Overview
The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:
where R is the gas constant (8.314 J/K·mol), T is the absolute temperature in kelvin, and Q is the reaction quotient. This equation may be used to predict the spontaneity of a process under any given set of conditions.
Reaction Quotient (Q)
The status of a reversible reaction is conveniently assessed by evaluating its reaction quotient, Q. For a reversible reaction described by
the reaction quotient is derived directly from the stoichiometry of the balanced equation as
where the subscript c denotes the use of molar concentrations in the expression. The concentration-based reaction quotient, Qc, is used for condensed phase equilibria. If the reactants and products are gaseous, a reaction quotient may be similarly derived using partial pressures:
Under standard conditions, the reactant and product solution concentrations are 1 M, or the pressure of gases is 1 bar, and Q is equal to 1. Therefore, under standard conditions
Under nonstandard conditions, Q must be calculated.
The numerical value of Q varies as a reaction proceeds towards equilibrium; therefore, it can serve as a useful indicator of the reaction’s status. To illustrate this point, consider the oxidation of sulfur dioxide:
Consider two different experimental scenarios, one in which this reaction is initiated with a mixture of reactants only, SO2 and O2, and another that begins with only the product, SO3. For the reaction that begins with a mixture of reactants only, Q is initially equal to zero:
As the reaction proceeds toward equilibrium in the forward direction, reactant concentrations decrease (as does the denominator of Qc), product concentration increases (as does the numerator of Qc), and the reaction quotient consequently increases. When equilibrium is achieved, the concentrations of reactants and products remain constant, as does the value of Qc.
If the reaction begins with only the product present, the value of Qc is initially undefined (immeasurably large or infinite):
In this case, the reaction proceeds toward equilibrium in the reverse direction. The product concentration and the numerator of Qc decrease with time, the reactant concentrations and the denominator of Qc increase, and the reaction quotient consequently decreases until it becomes constant at equilibrium.
This text is adapted from Openstax, Chemistry 2e, Chapter 16.4: Free Energy and Openstax, Chemistry 2e, Chapter 13.2: Equilibrium Constants.
Procedure
The standard free energy change for a reaction can only be determined if it occurs under standard state conditions—when both the reactants and products are in their standard states. However, most chemical reactions do not occur under these conditions.
Under any conditions—standard or nonstandard—the relative amount of products and reactants present in a reaction is described by the reaction quotient, Q.
For reactions occurring in solution, Q is calculated from the ratio of the product and reactant concentrations, with each reagent concentration raised to the power of its stoichiometric coefficient. For gaseous reactions, the partial pressures of the gases can be used in place of concentrations.
The free energy change of a reaction is equal to the sum of the standard-state free energy change for the reaction, delta G naught, and RT times the natural log of Q. Here, R is the universal gas constant in joules per mole-kelvin, and T is the temperature of the reaction in kelvin.
At constant temperature, the standard-state free energy has a fixed value, but Q varies because it depends on the composition of the reaction mixture.
Consider the synthesis of ammonia gas from nitrogen and hydrogen at 298 K. Under standard conditions, which for a gas is the pure gas at 1 atm, the partial pressures of all the components are equal to 1 atm, and the magnitude of the reaction quotient equals 1.
Thus, the free energy change of the reaction is equal to the standard free energy change of the reaction, −32.8 kJ/mol, and the forward reaction is spontaneous.
Under nonstandard conditions, the components of the reaction mixture may initially have partial pressures of 1.2 atmospheres of nitrogen, 3.6 atmospheres of hydrogen, and 0.60 atmospheres of ammonia.
As before, the reaction quotient can be determined from the values for the partial pressures. Substituting for Q into the equation, the free energy for the reaction is −45.3 kJ/mol, indicating a spontaneous reaction in the forward direction.
As the forward reaction proceeds, more ammonia is produced, and the reaction composition changes.
When the reactants and products are in equilibrium, the free energy change for the reaction is zero, and the value of RT times the natural log of Q is equal and opposite in sign to the standard free energy change.
Now, if the reaction mixture contains 0.02 atmospheres of nitrogen, 0.06 atmospheres of hydrogen, and 4.8 atmospheres of ammonia, Q is much larger, and the change in free energy is 5.6 kJ/mol.
A positive free energy change indicates that the reverse reaction is energetically favorable. Thus, under these conditions, ammonia decomposes to produce nitrogen and hydrogen.