7. Real Adsorbed Solution Theory

Todo

implement this into the code

The Gibbs adsorption isotherm is

(1)\[-a \mathrm{d}\Pi + \sum_i x_i \mathrm{d}\mu_i = 0\]

where \(a\) is the surface area per mole of adsorbate, \(\Pi\) is the spreading pressure, \(x_i\) is the adsorbed mole fraction of component i, and \(\mu_i\) is the adsorbed-phase chemical potential of component i.

For change in equilibrium conditions,

(2)\[\mathrm{d}\mu_i = \mathrm{d}\mu_i^\text{g} = RT \mathrm{d} \ln{\hat{f}_i^\text{g}}\]

where \(\hat{f}_i^\text{g}\) is the fugacity of component i in the gas phase. And the substituting surface area of the adsorbent is

(3)\[A = a \sum_i q_i\]

where \(q_i\) is the loading of component i. Substituting Equations (2) and (3) into Equation (1) yields

(4)\[\frac{A}{RT} \mathrm{d}\Pi = \sum_i q_i \mathrm{d} \ln{\hat{f}_i^\text{g}}\]

If we have a good description of the multicomponent isotherms,

\[q_i = F(\{\hat{f}_k\})\]

where \(F\) is an isotherm function, Equation (4) can be simplified to

\[\frac{A\Pi}{RT} = \sum_i \int_0^{\hat{f}_i^\text{g}}\frac{q_i}{f_i^\prime}\mathrm{d}f_i^\prime\]

where \(f_i^\prime\) is a dummy variable for integration.