Gibbs isotherm

Last updated

The Gibbs adsorption isotherm for multicomponent systems is an equation used to relate the changes in concentration of a component in contact with a surface with changes in the surface tension, which results in a corresponding change in surface energy. For a binary system, the Gibbs adsorption equation in terms of surface excess is

Contents

where

is the surface tension,
is the surface excess concentration of component i,
is the chemical potential of component i.

Adsorption

Different influences at the interface may cause changes in the composition of the near-surface layer. [1] Substances may either accumulate near the surface or, conversely, move into the bulk. [1] The movement of the molecules characterizes the phenomena of adsorption. Adsorption influences changes in surface tension and colloid stability. Adsorption layers at the surface of a liquid dispersion medium may affect the interactions of the dispersed particles in the media, and consequently these layers may play crucial role in colloid stability [2] The adsorption of molecules of liquid phase at an interface occurs when this liquid phase is in contact with other immiscible phases that may be gas, liquid, or solid [3]

Conceptual explanation of equation

Surface tension describes how difficult it is to extend the area of a surface (by stretching or distorting it). If surface tension is high, there is a large free energy required to increase the surface area, so the surface will tend to contract and hold together like a rubber sheet.

There are various factors affecting surface tension, one of which is that the composition of the surface may be different from the bulk. For example, if water is mixed with a tiny amount of surfactants (for example, hand soap), the bulk water may be 99% water molecules and 1% soap molecules, but the topmost surface of the water may be 50% water molecules and 50% soap molecules. In this case, the soap has a large and positive "surface excess". In other examples, the surface excess may be negative: For example, if water is mixed with an inorganic salt like sodium chloride, the surface of the water is on average less salty and more pure than the bulk average.

Consider again the example of water with a bit of soap. Since the water surface needs to have higher concentration of soap than the bulk, whenever the water's surface area is increased, it is necessary to remove soap molecules from the bulk and add them to the new surface. If the concentration of soap is increased a bit, the soap molecules are more readily available (they have higher chemical potential), so it is easier to pull them from the bulk in order to create the new surface. Since it is easier to create new surface, the surface tension is lowered. The general principle is:

When the surface excess of a component is positive, increasing the chemical potential of that component reduces the surface tension.

Next consider the example of water with salt. The water surface is less salty than bulk, so whenever the water's surface area is increased, it is necessary to remove salt molecules from the new surface and push them into bulk. If the concentration of salt is increased a bit (raising the salt's chemical potential), it becomes harder to push away the salt molecules. Since it is now harder to create the new surface, the surface tension is higher. The general principle is:

When the surface excess of a component is negative, increasing the chemical potential of that component increases the surface tension.

The Gibbs isotherm equation gives the exact quantitative relationship for these trends.

Location of surface and defining surface excess

Figure 1: Comparison between the real and idealized models of the surface Gibbs image 2.png
Figure 1: Comparison between the real and idealized models of the surface

Location of surface

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.

Definition of surface excess

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.

Figure 2: Variation in the concentration of components in the surface phase of the real model Gibbs image 3.png
Figure 2: Variation in the concentration of components in the surface phase of the real model

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:

where Γi 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

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:

The Gibbs adsorption isotherm equation

Derivation of the Gibbs adsorption equation

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:

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:

where U is the internal energy, p is the pressure, V is the volume, T is the temperature, S is the entropy, and μi 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:

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:

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

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:

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

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

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

The resulting equation is the Gibbs adsorption isotherm equation:

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:

Relation between surface tension and the surface excess concentration

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

where μi is the chemical potential of the i-th component, μio is the chemical potential of the i-th component at a reference state, R is the gas constant, T is the temperature, and ai is the activity of the i-th component.

Differentiation of the chemical potential equation results in:

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:

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

Gibbs equation for electrolyte adsorption

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

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

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

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 : Gibbs image 4.png

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.

Gibbs isotherm for ionic surfactants

Ionic surfactants require special considerations, as they are electrolytes:

where refers to the surface concentration of surfactant molecules, without considering the counter ion.

electrical microtome Microm HM 200. Electrical microtome.jpg
electrical microtome Microm HM 200.

Experimental methods

The extent of adsorption at a liquid interface can be evaluated using the surface tension concentration data and the Gibbs adsorption equation. [3] The microtome blade method is used to determine the weight and molal concentration of an interface. The method involves attaining a one square meter portion of air-liquid interface of binary solutions using a microtome blade.

Another method that is used to determine the extent of adsorption at an air-water interface is the emulsion technique, which can be used to estimate the relative surface excess with respect to water. [3]

Additionally, the Gibbs surface excess of a surface active component for an aqueous solution can be found using the radioactive tracer method. The surface active component is usually labeled with carbon-14 or sulfur-35. [3]

Related Research Articles

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

<span class="mw-page-title-main">Stress–energy tensor</span> Tensor describing energy momentum density in spacetime

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

<span class="mw-page-title-main">Beta distribution</span> Probability distribution

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.

