to define:

At vapor-liquid equilibrium, $\mu_i^{\text{liquid}} = \mu_i^{\text{vapor}}$. For an ideal solution in equilibrium with ideal vapor, this leads to Raoult's law:

Derivation 3: Activity and Activity Coefficients

For real (non-ideal) solutions, G. N. Lewis introduced the activity $a_i$to maintain the same mathematical form as ideal solutions:

The activity coefficient $\gamma_i$ measures the deviation from ideal behavior:

The one-parameter Margules model gives:

The parameter A can be positive (positive deviation) or negative (negative deviation). These expressions automatically satisfy the Gibbs-Duhem equation.

Derivation 4: Henry's Law

In the limit of infinite dilution ($x_i \to 0$), the solute molecules are surrounded entirely by solvent. The activity coefficient approaches a constant value $\gamma_i^\infty$, and the modified Raoult's law becomes:

Henry's law is the tangent to the real vapor pressure curve at $x_i = 0$, while Raoult's law is the tangent at $x_i = 1$. For the Margules model:

Henry's law is especially important for dissolved gases (O2, N2, CO2 in water) where the solute has no accessible pure liquid state at the system conditions.

Derivation 5: Fugacity and Fugacity Coefficient

Fugacity extends the concept of pressure to real gases, maintaining the ideal-gas form of the chemical potential:

The fugacity coefficient is derived from the equation of state:

where $Z = PV_m/(RT)$ is the compressibility factor. For a van der Waals gas:

When $\phi < 1$, attractive forces dominate (fugacity less than pressure). When $\phi > 1$, repulsive forces dominate. At low pressures, $\phi \to 1$and fugacity equals pressure (ideal gas limit).

Applications

Simulation: Solutions, Activity, and Fugacity

This simulation compares ideal and real solution behavior, demonstrates Raoult's and Henry's laws, plots activity coefficient models, and computes fugacity coefficients for CO2 using the van der Waals equation.

Colligative Properties

Colligative properties depend on the number of solute particles rather than their identity. They arise because solute lowers the chemical potential of the solvent.

All four colligative properties share the same physical origin: the reduction of solvent chemical potential by the presence of solute, $\Delta\mu_1 = RT\ln x_1 \approx -RT x_2$for dilute solutions.

Excess Thermodynamic Functions

Excess functions measure the deviation of a real solution from ideal behavior. They are defined as the difference between the actual mixing property and the ideal mixing property:

For the one-parameter Margules model with parameter A:

The two-parameter Margules model ($G^E = x_1 x_2 (A_{12}x_1 + A_{21}x_2) RT$) accommodates asymmetry. The van Laar, Wilson, NRTL, and UNIQUAC models offer increasing sophistication for strongly non-ideal systems.

Regular Solutions and the Hildebrand Model

Joel Hildebrand (1929) introduced the concept of regular solutions: mixtures where the excess entropy is zero ($S^E = 0$), so all non-ideality arises from enthalpy:

The Hildebrand solubility parameter $\delta = \sqrt{\Delta U_{\text{vap}}/V_m}$ allows estimation of the interchange energy:

The "like dissolves like" rule follows: substances with similar solubility parameters ($\delta_1 \approx \delta_2$) have small w and form nearly ideal solutions.