Give a mathematical expression showing relation between ${{K}_{p}}$ and ${{K}_{c}}$.

Hint: ${{K}_{p}}$ and ${{K}_{c}}$ are the equilibrium constants of an ideal gas mixture. The relationship between them is like the relationship between the equilibrium constants expressed in terms of pressure and concentration of an ideal gas mixture.

Complete step by step solution:
Let us begin evaluating the equilibrium constants i.e. ${{K}_{p}}$ and ${{K}_{c}}$. ${{K}_{p}}$ is the equilibrium constant which is used when equilibrium is to be expressed in terms of atmospheric pressure. Whereas, ${{K}_{c}}$ is the equilibrium constant which is used when equilibrium is to be expressed in terms of concentration (mostly molarity).
General chemical reaction is given as,
\[aA+bB\rightleftarrows cC+dD\]
where,
a is the mole of reactant A
b is the mole of reactant B
c is the mole of product C
d is the mole of product D
For example (for a gas phase reaction)-
\[2{{A}_{(g)}}+{{B}_{(g)}}\rightleftarrows 2{{C}_{(g)}}\]
Then ${{K}_{p}}$ is given as,
${{K}_{p}}=\dfrac{{{P}_{C}}^{2}}{{{P}_{A}}^{2}{{P}_{B}}}$
Simply, ${{K}_{p}}$ is the ratio of the product of product’s partial pressure raised to the powers of their respective coefficients to the product of reactant’s partial pressure raised to the powers of their respective coefficients.
And, ${{K}_{c}}$ is given as,
${{K}_{c}}=\dfrac{{{\left[ C \right]}^{2}}}{{{\left[ A \right]}^{2}}\left[ B \right]}$
Now, Ideal gas law is given as,
\[\begin{align}
& PV=nRT \\
& P=\dfrac{n}{V}RT \\
& P=CRT \\
\end{align}\]
Thus, can also be expressed as,
$\begin{align}
& {{K}_{p}}=\dfrac{{{\left[ C \right]}^{2}}{{\left( RT \right)}^{2}}}{{{\left[ A \right]}^{2}}\left( RT \right)\left[ B \right]\left( RT \right)} \\
& {{K}_{p}}=\dfrac{{{\left[ C \right]}^{2}}}{{{\left[ A \right]}^{2}}\left[ B \right]}\times \dfrac{{{\left( RT \right)}^{2}}}{{{\left( RT \right)}^{2}}\left( RT \right)} \\
\end{align}$
From we get,
${{K}_{p}}={{K}_{c}}\times {{\left( RT \right)}^{-1}}$
So, generally this can be stated as,
${{K}_{p}}={{K}_{c}}{{\left( RT \right)}^{\Delta n}}$ where,
$\Delta n$ is the change in number of moles of gas molecules. It is mostly the difference between the moles of product and moles of reactants.

Therefore, relation between ${{K}_{p}}$ and ${{K}_{c}}$ can be given as,
${{K}_{p}}={{K}_{c}}{{\left( RT \right)}^{\Delta n}}$ or,
${{K}_{c}}={{K}_{p}}{{\left( RT \right)}^{-\Delta n}}$

Note: We can express both the terms with respect to each other by one simpler reaction for gas phase molecules. Do note that these terms can only be related when the reaction is in the gas phase.