9.1 Partial molar quantities

Partial Molar Quantities

Definition and Significance

A partial molar quantity tells you how much a single component contributes to the total thermodynamic property of a mixture. More precisely, it's the rate of change of an extensive property (like volume or enthalpy) when you add an infinitesimal amount of one component, holding temperature, pressure, and the amounts of all other components fixed.

Why does this matter? Pure substances have straightforward molar properties, but when you mix components together, intermolecular interactions cause each species to behave differently than it would on its own. Partial molar quantities capture exactly that difference, and they're the foundation for describing non-ideal mixture behavior.

Common partial molar quantities include:

• Partial molar volume $\bar{V}_i$
• Partial molar enthalpy $\bar{H}_i$
• Partial molar entropy $\bar{S}_i$
• Partial molar Gibbs free energy $\bar{G}_i$, which is the chemical potential $\mu_i$

A central result connects these quantities to the total property of the mixture. For any extensive property $Y$:

$Y = \sum_i n_i \bar{Y}_i$

This follows from Euler's theorem for homogeneous functions of degree one. The total property equals the sum of each component's mole number times its partial molar quantity.

The Gibbs-Duhem equation then constrains how partial molar quantities can change simultaneously:

$\sum_i n_i \, d\bar{Y}_i = 0 \quad \text{(at constant } T, P\text{)}$

This means the partial molar quantities of different components in a mixture are not independent of each other. If one increases with composition, the other must adjust accordingly.

Relationship to Chemical Potentials

The partial molar Gibbs free energy has a special name: the chemical potential $\mu_i$. It's defined as:

$\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j \neq i}}$

Chemical potentials govern several key phenomena:

• Phase equilibrium: At equilibrium, the chemical potential of each component must be equal across all phases. If $\mu_i^{\alpha} \neq \mu_i^{\beta}$, material will spontaneously transfer from the phase of higher $\mu_i$ to the phase of lower $\mu_i$.
• Reaction spontaneity: The direction of a chemical reaction is determined by the chemical potentials of reactants and products.
• Diffusion: Mass transport occurs down gradients in chemical potential.

The dependence of $\mu_i$ on composition, temperature, and pressure encodes information about phase stability, solubility limits, and miscibility gaps.

Derivation of Partial Molar Properties

Thermodynamic Principles

The derivation rests on two mathematical tools:

1. Euler's theorem for homogeneous functions: Any extensive property $Y(T, P, n_1, n_2, \ldots)$ is homogeneous of degree one in the mole numbers. Euler's theorem then gives $Y = \sum_i n_i \bar{Y}_i$ directly.

2. Exact differentials: The total differential of $Y$ at constant $T$ and $P$ is:

$dY = \sum_i \left(\frac{\partial Y}{\partial n_i}\right)_{T,P,n_{j \neq i}} dn_i = \sum_i \bar{Y}_i \, dn_i$

Taking the full differential of the Euler relation $Y = \sum_i n_i \bar{Y}_i$ and comparing it with the expression above yields the Gibbs-Duhem equation:

$\sum_i n_i \, d\bar{Y}_i = 0 \quad \text{(at constant } T, P\text{)}$

This equation is not a definition but a constraint. It means you can measure the partial molar quantity of one component and calculate the other from it (in a binary system, for instance).

Expressions for Partial Molar Quantities

Each partial molar quantity follows the same pattern of definition:

Partial molar volume:

$\bar{V}_i = \left(\frac{\partial V}{\partial n_i}\right)_{T,P,n_{j \neq i}}$

This tells you how the total volume of the mixture changes when you add a small amount of component $i$. Note that $\bar{V}_i$ is generally not equal to the molar volume of pure $i$, because molecular packing changes in a mixture.

Partial molar enthalpy:

$\bar{H}_i = \left(\frac{\partial H}{\partial n_i}\right)_{T,P,n_{j \neq i}}$

This quantity is directly related to the heat of mixing. If $\bar{H}_i$ differs from the pure-component molar enthalpy $H_i^*$, the mixing process is exothermic or endothermic.

