Equilibrium constants, activities and approximations

Normalize the thermodynamic quantities

Thermodynamic activity a measures a species relative to its chosen standard state. It is dimensionless. For an appropriate concentration-based solute convention, a = γ_c(c/c°), where γ_c is a dimensionless activity coefficient and c° is the reference concentration. A dilute ideal approximation takes γ_c ≈ 1.

The reference is part of the definition, not a unit that can be silently discarded. A molality convention uses its own m/m° and coefficient. Using a numerical concentration in mol L⁻¹ without explaining the reference can hide both unit and nonideality assumptions.

The equation determines the product of activities

For the written reaction A + B ⇌ AB, define Q = a_AB/(a_A a_B). The thermodynamic standard equilibrium constant K° is its equilibrium value at the stated temperature and standard-state convention. A coefficient of 2 gives a squared activity; reactant factors appear in the denominator.

Here K° denotes the dimensionless thermodynamic constant throughout this chapter. Some older or practical quantities also called K_c or K_p use raw concentration or pressure powers and can carry units. Their numerical values depend on convention. Do not mix such an expression with our normalized activity expression without conversion.

Use partial pressure and the correct phase reference

For an ideal gas mixture, a_i ≈ p_i/p°, with each reacting species’ partial pressure and a specified standard pressure such as p° = 1 bar. For a real gas, the corresponding fugacity ratio supplies the thermodynamic activity. Total pressure is not the partial pressure of each constituent.

A pure condensed phase has activity 1 in its chosen pure-phase reference state. Relative to a fixed standard pressure, treating it as 1 near ordinary pressure is usually a stated approximation. A solute, impure solid or solid solution is not automatically a pure phase of unit activity. Omitting a pure solid factor does not mean its availability never matters.

Retain the reaction and the conditions

K° belongs to a written reaction, temperature and standard-state choices. Large or small values describe the equilibrium activity relation on that basis; they do not give a rate constant or uniquely specify every final amount without the material inventory.

The thermodynamic relation is ΔᵣG° = −RT ln K°, with R the molar gas constant and T in kelvin. The logarithm is valid because K° is dimensionless. Its connection to standard reaction Gibbs energy does not turn every raw concentration expression into the same numerical constant.

Worked examples

Try it yourself

1. What makes a concentration-based activity dimensionless?

   Show the answer and reasoning

   The concentration/reference ratio c/c°, multiplied by a dimensionless coefficient.

   The numerator and denominator use compatible concentration units.

2. For an ideal gas mixture, does a reacting gas use total mixture pressure directly?

   Show the answer and reasoning

   No.

   It uses its partial pressure relative to p°.

3. Is every solid automatically a pure phase with activity 1?

   Show the answer and reasoning

   No.

   Purity, phase composition and the chosen reference matter.

4. Does a large K° establish that equilibrium is reached quickly?

   Show the answer and reasoning

   No.

   K° constrains equilibrium activities; kinetic information determines the approach speed.

What to carry forward

• Write species, phases and stoichiometric powers first.
• Keep activities and their standard-state normalization explicit.
• Distinguish raw concentration constants from dimensionless thermodynamic K°.
• State ideality and pure-phase approximations before using them.