Extent of Reaction

Atomic Conservation

Let $\vec{\mathbf{n}}=\left[\begin{array}{c}{n}_1\\ \cdots \\ n_N\end{array}\right]$ be species moles and $\vec{\mathbf{b}}$ total elemental moles (C, H, O, N, …). With element–species matrix $\mathbf{A}$ (atoms of each element in each species),

$$\mathbf{A}\vec{\mathbf{n}}=\vec{\mathbf{b}},$$

constant in time for closed systems.

Extent of Reaction

For reactions with stoichiometric coefficients ${\nu }_i$ (products $+$, reactants $-$), define the extent $\xi$:

$$n_i=n_{i0}+{\nu }_i\xi .$$

Multiple independent reactions $\Rightarrow$ vector form $\vec{\mathbf{n}}={\vec{\mathbf{n}}}_0+\mathbf{N}\vec{\mathbf{\xi }}$, consistent with $\mathbf{A}\vec{\mathbf{n}}=\vec{\mathbf{b}}$.

Completion vs. Equilibrium (preview)

• Stoichiometric completion: choose $\xi$ so limiting reactant is exhausted (no kinetics/thermo limits).
• Equilibrium: $\xi ={\xi }_{\ast }(T,p)$ maximizing entropy (or minimizing $G$) subject to $\mathbf{A}\vec{\mathbf{n}}=\vec{\mathbf{b}}$.