## C Thermodynamic Relationships Involving the Equilibrium Constant^{1}

For the gas-phase reaction

$\begin{array}{cc}\text{A+}\frac{b}{a}\text{B}\rightleftarrows \frac{c}{a}+\frac{d}{a}\text{D}& \begin{array}{c}\left(\mathrm{2-2}\right)\end{array}\end{array}$

1. The true (dimensionless) equilibrium constant

$\begin{array}{c}RT\text{\hspace{0.17em}}\mathrm{ln}k=-\mathrm{\Delta}G\end{array}$

$\begin{array}{|c|}\hline K=\frac{{a}_{\text{C}}^{c\backslash a}{a}_{\text{D}}^{d/a}}{{a}_{\text{A}}{a}_{\text{B}}^{b/a}}\\ \hline\end{array}$

where *a*_{i} is the activity of species *i*

$${a}_{i}=\frac{{f}_{i}}{{f}_{i}^{\circ}}$$

where *f*_{i} = fugacity of species *i*

${f}_{i}^{\circ}$ = fugacity of species *i* at the standard state. For gases, the standard state is 1 bar, which is in the noise level of 1 atm, so we will use atm.

$${a}_{i}=\frac{{f}_{i}}{{f}_{i}^{\circ}}={\gamma}_{i}{P}_{i}$$

where *γ*_{i} is the activity coefficient

*K* = True equilibrium constant*K*_{y} = Activity equilibrium constant*K*_{p} = Pressure equilibrium constant*K*_{c} = Concentration equilibrium constant

*K*_{γ} has units of ${\left[\text{atm}\right]}^{-(d/a+c/a-\frac{b}{a}-1)}={[\text{atm ...}}^{}$