Relation between Kp and Kc

$K_P$ and $K_C$ describe the very same equilibrium, so they must be related to each other. For the general all-gas reaction $xA + yB \rightleftharpoons lC + mD$, starting from the ideal gas equation $PV = nRT$, i.e. $P = (n/V)RT$, and noting that active mass equals molar concentration $n/V$, gives

so each species' partial pressure can be written as $p_A^{\,x} = [A]^x(RT)^x$, and similarly for B, C and D. Substituting these into the $K_P$ expression,

Comparing with $K_C = \dfrac{[C]^l[D]^m}{[A]^x[B]^y}$, this gives the general relation

where $\Delta n_g$ is the difference between the total moles of gaseous products and the total moles of gaseous reactants, $(l+m)-(x+y)$. Three cases follow immediately:

• When $\Delta n_g = 0$: $K_P = K_C(RT)^0 = K_C$, e.g. $H_2(g)+I_2(g)\rightleftharpoons 2HI(g)$ and $N_2(g)+O_2(g)\rightleftharpoons 2NO(g)$.
• When $\Delta n_g$ is positive: $K_P = K_C(RT)^{+\text{ve}}$, so $K_P > K_C$, e.g. $2NH_3(g)\rightleftharpoons N_2(g)+3H_2(g)$ and $PCl_5(g)\rightleftharpoons PCl_3(g)+Cl_2(g)$.
• When $\Delta n_g$ is negative: $K_P = K_C(RT)^{-\text{ve}}$, so $K_P < K_C$, e.g. $2H_2(g)+O_2(g)\rightleftharpoons 2H_2O(g)$ and $2SO_2(g)+O_2(g)\rightleftharpoons 2SO_3(g)$. …