Fundamental relations for surfaces

In this section it will be shown that there are additional solubility effects associated with interfaces.

Consider:

$$\input{equations/29-2A}$$ (35-19)

since the dependent variables are all extensive, we can integrate (i.e. homogeneous degree $1$ in all of its variables), therefore

$$\input{equations/29-2B}$$ (35-20)

Taking the derivative (as was done when deriving the Gibbs-Duhem equation);

$$d\ensuremath{{U}^{\mbox{surf}}}=Td \ensuremath{{S}^{\mbox{surf}}}... ...remath{{N_i}^{\mbox{surf}}}+ \sum_{i=1}^C\ensuremath{{N_i}^{\mbox{surf}}}d\mu_i$$ (35-21)

comparing to Equation 35-19,

$$\input{equations/29-2C}$$ (35-22)

which expresses a relation between variations of the intensive degrees of freedom for a surface to remain in equilibrium.

Dividing through by the total surface area (so as to normalize by the area, creating derived intensive variables) and defining

$$\input{equations/29-3A}$$ (35-23)

as the entropy of the surface per area, then,

$$\input{equations/gibbs-abs}$$ (35-24)

where

$$\input{equations/Gamma-def}$$ (35-25)

is the standard notation for the excess surface concentration.

Holding everything (temperature, et cetera) constant except $\mu_1$, we get a relation that expresses the relation between the change in surface tension to the change in chemical potential of an absorbing species:

$$\input{equations/29-4A}$$ (35-26)

This is the ``Gibbs Absorption Isotherm.''

Note that if a species absorbs to the surface $\Gamma_i > 0$ and the surface tension decreases as the chemical potential of that species is increased.