8.5 Chemical Potential, Partial Molar Quantities, Fugacity, and Activity

Chemical Potential, Partial Molar Quantities, Fugacity, and Activity

Pure-substance thermodynamics must be extended to multicomponent, open, and chemically reacting systems where composition varies. On the PRC Chemist Licensure Examination, mastery of this domain requires an exact physical and mathematical understanding of partial molar properties, the Gibbs-Duhem equation, real-gas fugacity ($f$), and solution activity ($a$).

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1. Open Systems and Partial Molar Quantities

An extensive thermodynamic property $X$ of a multicomponent mixture depends on temperature, pressure, and the mole amounts of each constituent: $X = X(T, P, n_1, n_2, \dots, n_k)$.

The partial molar quantity ($\bar{X}_i$) of component $i$ is defined as the rate of change of property $X$ with respect to the addition of component $i$ at constant temperature, pressure, and amounts of all other components:

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

Euler's Homogeneous Function Theorem

Because extensive thermodynamic properties ($V, U, H, S, A, G$) are homogeneous functions of degree 1 with respect to mole numbers (i.e., scaling all mole numbers by $\lambda$ scales the property by $\lambda$: $X(T, P, \lambda n_1, \dots, \lambda n_k) = \lambda X(T, P, n_1, \dots, n_k)$ Euler's theorem of homogeneous functions requires that:

$X = \sum_{i=1}^k n_i \bar{X}_i$ Dividing both sides by total moles $n = \sum n_i$ gives the mean molar property: $X_m = \sum_{i=1}^k x_i \bar{X}_i$

Physical Demonstration: Partial Molar Volume ($\bar{V}_i$)

Total volume of a binary solution is given by: $V = n_1 \bar{V}_1 + n_2 \bar{V}_2$ The Ethanol-Water Anomaly: Pure water has a molar volume of $V_{m, H_2O} = 18.07\text{ cm}^3/\text{mol}$, while pure liquid ethanol has $V_{m, EtOH} = 58.68\text{ cm}^3/\text{mol}$. When $1.00\text{ mol}$ of ethanol ($58.68\text{ cm}^3$) is added to an infinitely large pool of pure water, the total volume increases by only $54.2\text{ cm}^3$ (thus $\bar{V}_{EtOH} = 54.2\text{ cm}^3/\text{mol}$ at infinite dilution).

Physical Rationale: Water's open, hydrogen-bonded tetrahedral cage structure accommodates smaller ethanol alkyl groups within interstitial cavities, accompanied by attractive water-ethanol dipole-dipole and hydrogen-bonding contractions.

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2. Chemical Potential ($\mu_i$) and the Gibbs-Duhem Equation

The chemical potential of component $i$ is the partial molar Gibbs free energy:

$\mu_i \equiv \bar{G}_i = \left(\frac{\partial G}{\partial n_i}\right)_{T, P, n_{j \neq i}}$ In an open system, the fundamental differential equation for Gibbs free energy becomes:

$dG = -S\,dT + V\,dP + \sum_{i=1}^k \mu_i\,dn_i$ Chemical potential represents the escaping tendency of a chemical species from a phase: at phase or chemical equilibrium, the chemical potential of each component must be identical in all accessible phases: $\mu_i^{(\alpha)} = \mu_i^{(\beta)} = \mu_i^{(\gamma)}$

Derivation of the Gibbs-Duhem Equation

Applying Euler's theorem to Gibbs free energy: $G = \sum_{i=1}^k n_i \mu_i$. Differentiating this expression completely using the product rule: $dG = \sum_{i=1}^k n_i\,d\mu_i + \sum_{i=1}^k \mu_i\,dn_i$ Equating this to the fundamental equation $dG = -S\,dT + V\,dP + \sum \mu_i\,dn_i$: $\sum_{i=1}^k n_i\,d\mu_i + \sum_{i=1}^k \mu_i\,dn_i = -S\,dT + V\,dP + \sum_{i=1}^k \mu_i\,dn_i$ Canceling $\sum \mu_i\,dn_i$ yields the general Gibbs-Duhem Equation:

$-S\,dT + V\,dP - \sum_{i=1}^k n_i\,d\mu_i = 0$

At constant temperature ($dT = 0$) and constant pressure ($dP = 0$):

$\sum_{i=1}^k n_i\,d\mu_i = 0 \iff \sum_{i=1}^k x_i\,d\mu_i = 0$

Significance for Binary Mixtures

For a two-component mixture ($x_1 + x_2 = 1$): $x_1\,d\mu_1 + x_2\,d\mu_2 = 0 \implies d\mu_2 = -\frac{x_1}{x_2}\,d\mu_1$ Thermodynamic Consequence: The chemical potentials of components in a solution cannot vary independently. If component 1 undergoes an increase in chemical potential ($d\mu_1 > 0$), the chemical potential of component 2 must simultaneously decrease ($d\mu_2 < 0$). This constraint underpins the Duhem-Margules equation relating vapor pressures of binary mixtures: $\frac{d \ln P_1}{d \ln x_1} = \frac{d \ln P_2}{d \ln x_2}$

