Equilibrium Composition

Mass-Action Form (ideal gas)

For reaction ${\sum }_i{\nu }_i\mathrm{A}{}_{\mathrm{i}}=0$ and ideal gas at $(T,p)$, define

$$K_p(T)=\prod _i{\left(\frac{p_i}{p^{\circ }}\right)}^{{\nu }_i},$$

or in mole fractions (total pressure $p$):

$$K_x(T,p)=\prod _i{x}_i^{{\nu }_i}{\left(\frac{p}{p^{\circ }}\right)}^{{\sum }_i{\nu }_i}.$$

Link to standard Gibbs energy:

$$K(T)=\exp \left(-\frac{\Delta G^{\circ }(T)}{RT}\right),\Delta G^{\circ }=\sum _i{\nu }_i{\mu }_i^{\circ }(T).$$

Solution Strategy

1. Pick state $(T,p)$ and compute $K(T)$.
2. Use atom balances $\mathbf{A}\vec{\mathbf{n}}=\vec{\mathbf{b}}$ (or extent variables) and the mass-action relation(s) to solve for $\{x_i\}$.
3. Recover $n_i=x_{i}n$; update $\bar{W},R$ for performance links.

NOTE

For multiple reactions, solve a coupled nonlinear system in $\{x_i\}$ (or $\vec{\mathbf{\xi }}$). Numerical solvers with positivity constraints on $n_i$ are standard.