Equilibrium chemistry

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.

Thermodynamic equilibrium

A chemical system is said to be in equilibrium when the quantities of the chemical entities involved do not and cannot change in time without the application of an external influence. In this sense a system in chemical equilibrium is in a stable state. The system at chemical equilibrium will be at a constant temperature, pressure or volume and a composition. It will be insulated from exchange of heat with the surroundings, that is, it is a closed system. A change of temperature, pressure constitutes an external influence and the equilibrium quantities will change as a result of such a change. If there is a possibility that the composition might change, but the rate of change is negligibly slow, the system is said to be in a metastable state. The equation of chemical equilibrium can be expressed symbolically as
The sign means "are in equilibrium with". This definition refers to macroscopic properties. Changes do occur at the microscopic level of atoms and molecules, but to such a minute extent that they are not measurable and in a balanced way so that the macroscopic quantities do not change. Chemical equilibrium is a dynamic state in which forward and backward reactions proceed at such rates that the macroscopic composition of the mixture is constant. Thus, equilibrium sign symbolizes the fact that reactions occur in both forward and backward directions.
A steady state, on the other hand, is not necessarily an equilibrium state in the chemical sense. For example, in a radioactive decay chain the concentrations of intermediate isotopes are constant because the rate of production is equal to the rate of decay. It is not a chemical equilibrium because the decay process occurs in one direction only.
Thermodynamic equilibrium is characterized by the free energy for the whole system being a minimum. For systems at constant volume the Helmholtz free energy is minimum and for systems at constant pressure the Gibbs free energy is minimum. Thus a metastable state is one for which the free energy change between reactants and products is not minimal even though the composition does not change in time.
The existence of this minimum is due to the free energy of mixing of reactants and products being always negative. For ideal solutions the enthalpy of mixing is zero, so the minimum exists because the entropy of mixing is always positive. The slope of the reaction free energy, δG_r with respect to the reaction coordinate, ξ, is zero when the free energy is at its minimum value.

Equilibrium constant

is the partial molar free energy. The potential, μ_i, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, N_i:
A general chemical equilibrium can be written as
n_j are the stoichiometric coefficients of the reactants in the equilibrium equation, and m_j are the coefficients of the products. The value of δG_r for these reactions is a function of the chemical potentials of all the species.
The chemical potential, μ_i, of the ith species can be calculated in terms of its activity, a_i.
μ is the standard chemical potential of the species, R is the gas constant and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δG_r = 0:
Rearranging the terms,
This relates the standard Gibbs free energy change, ΔG^o to an equilibrium constant, K, the reaction quotient of activity values at equilibrium.
It follows that any equilibrium of this kind can be characterized either by the standard free energy change or by the equilibrium constant. In practice concentrations are more useful than activities. Activities can be calculated from concentrations if the activity coefficient are known, but this is rarely the case. Sometimes activity coefficients can be calculated using, for example, Pitzer equations or Specific ion interaction theory. Otherwise conditions must be adjusted so that activity coefficients do not vary much. For ionic solutions this is achieved by using a background ionic medium at a high concentration relative to the concentrations of the species in equilibrium.
If activity coefficients are unknown they may be subsumed into the equilibrium constant, which becomes a concentration quotient. Each activity a_i is assumed to be the product of a concentration, , and an activity coefficient, γ_i:
This expression for activity is placed in the expression defining the equilibrium constant.
By setting the quotient of activity coefficients, Γ, equal to one, the equilibrium constant is defined as a quotient of concentrations.
In more familiar notation, for a general equilibrium
This definition is much more practical, but an equilibrium constant defined in terms of concentrations is dependent on conditions. In particular, equilibrium constants for species in aqueous solution are dependent on ionic strength, as the quotient of activity coefficients varies with the ionic strength of the solution.
The values of the standard free energy change and of the equilibrium constant are temperature dependent. To a first approximation, the van 't Hoff equation may be used.
This shows that when the reaction is exothermic, then K decreases with increasing temperature, in accordance with Le Châtelier's principle. The approximation involved is that the standard enthalpy change, ΔH^o, is independent of temperature, which is a good approximation only over a small temperature range. Thermodynamic arguments can be used to show that
where C_p is the heat capacity at constant pressure.

Equilibria involving gases

When dealing with gases, fugacity, f, is used rather than activity. However, whereas activity is dimensionless, fugacity has the dimension of pressure. A consequence is that chemical potential has to be defined in terms of a standard pressure, p^o:
By convention p^o is usually taken to be 1 bar.
Fugacity can be expressed as the product of partial pressure, p, and a fugacity coefficient, Φ:
Fugacity coefficients are dimensionless and can be obtained experimentally at specific temperature and pressure, from measurements of deviations from ideal gas behaviour. Equilibrium constants are defined in terms of fugacity. If the gases are at sufficiently low pressure that they behave as ideal gases, the equilibrium constant can be defined as a quotient of partial pressures.
An example of gas-phase equilibrium is provided by the Haber–Bosch process of ammonia synthesis.
This reaction is strongly exothermic, so the equilibrium constant decreases with temperature. However, a temperature of around 400 °C is required in order to achieve a reasonable rate of reaction with currently available catalysts. Formation of ammonia is also favoured by high pressure, as the volume decreases when the reaction takes place. The same reaction, nitrogen fixation, occurs at ambient temperatures in nature, when the catalyst is an enzyme such as nitrogenase. Much energy is needed initially to break the nitrogen–nitrogen triple bond even though the overall reaction is exothermic.
Gas-phase equilibria occur during combustion and were studied as early as 1943 in connection with the development of the V2 rocket engine.
The calculation of composition for a gaseous equilibrium at constant pressure is often carried out using ΔG values, rather than equilibrium constants.

Multiple equilibria

Two or more equilibria can exist at the same time. When this is so, equilibrium constants can be ascribed to individual equilibria, but they are not always unique. For example, three equilibrium constants can be defined for a dibasic acid, H₂A.
The three constants are not independent of each other and it is easy to see that. The constants K₁ and K₂ are stepwise constants and β is an example of an overall constant.

Speciation

The concentrations of species in equilibrium are usually calculated under the assumption that activity coefficients are either known or can be ignored. In this case, each equilibrium constant for the formation of a complex in a set of multiple equilibria can be defined as follows
The concentrations of species containing reagent A are constrained by a condition of mass-balance, that is, the total concentration, which is the sum of all species' concentrations, must be constant. There is one mass-balance equation for each reagent of the type
There are as many mass-balance equations as there are reagents, A, B..., so if the equilibrium constant values are known, there are n mass-balance equations in n unknowns, , ..., the so-called free reagent concentrations. Solution of these equations gives all the information needed to calculate the concentrations of all the species.
Thus, the importance of equilibrium constants lies in the fact that, once their values have been determined by experiment, they can be used to calculate the concentrations, known as the speciation, of mixtures that contain the relevant species.