A

Thermodynamic excess properties defined relative to a Gibbs surface.

Surface excess energy ( $U^{\sigma}$) is defined by

$$U^{\sigma}=U-U^{\alpha}-U^{\beta}=U-V^{\alpha}\left(\frac{U^{\alp... ...m}{V^{\alpha}_m}\right) -V^{\beta}\left(\frac{U^{\beta}_m}{V^{\beta}_m}\right),$$

where $V^{\alpha}_m$ and $V^{\beta}_m$ satisfy the condition

$$V^{\alpha}_m+V^{\beta}_m=V,$$

the total volume of the system.

$(U^{\alpha}_m/V^{\alpha}_m)$ and $(U^{\beta}_m/V^{\beta}_m)$ are the energy densities in the two bulk phases where $U^{\alpha}_m$ and $U^{\beta}_m$ are the mean molar energies and $V^{\alpha}_m$ and $V^{\beta}_m$ are the mean molar volumes of the two phases.

Surface excess entropy ( $S^{\sigma}$) is defined by

$$S^{\sigma}=S-S^{\alpha}-S^{\beta}=S-V^{\alpha}\left(\frac{S^{\alp... ...m}{V^{\alpha}_m}\right) -V^{\beta}\left(\frac{S^{\beta}_m}{V^{\beta}_m}\right).$$

$(S^{\alpha}_m/V^{\alpha}_m)$ and $(S^{\beta}_m/V^{\beta}_m)$ are the entropy densities in the two bulk phases, where $S^{\alpha}_m$ and $S^{\beta}_m$ are the mean molar entropies of the two phases.

Surface excess Helmholtz energy ( $A^{\sigma}$) is defined by

$$A^{\sigma}=U^{\sigma}-TS^{\sigma}.$$

Surface excess enthalpy ( $H^{\sigma}$) is defined by

$$H^{\sigma}=U^{\sigma}-{\gamma}A_s.$$

Surface excess Gibbs energy ( $G^{\sigma}$) is defined by

$$G^{\sigma}=H^{\sigma}-TS^{\sigma}=A^{\sigma}-{\gamma}A_s.$$

When the thermodynamics of surfaces is discussed in terms of excess quantities, $V^{\sigma}=0$. There is thus only one way of defining the excess surface enthalpy and excess surface Gibbs energy (cf. case B, below).

The corresponding excess quantities per unit area may be denoted by lower case letters:

$$u^{\sigma}=U^{\sigma}/A_s,$$

$$s^{\sigma}=S^{\sigma}/A_s,$$

$$a^{\sigma}=A^{\sigma}/A_s,$$

$$h^{\sigma}=H^{\sigma}/A_s,$$

$$g^{\sigma}=G^{\sigma}/A_s.$$

Quantities invariant to the choice of dividing surface may be defined as follows:

Relative (excess) surface energy (with respect to component 1)

$$U^{{\sigma}(1)}=U^{\sigma}-n^{\sigma}_1\left[ \left(\frac{U^{\alp... ...\left(\frac{U^{\beta}_m}{V^{\beta}_m}\right)\right]/(c^{\alpha}_1-c^{\beta}_1).$$

Analogous equations hold for $S^{{\sigma}(1)}$, $A^{{\sigma}(1)}$, $H^{{\sigma}(1)}$ and $G^{{\sigma}(1)}$.

Reduced (excess) surface energy

$$U^{{\sigma}(n)}=U^{\sigma}-n^{\sigma}\left[ \left(\frac{U^{\alpha... ...t) -\left(\frac{U^{\beta}_m}{V^{\beta}_m}\right)\right]/(c^{\alpha}-c^{\beta}),$$

where $n^{\sigma}$ is the total adsorption relative to an arbitrary choice of dividing surface and $c^{\alpha}$ and $c^{\beta}$ are the total concentrations of the two phases.

Analogous equations hold for $S^{{\sigma}(n)}$, $A^{{\sigma}(n)}$, $H^{{\sigma}(n)}$ and $G^{{\sigma}(n)}$.