Derive the relation between ΔH and ΔU for gaseous phase reaction.

Answer: \(\Delta H = \Delta U + \Delta n_g RT\)

By definition \(H = U + PV\). At constant pressure, \(\Delta H = \Delta U + P\Delta V\) … (1)

For a reaction involving gases, let \(V_1\) be the volume of \(n_1\) moles of gaseous reactants and \(V_2\) the volume of \(n_2\) moles of gaseous products at the same \(P\) and \(T\). From the ideal gas equation, \(PV_1 = n_1RT\) and \(PV_2 = n_2RT\).

\(P\Delta V = P(V_2 - V_1) = (n_2 - n_1)RT = \Delta n_g RT\)

Substituting in (1): \(\Delta H = \Delta U + \Delta n_g RT\), where \(\Delta n_g\) = moles of gaseous products − moles of gaseous reactants. If \(\Delta n_g = 0\), \(\Delta H = \Delta U\).