Partial molar Gibbs free energy (chemical potential):

$\mu_i = \bar{G}_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j \neq i}}$

This is the most thermodynamically significant partial molar quantity because it determines equilibrium and spontaneity.

Calculation of Partial Molar Quantities

Experimental Methods

For binary mixtures, two graphical techniques are commonly used:

Method of intercepts (tangent-intercept method):

1. Measure the total molar property $Y_m = Y/(n_1 + n_2)$ as a function of mole fraction $x_1$.
2. At the composition of interest, draw the tangent line to the $Y_m$ vs. $x_1$ curve.
3. The intercept of the tangent at $x_1 = 1$ gives $\bar{Y}_1$, and the intercept at $x_1 = 0$ gives $\bar{Y}_2$.

This works because of the relationship:

$\bar{Y}_1 = Y_m + (1 - x_1)\frac{dY_m}{dx_1}$

$\bar{Y}_2 = Y_m - x_1\frac{dY_m}{dx_1}$

Slope method:

1. Plot the total extensive property $Y$ against $n_1$ while holding $n_2$ constant.
2. The slope of the tangent at the desired composition gives $\bar{Y}_1$ directly.

For multicomponent mixtures, you vary the amount of one component while holding all others fixed, and evaluate the slope $(\partial Y / \partial n_i)_{T,P,n_{j \neq i}}$ at the composition of interest.

Equations of State

When experimental data aren't available, equations of state provide an alternative route:

1. Start with an equation of state (e.g., Peng-Robinson, Redlich-Kwong) that relates $P$, $V$, $T$, and composition through mixing rules for the equation parameters.
2. Derive an expression for the total extensive property (e.g., volume) as a function of $T$, $P$, and $\{n_i\}$.
3. Differentiate with respect to $n_i$ at constant $T$, $P$, and $n_{j \neq i}$ to obtain the partial molar quantity.

The accuracy of this approach depends on how well the equation of state and its mixing rules represent the actual mixture. Cubic equations of state work well for non-polar and slightly polar systems at moderate conditions but can struggle with strongly associating or polar mixtures.

Dependence of Partial Molar Quantities

Composition Dependence

In an ideal mixture, partial molar quantities equal the pure-component molar quantities at the same $T$ and $P$:

$\bar{V}_i^{\text{ideal}} = V_i^* \qquad \bar{H}_i^{\text{ideal}} = H_i^*$

Real mixtures deviate from this. The deviation is quantified by excess functions:

$Y^E = Y_{\text{mixture}} - Y_{\text{ideal mixture}} = \sum_i n_i (\bar{Y}_i - Y_i^*)$

For example, the excess volume $V^E$ tells you whether the mixture is more tightly or loosely packed than you'd predict from the pure components. A negative $V^E$ indicates stronger attractive interactions between unlike molecules.

Excess functions and partial molar quantities are linked through the Gibbs-Duhem equation, so measuring one constrains the other. This composition dependence is directly useful for designing separation processes (distillation, extraction) and predicting miscibility.

Temperature and Pressure Dependence

Temperature dependence is governed by partial molar heat capacities:

$\left(\frac{\partial \bar{H}_i}{\partial T}\right)_{P, n_j} = \bar{C}_{P,i}$

Partial molar heat capacities reflect how the energetics of molecular interactions in the mixture change with temperature.

Pressure dependence is governed by partial molar volumes:

$\left(\frac{\partial \mu_i}{\partial P}\right)_{T, n_j} = \bar{V}_i$

This relation is important for understanding how high pressures shift equilibria and phase boundaries. For condensed phases, $\bar{V}_i$ is small and the pressure dependence is weak. For gas-phase mixtures, the pressure dependence is significant.

Understanding both dependencies is essential for predicting mixture behavior across the range of conditions encountered in industrial processes (high-pressure reactors, cryogenic separations) and natural systems (geochemical fluids, biological environments).