The relationship between $K_c$ and $K_p$

What is the relationship between the equilibrium constant in terms of concentration $K_c$ and the equilibrium constant in terms of partial pressure $K_p$?

For an ideal gas,

$p=\frac{n}{V}RT\; \; \Rightarrow\; \; p=[concentration]RT\; \; \; \; \; \; \; \; 1$

Substitute eq1 in the general equilibrium constant expression in terms of concentration,

$K_c=\frac{[C]^p[D]^q...}{[A]^m[B]^n...}=\frac{\left ( \frac{p_C}{RT} \right )^p\left ( \frac{p_D}{RT} \right )^q...}{\left ( \frac{p_A}{RT} \right )^m\left ( \frac{p_B}{RT} \right )^n...}$

$(RT)^{(p+q+...)-(m+n+...)}K_c=\frac{p_C^{\; \; \; p}p_D^{\; \; \; q}...}{p_A^{\; \; \; m}p_B^{\; \; \; n}...}\; \; \; \; \; \; \; \; 2$

Substitute the general equilibrium constant expression in terms of partial pressures, K_p, in eq2

$K_p=K_c(RT)^{\Delta n}$

where Δn = sum of stoichiometric coefficients of products – sum of stoichiometric coefficients of reactants.

Note that K_p= K_c if Δn = 0.

The relationship between \(K_c\) and \(K_p\)

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