<span class="mw-page-title-main">Surface energy</span> Excess energy at the surface of a material relative to its interior

In surface science, surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energetically favorable than the bulk of the material, otherwise there would be a driving force for surfaces to be created, removing the bulk of the material by sublimation. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding between the two created surfaces.

In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed directly in terms of simple concentrations or partial pressures of the substances present e.g. Raoult's law. Deviations from ideality are accommodated by modifying the concentration by an activity coefficient. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a fugacity coefficient.

<span class="mw-page-title-main">Wetting</span> Ability of a liquid to maintain contact with a solid surface

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. This happens in presence of a gaseous phase or another liquid phase not miscible with the first one. The degree of wetting (wettability) is determined by a force balance between adhesive and cohesive forces. There are two types of wetting: non-reactive wetting and reactive wetting.

<span class="mw-page-title-main">Electromagnetic tensor</span> Mathematical object that describes the electromagnetic field in spacetime

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below.

The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Paul Dirac in 1928.

<span class="mw-page-title-main">Gibbs–Duhem equation</span> Equation in thermodynamics

In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:

<span class="mw-page-title-main">Mathematical descriptions of the electromagnetic field</span> Formulations of electromagnetism

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In materials science, segregation is the enrichment of atoms, ions, or molecules at a microscopic region in a materials system. While the terms segregation and adsorption are essentially synonymous, in practice, segregation is often used to describe the partitioning of molecular constituents to defects from solid solutions, whereas adsorption is generally used to describe such partitioning from liquids and gases to surfaces. The molecular-level segregation discussed in this article is distinct from other types of materials phenomena that are often called segregation, such as particle segregation in granular materials, and phase separation or precipitation, wherein molecules are segregated in to macroscopic regions of different compositions. Segregation has many practical consequences, ranging from the formation of soap bubbles, to microstructural engineering in materials science, to the stabilization of colloidal suspensions.

<span class="mw-page-title-main">Langmuir adsorption model</span> Model describing the adsorption of a mono-layer of gas molecules on an ideal flat surface

The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressure i.e. at these conditions the adsorbate's partial pressure is related to its volume V adsorbed onto a solid adsorbent. The adsorbent, as indicated in the figure, is assumed to be an ideal solid surface composed of a series of distinct sites capable of binding the adsorbate. The adsorbate binding is treated as a chemical reaction between the adsorbate gaseous molecule and an empty sorption site S. This reaction yields an adsorbed species with an associated equilibrium constant :

Equilibrium chemistry is concerned with systems in chemical equilibrium. The unifying principle is that the free energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero. This principle, applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid–base, host–guest, metal–complex, solubility, partition, chromatography and redox equilibria.

Adsorption is the adhesion of ions or molecules onto the surface of another phase. Adsorption may occur via physisorption and chemisorption. Ions and molecules can adsorb to many types of surfaces including polymer surfaces. A polymer is a large molecule composed of repeating subunits bound together by covalent bonds. In dilute solution, polymers form globule structures. When a polymer adsorbs to a surface that it interacts favorably with, the globule is essentially squashed, and the polymer has a pancake structure.

The potential theory of Polanyi, also called Polanyi adsorption potential theory, is a model of adsorption proposed by Michael Polanyi where adsorption can be measured through the equilibrium between the chemical potential of a gas near the surface and the chemical potential of the gas from a large distance away. In this model, he assumed that the attraction largely due to Van Der Waals forces of the gas to the surface is determined by the position of the gas particle from the surface, and that the gas behaves as an ideal gas until condensation where the gas exceeds its equilibrium vapor pressure. While the adsorption theory of Henry is more applicable in low pressure and BET adsorption isotherm equation is more useful at from 0.05 to 0.35 P/Po, the Polanyi potential theory has much more application at higher P/Po (~0.1–0.8).

References

  1. 1 2 3 4 Shchukin, E. D., Pertsov, A. V., Amelina E. A. and Zelenev, A. S. Colloid and Surface Chemistry. 1st ed. Mobius D. and Miller R. Vol. 12. Amsterdam: Elsevier Science B.V. 2001.
  2. 1 2 3 Hiemenz, Paul C. and Rajagopalan, Raj. Principles of Colloid and Surface Chemistry. 3rd ed. New York: Marcel Dekker, Inc, 1997.
  3. 1 2 3 4 5 6 7 Chattoraj, D. K. and Birdi, K. S. Adsorption and the Gibbs Surface Excess. New York: Plenum Publishing Company, 1984.
  4. Callen, Herbert B. Thermodynamics and an Introduction to Thermostatics. 2nd ed. Canada: John Wiley & Sons, Inc, 1985.
  5. Petersen, Poul B.; Saykally, Richard J. (2006). "On the Nature of Ions at the Liquid Water Surface". Annual Review of Physical Chemistry. 57 (1): 333–364. doi:10.1146/annurev.physchem.57.032905.104609.