Derive the relation ${K_p} = {K_c}{(RT)^{\Delta ng}}$ for a general chemical equilibrium reaction.

Hint: We have to remember that in a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system.

Complete answer:
We must remember that the equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change.
Let the gaseous reaction is a state of equilibrium is:
$aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)$
Let ${p_A},{p_{B,}}{p_{C,}}{p_D}$ be the partial pressure of A, B, C and D respectively.
Therefore, we know that, the values of ${K_c}$ and ${K_p}$ are:
${K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}}$ → (1)
${K_p} = \dfrac{{{p_c}^c{p_D}^d}}{{{p_A}^a{p_B}^b}}$ → (2)
From an ideal gas equation, we know that
PV= nRT
⇒P =$\dfrac{n}{V}$ ​RT = C R T
Here, C is referred to as the concentration.
Therefore,
${p_A}$​= [A] R T
${p_B}$​= [B] R T
${p_C}$​= [C] R T
${p_D}$​= [D] R T
Now, we will substitute the values in equation (2), so we get
${K_p} = \dfrac{{{{[C]}^c}{{(RT)}^c}{{[D]}^d}{{(RT)}^d}}}{{{{[A]}^a}{{(RT)}^a}{{[B]}^b}{{(RT)}^b}}}$
$ \Rightarrow {K_p} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}}{(RT)^{[(c + d) - (a - b)]}}$
From eq (1).
$ \Rightarrow {K_p} = {K_c}{(RT)^{\Delta {n_g}}}$
Here,
$\Delta ng$ is the total no. of moles of gaseous product – the total no. of moles of gaseous a reactant. So, the relation between ${K_c}$ and ${K_p}$ is:
${K_p} = {K_c}{(RT)^{\Delta ng}}$ Hence, Proved.

Note:
In the end, we need to note that in equilibrium, the reactants and the products are present in two or more that two phases, it is called a heterogeneous equilibrium. Whereas, in an equilibrium when all the reactants and the products are present in the same phase it is called homogeneous equilibrium.