Derivation of the Gibbs adsorption equation For a two-phase system consisting of the and phase in equilibrium with a surface dividing the phases, the total
Gibbs free energy of a system can be written as: : G = G^{\alpha}\,+ G^{\beta}\, + G^{\mathrm{S}}\,, where is the Gibbs free energy. The equation of the Gibbs Adsorption Isotherm can be derived from the “particularization to the thermodynamics of the Euler theorem on homogeneous first-order forms.” The Gibbs free energy of each phase , phase , and the surface phase can be represented by the equation: : G = U+pV-TS+\sum_{i=1}^k \mu_i \,\mathrm{n}_i \,, where is the
internal energy, is the pressure, is the volume, is the temperature, is the entropy, and is the chemical potential of the -th component. By taking the
total derivative of the Euler form of the Gibbs equation for the phase, phase and the surface phase: : \mathrm{d}G = \sum_{\alpha,\beta,S}\,\left( \mathrm{d}U +p\mathrm{d}V\,+ V\mathrm{d}p\,-T\mathrm{d}S\,-S\mathrm{d}T\,+\sum_{i=1}^k \mu_i \,\mathrm{d}n_i\,+ \sum_{i=1}^k \mathrm{n}_i \,\mathrm{d}\mu_i\,\right) +A\mathrm{d}\gamma\, + \gamma\mathrm{d}A\,, where is the area of the dividing surface, and is the
surface tension. For reversible processes, the
first law of thermodynamics requires that: : \mathrm{d}U = \delta\,q + \delta\,w\,, where is the heat energy and is the work. : \delta\,q + \delta\,w = \sum_{\alpha,\beta,S}\,\left( T\mathrm{d}S\, - p\mathrm{d}V\, -\delta\,w_{\text{non-pV}}\right)\,. Substituting the above equation into the total derivative of the Gibbs energy equation and by utilizing the result is equated to the non-pressure volume work when surface energy is considered: : \mathrm{d}G = \sum_{\alpha,\beta,S}\,\left( V\mathrm{d}p\,-S\mathrm{d}T\,+\sum_{i=1}^k \mu_i \,\mathrm{d}n_i\,+ \sum_{i=1}^k \mathrm{n}_i \,\mathrm{d}\mu_i\,\right) +A\mathrm{d}\gamma\,, by utilizing the fundamental equation of Gibbs energy of a multicomponent system: : \mathrm{d}G = V\mathrm{d}p\,-S\mathrm{d}T\,+\sum_{i=1}^k \mu_i \,\mathrm{d}n_i\,. The equation relating the phase, phase and the surface phase becomes: : \sum_{i=1}^k \mathrm{n_i}^\alpha\,\mathrm{d}\mu_i\,+\sum_{i=1}^k \mathrm{n_i}^\beta\,\mathrm{d}\mu_i\,+\sum_{i=1}^k \mathrm{n_i}^\mathrm{S}\,\mathrm{d}\mu_i\, +A\mathrm{d}\gamma\, = 0\,. When considering the bulk phases ( phase, phase), at equilibrium at constant temperature and pressure the
Gibbs–Duhem equation requires that: : \sum_{i=1}^k \mathrm{n_i}^\alpha\,\mathrm{d}\mu_i\,+ \sum_{i=1}^k \mathrm{n_i}^\beta\,\mathrm{d}\mu_i\, = 0\,. The resulting equation is the Gibbs adsorption isotherm equation: : \sum_{i=1}^k \mathrm{n_i}^\mathrm{S}\,\mathrm{d}\mu_i\, +A\mathrm{d}\gamma\, = 0\,. The Gibbs adsorption isotherm is an equation which could be considered an
adsorption isotherm that connects
surface tension of a solution with the concentration of the solute. For a binary system containing two components the Gibbs Adsorption Equation in terms of surface excess is: : -\mathrm{d}\gamma\ = \Gamma_1\mathrm{d}\mu_1\, + \Gamma_2\mathrm{d}\mu_2\,.
Relation between surface tension and the surface excess concentration The chemical potential of species in solution, \mu_i, depends on its activity a_i by the following equation: : \mu_i = {\mu_i}^o + RT \ln a_i\,, where {\mu_i}^o is the chemical potential of the -th component at a reference state, is the
gas constant and is the temperature. Differentiation of the chemical potential equation results in: : \mathrm{d}\mu_i = RT \frac{\mathrm{d}a_i}{a_i} = RT \mathrm{d}\ln fC_i\,, where is the activity coefficient of component , and is the concentration of species in the bulk phase. If the solutions in the and phases are dilute (rich in one particular component ) then
activity coefficient of the component approaches unity and the Gibbs isotherm becomes: : \Gamma_i = - \frac{1}{RT} \left( \frac{\partial \gamma}{\partial \ln C_i} \right)_{T,p} \,. The above equation assumes the interface to be bidimensional, which is not always true. Further models, such as Guggenheim's, correct this flaw. ==Ionic dissociation effects==