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Thermodynamic properties of an interfacial layer

The following definitions are useful only when $ V^s$ can be assessed unequivocally on the basis of a physica1 model of the interfacial layer, or when $ V^s$ can be taken as negligibly small.

Interfacial energy ($ U^s$) is defined by

$\displaystyle U^s = U - U^{\alpha}- U^{\beta}= U - V^{\alpha}\left(\frac{U^{\al...
-V^{\beta}\left(\frac{U^{\beta}_m}{V^{\beta}_m}\right), $

where $ U$ is the total energy of the system, and $ U^{\alpha}$ and $ U^{\beta}$ are the energies attributed to the bulk phases $ {\alpha}$ and $ {\beta}$ of volumes $ V^{\alpha}$ and $ V^{\beta}$ subject to the condition

$\displaystyle V = V^{\alpha}+V^{\beta}+V^s, $

where $ V$ is the total volume.

Interfacial entropy ($ S^s$) is defined by

$\displaystyle S^s = S - S^{\alpha}- S^{\beta}= S - V^{\alpha}\left(\frac{S^{\al...
-V^{\beta}\left(\frac{S^{\beta}_m}{V^{\beta}_m}\right), $

where $ S$ is the total entropy of the system.

Interfacial Helmholtz energy ($ A^s$) is defined by

$\displaystyle A^s = U^s - TS^s. $

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

$\displaystyle u^s = U^s/A_s, $

$\displaystyle s^s = S^s/A_s, $

$\displaystyle a^s = A^s/A_s. $

Enthalpy and Gibbs energy. When the state of a system depends upon more than one pair of conjugate mechanical (or electrical) variables, i.e. more than $ (p, V)$, then it is possible to derive several sets of functions having the character of enthalpies and Gibbs energies. These functions are related in the following way

Energy [r]^-xY [d]^-TS & Enthalpy[d]^-TS
Helmholtz energy [r]^-xY & Gibbs energy

where $ x$ is an intensive mechanical (or electrical) variable and $ Y$ the conjugate extensive variable.

The properties of interfacial layers depend on both $ (p, V^s)$ and $ ({\gamma}, A_s)$. In defining an enthalpy in terms of the corresponding energy either $ -pV^s$, $ {\gamma}A_s$ or $ -(pV^s -{\gamma}A_s)$ may be subtracted from the energy function. There are thus three possible definitions of interfacial enthalpy:

$\displaystyle \mathscr{H}^s$ $\displaystyle =$ $\displaystyle U^s + pV^s,$  
$\displaystyle \hat{H}^s$ $\displaystyle =$ $\displaystyle U^s - {\gamma}A_s,$  
$\displaystyle H^s$ $\displaystyle =$ $\displaystyle U^s + pV^s - {\gamma}A_s,$  

and three definitions of the interfacial Gibbs energy

\mathscr{G}^s & = & A^s+pV^s & = &\maths...

No distinguishing names have been suggested for these different functions.

A possible nomenclature, if one is needed, could be

$\displaystyle \mathscr{H}$ $\displaystyle =$ $\displaystyle pV$-enthalpy$\displaystyle ,$  
$\displaystyle \hat{H}$ $\displaystyle =$ $\displaystyle {\gamma}A$-enthalpy$\displaystyle ,$  
$\displaystyle H$ $\displaystyle =$ $\displaystyle pV{\gamma}A$-enthalpy$\displaystyle ,$  
$\displaystyle \mathscr{G}$ $\displaystyle =$ $\displaystyle pV$-Gibbs energy$\displaystyle ,$  
$\displaystyle \hat{G}$ $\displaystyle =$ $\displaystyle {\gamma}A$-Gibbs energy$\displaystyle ,$  
$\displaystyle G$ $\displaystyle =$ $\displaystyle pV{\gamma}A$-Gibbs energy$\displaystyle .$  

Of these $ \hat{H}^s$ and $ \hat{G}^s$ are not often used.

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Next: Surface chemical potentials Up: Thermodynamic properties Previous: A   Contents