Gibbs isotherm

In the presence of two phases (α and β), the surface (surface phase) is located in between the phase α and phase β. Experimentally, it is difficult to determine the exact structure of an inhomogeneous surface phase that is in contact with a bulk liquid phase containing more than one solute.[2] Inhomogeneity of the surface phase is a result of variation of mole ratios.[1] A model proposed by Josiah Willard Gibbs proposed that the surface phase as an idealized model that had zero thickness. In reality, although the bulk regions of α and β phases are constant, the concentrations of components in the interfacial region will gradually vary from the bulk concentration of α to the bulk concentration of β over the distance x. This is in contrast to the idealized Gibbs model where the distance x takes on the value of zero. The diagram to the right illustrates the differences between the real and idealized models.

In the idealized model, the chemical components of the α and β bulk phases remain unchanged except when approaching the dividing surface.[3] The total moles of any component (Examples include: water, ethylene glycol etc.) remains constant in the bulk phases but varies in the surface phase for the real system model as shown below.

In the real system, however, the total moles of a component varies depending on the arbitrary placement of the dividing surface. The quantitative measure of adsorption of the i-th component is captured by the surface excess quantity.[1] The surface excess represents the difference between the total moles of the i-th component in a system and the moles of the i-th component in a particular phase (either α or β) and is represented by:

${\displaystyle \Gamma _{i}={\frac {{n_{i}}^{\text{TOTAL}}-{n_{i}}^{\alpha }\,-{n_{i}}^{\beta }\,}{A}}\,,}$

where Γ_i is the surface excess of the i-th component, n are the moles, α and β are the phases, and A is the area of the dividing surface.

Γ represents excess of solute per unit area of the surface over what would be present if the bulk concentration prevailed all the way to the surface, it can be positive, negative or zero. It has units of mol/m².

Relative Surface Excess quantities are more useful than arbitrary surface excess quantities.[3] The Relative surface excess relates the adsorption at the interface to a solvent in the bulk phase. An advantage of using the relative surface excess quantities is that they don't depend on the location of the dividing surface. The relative surface excess of species i and solvent 1 is therefore:

${\displaystyle \Gamma _{i}^{1}=\Gamma _{i}-\Gamma _{1}\,\left({\frac {{C_{i}}^{\alpha }\,-{C_{i}}^{\beta }\,}{{C_{1}}^{\alpha }\,-{C_{1}}^{\beta }\,}}\right)\,.}$

For a two-phase system consisting of the α and β phase in equilibrium with a surface S dividing the phases, the total Gibbs free energy of a system can be written as:

${\displaystyle G=G^{\alpha }\,+G^{\beta }\,+G^{\mathrm {S} }\,,}$

where G 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.”[4] The Gibbs free energy of each phase α, phase β, and the surface phase can be represented by the equation:

${\displaystyle G=U+pV-TS+\sum _{i=1}^{k}\mu _{i}\,\mathrm {n} _{i}\,,}$

where U is the internal energy, p is the pressure, V is the volume, T is the temperature, S is the entropy, and μ_i is the chemical potential of the i-th component.

By taking the total derivative of the Euler form of the Gibbs equation for the α phase, β phase and the surface phase:

${\displaystyle \mathrm {d} G=\sum _{\alpha ,\beta ,S}\,\left(\mathrm {d} U+p\mathrm {d} V\,+V\mathrm {d} p\,-T\mathrm {d} S\,-S\mathrm {d} T\,+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} n_{i}\,+\sum _{i=1}^{k}\mathrm {n} _{i}\,\mathrm {d} \mu _{i}\,\right)+A\mathrm {d} \gamma \,+\gamma \mathrm {d} A\,,}$

where A is the area of the dividing surface, and γ is the surface tension.

For reversible processes, the first law of thermodynamics requires that:

${\displaystyle \mathrm {d} U=\delta \,q+\delta \,w\,,}$

where q is the heat energy and w is the work.

${\displaystyle \delta \,q+\delta \,w=\sum _{\alpha ,\beta ,S}\,\left(T\mathrm {d} S\,-p\mathrm {d} V\,-\delta \,w_{\text{non-pV}}\right)\,.}$

Substituting the above equation into the total derivative of the Gibbs energy equation and by utilizing the result γdA is equated to the non-pressure volume work when surface energy is considered:

${\displaystyle \mathrm {d} G=\sum _{\alpha ,\beta ,S}\,\left(V\mathrm {d} p\,-S\mathrm {d} T\,+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} n_{i}\,+\sum _{i=1}^{k}\mathrm {n} _{i}\,\mathrm {d} \mu _{i}\,\right)+A\mathrm {d} \gamma \,,}$

by utilizing the fundamental equation of Gibbs energy of a multicomponent system:

${\displaystyle \mathrm {d} G=V\mathrm {d} p\,-S\mathrm {d} T\,+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} n_{i}\,.}$

The equation relating the α phase, β phase and the surface phase becomes:

