When the addition of a differential amount \(\mathrm{d}n_{i}^{\sigma }\) or \(\mathrm{d}n_{i}^{\text{s}}\) is effected at constant gas volume, the differential molar energy of adsorption of component \(i\), \(\Delta _{a}U_{i}^{\sigma }\) or \(\Delta _{a}U_{i}^{\text{s}}\), is defined as: \[\Delta _{a}U_{i}^{\sigma }=U_{i}^{\sigma }- U_{i}^{\text{g}}\] or \[\Delta _{a}U_{i}^{\text{s}} = U_{i}^{\text{s}}- U_{i}^{\text{g}}\] where the differential molar surface excess energy, \(U_{i}^{\sigma }\), is given by \[U_{i}^{\sigma }=(\frac{\partial U^{\sigma }}{\partial n_{i}^{\text{s}}})_{T,m,n_{j}^{\sigma }}=(\frac{\partial U}{\partial n_{i}^{\sigma }})_{T,m,V^{\text{g}},p_{i},n_{j}^{\sigma }}\] and the differential molar interfacial energy, \(U_{i}^{s}\), by \[U_{i}^{\text{s}}=(\frac{\partial U}{\partial n_{i}^{\text{s}}})_{T,m,V^{\text{g}},p_{i},n_{j}^{\sigma }}=(\frac{\partial U}{\partial n_{i}^{\text{s}}})_{T,m,V^{\text{g}},V^{\text{s}},p_{i},n_{j}^{\text{s}}}\] \(U_{i}^{\text{g}}\) is the differential molar energy of component \(i\) in the gas phase, i.e. \[(\frac{\partial U}{\partial n_{i}^{\text{g}}})_{T,V,n_{i}^{\text{g}}}\]