Equilibrium reactions: a primer

Notation and definitions are described in Nomenclature.

Consider the hypothetical reactions among species B, C and D, where , and are the reaction coefficients. This means that moles of B can be created by removing moles of C and moles of D, and vice versa.

The free energy

Now consider the change of free energy, , corresponding to an increase in the number of moles of B, . It is The first line expresses that when the number of moles of B is changed, the number of moles of C and D must change too, by the equilibrium reaction (the partial derivatives indicate "keep everything else fixed"). The second line introduces the correct stoichiometry. That is: an increase in must be accompanied by a corresponding decrease in and .

"Equilibrium" means a minimum in the free energy, so the above derivative must be zero. Also, the total derivatives with respect to and must be zero, but they give the same result. The partial derivatives, e.g. , themselves shouldn't be zero, as the system cannot ever evolve along the direction of changing while keeping and fixed.

Chemical potential

Introduce the chemical potentials etc. Here, the partial derivative is taken with constant and , as well as constant temperature and pressure. Then equilibrium reads Given the functional forms of the , and, say and , we can find using this formula.

Ideal solutions

The "theory of ideal solutions" states that for each constituent where

  • [J.mol] is the chemical potential

  • [J.mol] is the constant number, independent of temperature, pressure, other constituents, etc, but which depends on the reaction in question

  • J.K.mol is the gas constant

  • [K] is temperature

  • is the natural logarithm

  • [dimensionless] is the mole fraction of the constituent

Mass action for ideal solutions

Equilibrium then reads The first term involves the reaction's standard free energy, and allows introduction of the so-called equilibrium constant for the reaction, which is defined to be Note that the equilibrium "constant" is not constant: it at least depends on temperature. It is also reaction dependent.

Finally, equilibrium can be stated as This is called "mass action".

Non-ideal solutions

The chemical potential in the non-ideal case is written as where most parameters have been defined above, but

  • [dimensionless] is the constituent's activity, which depends on temperature and potentially other things.

Mass action for non-ideal solutions

This is