In which of the following equilibria, K_p and K_c are NOT equal?

The relationship between the equilibrium constant expressed in terms of partial pressures ($K_p$) and the equilibrium constant expressed in terms of molar concentrations ($K_c$) is given by the following equation:
$K_p=K_c{\left(RT\right)}^{\Delta n_g}$
where:
• $R$ is the universal gas constant,
• $T$ is the absolute temperature in Kelvin, and
• $\Delta n_g$ is the change in the number of moles of gaseous products and gaseous reactants:
$\Delta n_g=\sum n_{\text{products (g)}}-\sum n_{\text{reactants (g)}}$

For $K_p$ and $K_c$ to be equal:
${\left(RT\right)}^{\Delta n_g}=1$
which occurs when $\Delta n_g=0$.
Conversely, $K_p$ and $K_c$ are NOT equal when $\Delta n_g\ne 0$.

Let us evaluate $\Delta n_g$ for each given option:

Option 1: ${\mathrm{PCl}}_{5\left(g\right)}\rightleftharpoons {\mathrm{PCl}}_{3\left(g\right)}+{\mathrm{Cl}}_{2\left(g\right)}$
• Gaseous moles of products = 1 (from ${\mathrm{PCl}}_3$) + 1 (from ${\mathrm{Cl}}_2$) = 2
• Gaseous moles of reactants = 1 (from ${\mathrm{PCl}}_5$)
• $\Delta n_g=2-1=1$
Since $\Delta n_g\ne 0$, $K_p\ne K_c$ for this reaction.

Option 2: $H_{2\left(g\right)}+I_{2\left(g\right)}\rightleftharpoons 2{\mathrm{HI}}_{\left(g\right)}$
• Gaseous moles of products = 2
• Gaseous moles of reactants = 1 + 1 = 2
• $\Delta n_g=2-2=0$
Here, $K_p=K_c$.

Option 3: ${\mathrm{CO}}_{\left(g\right)}+H_2{O}_{\left(g\right)}\rightleftharpoons {\mathrm{CO}}_{2\left(g\right)}+H_{2\left(g\right)}$
• Gaseous moles of products = 1 + 1 = 2
• Gaseous moles of reactants = 1 + 1 = 2
• $\Delta n_g=2-2=0$
Here, $K_p=K_c$.

Option 4: $2{\mathrm{BrCl}}_{\left(g\right)}\rightleftharpoons {\mathrm{Br}}_{2\left(g\right)}+{\mathrm{Cl}}_{2\left(g\right)}$
• Gaseous moles of products = 1 + 1 = 2
• Gaseous moles of reactants = 2
• $\Delta n_g=2-2=0$
Here, $K_p=K_c$.

Therefore, the equilibrium in which $K_p$ and $K_c$ are not equal is:
${\mathrm{PCl}}_{5\left(g\right)}\rightleftharpoons {\mathrm{PCl}}_{3\left(g\right)}+{\mathrm{Cl}}_{2\left(g\right)}$