${\displaystyle \sum _{i=1}^{k}\mathrm {n_{i}} ^{\alpha }\,\mathrm {d} \mu _{i}\,+\sum _{i=1}^{k}\mathrm {n_{i}} ^{\beta }\,\mathrm {d} \mu _{i}\,+\sum _{i=1}^{k}\mathrm {n_{i}} ^{\mathrm {S} }\,\mathrm {d} \mu _{i}\,+A\mathrm {d} \gamma \,=0\,.}$

When considering the bulk phases (α phase, β phase), at equilibrium at constant temperature and pressure the Gibbs–Duhem equation requires that:

${\displaystyle \sum _{i=1}^{k}\mathrm {n_{i}} ^{\alpha }\,\mathrm {d} \mu _{i}\,+\sum _{i=1}^{k}\mathrm {n_{i}} ^{\beta }\,\mathrm {d} \mu _{i}\,=0\,.}$

The resulting equation is the Gibbs adsorption isotherm equation:

${\displaystyle \sum _{i=1}^{k}\mathrm {n_{i}} ^{\mathrm {S} }\,\mathrm {d} \mu _{i}\,+A\mathrm {d} \gamma \,=0\,.}$

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:

${\displaystyle -\mathrm {d} \gamma \ =\Gamma _{1}\mathrm {d} \mu _{1}\,+\Gamma _{2}\mathrm {d} \mu _{2}\,.}$

The chemical potential of species i in solution depends on the activity a using the following equation:[2]

${\displaystyle \mu _{i}={\mu _{i}}^{o}+RT\ln a_{i}\,,}$

where μ_i is the chemical potential of the i-th component, μ_i^o is the chemical potential of the i-th component at a reference state, R is the gas constant, T is the temperature, and a_i is the activity of the i-th component.

Differentiation of the chemical potential equation results in:

${\displaystyle \mathrm {d} \mu _{i}=RT{\frac {\mathrm {d} a_{i}}{a_{i}}}=RT\mathrm {d} \ln fC_{i}\,,}$

where f is the activity coefficient of component i, and C is the concentration of species i in the bulk phase.

If the solutions in the α and β phases are dilute (rich in one particular component i) then activity coefficient of the component i approaches unity and the Gibbs isotherm becomes:

${\displaystyle \Gamma _{i}=-{\frac {1}{RT}}\left({\frac {\partial \gamma }{\partial \ln C_{i}}}\right)_{T,p}\,.}$

The above equation assumes the interface to be bidimensional, which is not always true. Further models, such as Guggenheim's, correct this flaw.

Consider a system composed of water that contains an organic electrolyte RNaz and an inorganic electrolyte NaCl that both dissociate completely such that:

${\displaystyle \mathrm {R} \,\mathrm {Na_{\text{z}}} =\mathrm {R} ^{\text{z-}}+\mathrm {z} \,\mathrm {Na^{\text{+}}} \,.}$

The Gibbs Adsorption equation in terms of the relative surface excess becomes:

${\displaystyle -\mathrm {d} \gamma =\Gamma _{\text{RNaz}}^{W}\,\mathrm {d} \mu _{\text{RNaz}}\,+\Gamma _{\text{NaCl}}^{W}\,\mathrm {d} \mu _{\text{NaCl}}\,.}$

The Relation Between Surface Tension and The Surface Excess Concentration becomes:

${\displaystyle -\mathrm {d} \gamma =mRT\Gamma _{i}^{W}\,\mathrm {d} \ln C_{i}\,,}$

where m is the coefficient of the Gibbs adsorption.[3] Values of m are calculated using the Double layer (interfacial) models of Helmholtz, Gouy, and Stern.

Substances can have different effects on surface tension as shown :

• No effect, for example sugar
• Increase of surface tension, inorganic salts
• Decrease surface tension progressively, alcohols
• Decrease surface tension and, once a minimum is reached, no more effect: surfactants

Therefore, the Gibbs isotherm predicts that inorganic salts have negative surface concentrations. However, this view has been challenged extensively in recent years due to a combination of more precise interfacially sensitive experiments and theoretical models, both of which predict an increase in surface propensity of the halides with increasing size and polarizability.[5] As such, surface tension is not a reliable method for determining the relative propensity of ions toward the air-water interface.

A method for determining surface concentrations is needed in order to prove the validity of the model: two different techniques are normally used: ellipsometry and following the decay of ¹⁴C present in the surfactant molecules.

Ionic surfactants require special considerations, as they are electrolytes:

• In absence of extra electrolytes

${\displaystyle \Gamma _{S}=-{\frac {1}{2RT}}\left({\frac {\partial \gamma }{\partial \ln C}}\right)_{T,p}\,,}$

where ${\displaystyle \Gamma _{S}}$ refers to the surface concentration of surfactant molecules, without considering the counter ion.

• In presence of added electrolytes

${\displaystyle \Gamma _{S}=-{\frac {1}{RT}}\left({\frac {\partial \gamma }{\partial \ln C}}\right)_{T,p}\,.}$