To determine in which equilibrium $K_p$ and $K_c$ are not equal, we need to consider the relationship between $K_p$ and K_c.} \\ The relationship is given by:$\\ K_p = K_c (RT)^{\Delta n} \\$where $\Delta n$ is the change in the number of moles of gas, $R$ is the gas constant, and $T$ is the temperature in Kelvin.$\\$Let's analyze each option:$\\$Option 1: $$CO(g) + H$_2$O(g) $\rightleftharpoons$ CO$_2$(g) + H$_2$(g)$\\ \Delta n = (1 + 1) - (1 + 1) = 0 \\$Since $\Delta n = 0,$ $K_p = K_c. \\$Option 2: $$2 BrCl(g) $\rightleftharpoons$ Br$_2$(g) + Cl$_2$(g)$\\ \Delta n = (1 + 1) - 2 = 0 \\$Since $\Delta n = 0,$ $K_p = K_c. \\$Option 3: $$PCl$_5$(g) $\rightleftharpoons$ PCl$_3$(g) + Cl$_2$(g)$\\ \Delta n = (1 + 1) - 1 = 1 \\$Since $\Delta n \neq 0,$ $K_p \neq K_c. \\$Option 4: $$H$_2$(g) + I$_2$(g) $\rightleftharpoons$ 2 HI(g)$\\ \Delta n = 2 - (1 + 1) = 0 \\$Since $\Delta n = 0,$ $K_p = K_c. \\$Therefore, the correct option is 3, where $K_p$ and $K_c$ are not equal.