The relationship between enthalpy change ($\Delta H$) and internal energy change ($\Delta E$ or $\Delta U$) is given by the equation: $\Delta H = \Delta E + \Delta n_g RT$. For the given reaction: $\text{C}_2\text{H}_5\text{OH}(l) + 3\text{O}_2(g) \rightarrow 2\text{CO}_2(g) + 3\text{H}_2\text{O}(l)$ The change in the number of gaseous moles, $\Delta n_g = n_p(g) - n_r(g) = 2 - 3 = -1$. Substituting $\Delta n_g$ in the equation, we get: $\Delta H = \Delta E + (-1)RT \Rightarrow \Delta H = \Delta E - RT$.