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3. Real Gases: Van der Waals, Compressibility Factor, and Fugacity

The Van der Waals Equation of State

To account for non-ideal gas behavior, Johannes van der Waals introduced corrections for intermolecular forces ($a$) and finite molecular volume ($b$):

$\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \iff \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$

• Parameter $a$ (Cohesive / Attractive Parameter): Units: $\text{bar}\cdot\text{L}^2\cdot\text{mol}^{-2}$ or $\text{Pa}\cdot\text{m}^6\cdot\text{mol}^{-2}$. Accounts for attractive van der Waals forces between molecules, reducing the pressure exerted on the container walls.
• Parameter $b$ (Excluded Volume / Co-volume): Units: $\text{L}\cdot\text{mol}^{-1}$ or $\text{m}^3\cdot\text{mol}^{-1}$. Represents the non-compressible volume occupied by 1 mole of molecules ($b = 4 N_A \cdot v_{molecule}$). Incompressible hard cores prevent the gas volume from reaching zero at high pressure.

The Compressibility Factor ($Z$)

The compressibility factor quantifies deviation from ideal gas behavior:

$Z \equiv \frac{P V_m}{RT}$

The Boyle Temperature ($T_B$)

The Boyle temperature is the temperature at which the initial slope of $Z$ versus $P$ vanishes as $P \rightarrow 0$: $\lim_{P \to 0} \left(\frac{\partial Z}{\partial P}\right)_T = 0$ For a van der Waals gas, expanding $Z$ in terms of virial coefficients yields:

$T_B = \frac{a}{Rb}$ At $T = T_B$, attractive and repulsive forces cancel over a broad pressure range, causing the real gas to obey the ideal gas law up to substantial pressures.

Fugacity ($f$) and Fugacity Coefficient ($\phi$)

For an ideal gas, chemical potential is $\mu_i = \mu_i^\circ + RT \ln(\frac{P_i}{P^\circ})$. Gilbert N. Lewis introduced fugacity ($f$) as an "effective thermodynamic pressure" to retain this exact mathematical form for real gases:

$\mu_i = \mu_i^\circ + RT \ln\left(\frac{f_i}{P^\circ}\right) \quad \text{with boundary condition: } \lim_{P \to 0} \frac{f}{P} = 1$

The dimensionless fugacity coefficient ($\phi$) is: $\phi \equiv \frac{f}{P}$ $\mu = \mu^\circ + RT \ln\left(\frac{\phi P}{P^\circ}\right) = \mu_{ideal} + RT \ln \phi$

Exact Thermodynamic Evaluation of $\phi$

From the fundamental relation $d\mu = V_m\,dP$ at constant $T$: $\ln \phi = \int_{0}^{P} \frac{Z - 1}{P}\,dP$

Worked Step-by-Step Fugacity Calculation

Scenario: A gas at $300.0\text{ K}$ and $P = 50.0\text{ bar}$ has its compressibility factor modeled by the truncated virial equation: $Z = 1 + B'P \quad \text{where } B' = -2.00 \times 10^{-3}\text{ bar}^{-1}$ Calculate the fugacity coefficient $\phi$ and the fugacity $f$.

Step 1: Evaluate the integral for $\ln \phi$: $\frac{Z - 1}{P} = \frac{(1 + B'P) - 1}{P} = B' = -2.00 \times 10^{-3}\text{ bar}^{-1}$ $\ln \phi = \int_{0}^{50.0} B'\,dP = B' \cdot P = (-2.00 \times 10^{-3}\text{ bar}^{-1})(50.0\text{ bar}) = -0.1000$

Step 2: Solve for $\phi$ and $f$: $\phi = e^{-0.1000} = 0.90484 \approx 0.905$ $f = \phi \cdot P = (0.90484)(50.0\text{ bar}) = 45.24\text{ bar} \approx 45.2\text{ bar}$ Thermodynamic Deduction: Because attractive forces dominate ($Z < 1$), the fugacity coefficient is less than unity ($\phi = 0.905$). The real gas exerts a chemical escaping tendency equivalent to an ideal gas at only $45.2\text{ bar}$ rather than $50.0\text{ bar}$.

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4. Non-Ideal Solutions: Activity, Activity Coefficients, and Standard States

Activity ($a_i$) and Activity Coefficient ($\gamma_i$)

To preserve the thermodynamic equations of ideal solutions (where $\mu_i = \mu_i^* + RT \ln x_i$), Lewis introduced activity ($a_i$) as "effective concentration":

$\mu_i = \mu_i^\circ + RT \ln a_i$

The activity coefficient quantifies departure from ideality:

• Mole Fraction Scale: $a_i = \gamma_i x_i$
• Molality Scale: $a_i = \gamma_m \left(\frac{m_i}{m^\circ}\right) \quad [m^\circ = 1\text{ mol/kg}]$

Standard State Conventions

The Lewis-Randall Rule

The Lewis-Randall rule estimates the fugacity of a component in a real gas or liquid mixture assuming solution ideality without requiring empirical mixture parameters: $f_i = x_i f_i^*$ where $f_i^*$ is the fugacity of pure component $i$ at the mixture's temperature and total pressure.