Advancement describes progress towards the achievement of a goal. In Chemistry, the advancement is the amount of chemical change away from the initial conditions and toward equilibrium. Equilibrium advancement is determined by the equilibrium constant and the initial conditions. Advancement is formally a count of molecular events, so it has the units of molecules or moles. Sometimes, for constant volume (isochoric) processes, the advancement may be given the units of moles/liter or even atmospheres. Care must be taken with these non-standard definitions.

Consider the dimerization of species A

If initially, n_A⁰ moles of momomer A is added to a vessel of volume V, the initial concentration of species A is

Equilibrium is established when

Since there is no dimer, D, initially present, the equilibrium number of moles of the dimer is simply the number of moles of reaction that spontaneously proceed in the forward direction to reach equilibrium, the equilibrium advancement. The equilibrium concentration is this divided by the volume of the system.

Here x_e stands for the equilibrium advancement. The equilibrium concentration of the monomer is reduced from its initial value by the extent of the reaction

The equilibrium concentrations must satisfy the equilibrium relation

Which allows for the solution of the value of the equilibrium advancement, in general.

or, in a familiar quadratic form in the unknown x_e:

Two roots for x_e results from this relation, which can be obtained analytically from the quadratic formula:

In this case a=4, b=-(4n_A⁰ + V/K_eq), and c = (n_A⁰)², so

But, of the two roots for x_e, only one represents a physically reasonable solution to the problem. Which one? Only one root [proof?] will result in no negative concentrations for all the species.

Problem: What is the expression for the equilibrium advancement if BOTH species A and D are present initially?

SOL'N

Consider the dimerization of species A

If initially, n_A⁰ moles of momomer A and n_D⁰ moles of the dimer D are added to a vessel of volume V, the initial concentrations are

Equilibrium is established when

The dimer, D, is initially present so its concentration is

The equilibrium concentration of the monomer is reduced from its initial value by the extent of the reaction

The equilibrium concentrations must satisfy the equilibrium relation

Which allows for the solution of the value of the equilibrium advancement, in general.

or, in a familiar quadratic form in the unknown x_e:

Two roots for x_e results from this relation, which can be obtained analytically from the quadratic formula:

In this case a=4, b= -(4n_A⁰ + V/K_eq), and c = ((n_A⁰)² - Vn_D⁰ /K_eq), so

But, of the two roots for x_e, only one represents a physically reasonable solution to the problem. Which one? Only one root [proof?] will result in no negative concentrations for all the species.

Lets evaluate this equation for some simple numerical examples. Assume the equilibrium constant Keq is 10, which means that the association of the monomer to form dimer is highly favored.

Note: in all the above cases, the root with the negative sign in front of the square root is the physically significant solution.