9.1 Chemical potential and Gibbs-Duhem equation

Chemical Potential and Gibbs Free Energy

Chemical potential and Gibbs free energy

Chemical potential tells you how the energy of a system changes when you add a tiny bit of one substance while keeping everything else fixed. It's the central quantity for predicting which direction reactions go and when phases are in equilibrium.

The chemical potential of component $i$, written $\mu_i$, is defined as the partial molar Gibbs free energy:

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

This says: hold temperature, pressure, and the amounts of all other components constant, then see how $G$ changes as you add one mole of component $i$. That rate of change is $\mu_i$.

For a system with multiple components, the total Gibbs free energy is the sum of each component's contribution:

$G = \sum_{i} \mu_i \, n_i$

This expression follows from the fact that $G$ is an extensive property (it scales with system size), so it can be built up from the partial molar quantities. A few things to note:

• $\mu_i$ generally depends on temperature, pressure, and composition, so it isn't a fixed number for a given substance.
• Matter spontaneously moves from regions of higher chemical potential to regions of lower chemical potential. At equilibrium, the chemical potential of each component is the same in every phase.
• This is what governs familiar processes like melting, boiling, dissolution, and vapor-liquid equilibrium.

Gibbs-Duhem Equation and Its Applications

Derivation of the Gibbs-Duhem equation

The Gibbs-Duhem equation places a constraint on how the chemical potentials in a mixture can change. You can't independently vary every $\mu_i$; they're linked. Here's how the derivation works.

1. Start with the total differential of Gibbs free energy for an open system:

$dG = -S\,dT + V\,dP + \sum_{i} \mu_i \, dn_i$

1. Separately, because $G = \sum_i \mu_i n_i$, you can also take its total differential using the product rule:

$dG = \sum_{i} \mu_i \, dn_i + \sum_{i} n_i \, d\mu_i$

1. Set these two expressions for $dG$ equal to each other:

$-S\,dT + V\,dP + \sum_{i} \mu_i \, dn_i = \sum_{i} \mu_i \, dn_i + \sum_{i} n_i \, d\mu_i$

1. The $\sum_i \mu_i \, dn_i$ terms cancel, leaving:

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

This is the general form of the Gibbs-Duhem equation. It holds for any single-phase system.

1. At constant temperature and pressure ($dT = 0$, $dP = 0$), it reduces to:

$\sum_{i} n_i \, d\mu_i = 0$

Dividing through by the total number of moles $n_t$ gives the mole-fraction form:

$\sum_{i} x_i \, d\mu_i = 0$

The physical meaning: if one component's chemical potential goes up, at least one other component's chemical potential must go down to compensate. The chemical potentials in a mixture are not independent of each other.

Applications of the Gibbs-Duhem equation

For a binary system (components 1 and 2) at constant $T$ and $P$, the equation becomes:

$x_1 \, d\mu_1 + x_2 \, d\mu_2 = 0$

Rearranging:

$d\mu_2 = -\frac{x_1}{x_2} \, d\mu_1$

This is extremely useful in practice. If you measure how $\mu_1$ changes with composition (say, from vapor pressure data for the more volatile component), you can calculate $\mu_2$ by integration without needing a separate measurement.

Other key applications:

• Thermodynamic consistency checks. If someone reports activity coefficient data for both components in a binary mixture, the Gibbs-Duhem equation lets you verify whether the data are internally consistent.
• Relating activity coefficients. Changes in $\mu_i$ can be rewritten in terms of activity coefficients ($\gamma_i$), so the equation constrains how $\ln \gamma_1$ and $\ln \gamma_2$ vary with composition.
• Deriving other relations. The Clausius-Clapeyron equation (relating vapor pressure to temperature) and the Duhem-Margules equation (relating partial pressures in a binary mixture) both follow from the Gibbs-Duhem framework.
• Understanding colligative properties and azeotropes. The interdependence of chemical potentials explains why, for instance, adding a solute lowers the solvent's vapor pressure.

Factors affecting chemical potential

Temperature. The temperature dependence is:

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

where $\bar{S}_i$ is the partial molar entropy of component $i$. Since entropy is positive, increasing temperature decreases the chemical potential. Components with larger partial molar entropies (like gases) are more sensitive to temperature changes.

Pressure. The pressure dependence is:

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

where $\bar{V}_i$ is the partial molar volume. Since molar volumes are positive, increasing pressure raises the chemical potential. This effect is large for gases (large $\bar{V}_i$) and small for condensed phases.

Composition. This is where the system's ideality matters:

• In an ideal mixture, the chemical potential depends on mole fraction through a logarithmic relation:

$\mu_i = \mu_i^0 + RT \ln x_i$

Here $\mu_i^0$ is the chemical potential of pure component $i$ at the same $T$ and $P$, and $R$ is the gas constant. Because $x_i < 1$, the $\ln x_i$ term is always negative, so mixing always lowers the chemical potential in an ideal system.

• In non-ideal mixtures, you replace $x_i$ with the activity $a_i = \gamma_i x_i$, where $\gamma_i$ is the activity coefficient:

$\mu_i = \mu_i^0 + RT \ln(\gamma_i x_i)$

The activity coefficient captures deviations from ideality caused by differences in intermolecular forces (e.g., hydrogen bonding, size disparity). When $\gamma_i = 1$, you recover the ideal case.