For which of the following gaseous reactions is the change in...

The relationship between change in enthalpy ($\Delta H$) and change in internal energy ($\Delta U$) for a gaseous reaction at constant temperature is given by:

$\Delta H = \Delta U + \Delta n_g RT$

where $\Delta n_g$ is the difference between the number of moles of gaseous products and the number of moles of gaseous reactants.

For $\Delta H$ to be exactly equal to $\Delta U$, the value of $\Delta n_g$ must be zero.

Evaluating $\Delta n_g$ for the given reactions:

For $PCl_5(g) \rightarrow PCl_3(g) + Cl_2(g)$, $\Delta n_g = (1 + 1) - 1 = 1$

For $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$, $\Delta n_g = 2 - (1 + 3) = -2$

For $N_2(g) + O_2(g) \rightarrow 2NO(g)$, $\Delta n_g = 2 - (1 + 1) = 0$

For $2SO_2(g) + O_2(g) \rightarrow 2SO_3(g)$, $\Delta n_g = 2 - (2 + 1) = -1$

Since $\Delta n_g = 0$ for the reaction $N_2(g) + O_2(g) \rightarrow 2NO(g)$, $\Delta H = \Delta U$ for this reaction.

Answer: $N_2(g) + O_2(g) \rightarrow 2NO(g)$