$$\text { Kp and } Kc \text { are equilibrium constants of } \\$$
$$\text { a gaseous mixture. } \\$$
$$\text { Ke is equilibrium constant when the } \\$$
$$\text { concentration are in molarity. } \\$$
$$\text { Kp is equilibrium constant when the } \\$$
$$\text { concentration are in atm. }$$
$$\text { Derivation } \\$$
$$\text { for } a A+b B=c C+d D \\$$
$$k p=\dfrac{(P C)^{C}(P D)^{d}}{\left(P_{A}\right)^{a}\left(P_{B}\right)^{b}} \text { , $$where, } A, B, \text { cand } D \\$$
$$\text { By ideal gas eqn., } \\$$
$$P V-n R T \\$$
$$P-\dfrac{n}{V} R T=C R T$$
$$K_{p}=\dfrac{[C]^{C}[D]^{d}[R T]^{C+d}}{[A]^{a}[B]^{b}[R T]^{a}+b}$$
$$K_{C}=\dfrac{[C]^{C}[D]^{d}}{[A]^{a}[B]^{b}}$$
$$\Rightarrow K p=K_{C}[R T]^{\Delta n g}$$
$${\Delta n g}$$ (gaseous moles) $$=(c+d)-(a+b)$$
$$K p=K_{C}(R T)^{\Delta n g}$$