Derive the relation between $K_{p}$ and $K_{c}$ for a general chemical equilibrium reaction.

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Relation between $K_{c}$ & $K_{p}$ for a reaction.

$a A + b B ⇌ c C + d D$

$K_{c} = \frac{K_{f}}{K_{b}} = \frac{[C]^{c} [D]^{d}}{[A]^{a} [B]^{b}}$ ____(1)

$K_{p} = \frac{[P_{C}]^{c} [P_{D}]^{d}}{[P_{A}]^{a} [P_{B}]^{b}}$

As we know, $P V = n R T$

$P = \frac{n}{v} R T \Rightarrow P = C R T$ ${c = \frac{n}{v}}$

So, $P_{A} = C_{A} R T, P_{C} = C_{C} R T$
$P_{B} = C_{B} R T, P_{D} = C_{D} R T$

$K_{p} = \frac{[C_{C} R T]^{c} [C_{D} R T]^{d}}{[C_{A} R T]^{a} [C_{B} R T]^{b}} = \frac{[C]^{c} [D]^{d}}{[A]^{a} [B]^{b}} \times [R T]^{(c + d) - (a + b)}$

$K_{p} = K_{c} (R T)^{Δ n_{g}}$ where $Δ n_{g} = c + d - a - b$

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