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. $(\frac{\partial U}{\partial n_{i}^{\text{g}}})_{T,V,n_{i}^{\text{g}}}$