Elements of Chemical Reaction Engineering, 7th Edition

C. Thermodynamic Relationships Involving the Equilibrium Constant¹

¹ For the limitations and for further explanation of these relationships, see, for example, K. Denbigh, The Principles of Chemical Equilibrium, 3rd ed. Cambridge: Cambridge University Press, 1971, p. 138.

For the gas-phase reaction

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

1. The true (dimensionless) equilibrium constant

   RTlnK = –ΔG

   $\boxed{K={\displaystyle \frac{a_{\text{C}}^{c/a}{a}_{\text{D}}^{d/a}}{a_{\text{A}}{a}_{\text{B}}^{b/a}}}}$

   where a_i is the activity of species i

   $a_i={\displaystyle \frac{f_i}{f_i^{\circ }}}$

   where fi	=	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.

   $a_i={\displaystyle \frac{f_i}{f_i^{\circ }}}={\gamma }_i{P}_i$

   where γ_i is the activity coefficient

   $K={\displaystyle \frac{{\gamma }_{\text{C}}^{c/a}{\gamma }_{\text{D}}^{d/a}}{{\displaystyle \underset{K_{\gamma }}{\underbrace{{\gamma }_{\text{A}}{\gamma }_{\text{B}}^{b/a}}}}}}·{\displaystyle \frac{P_{\text{C}}^{c/a}{P}_D^{d/a}}{\underset{K_p}{\underbrace{P_{\text{A}}{P}_{\text{B}}^{b/a}}}}}=K_{\gamma }{K}_p$

   K_γ has units of ...