Understanding the Calculation of the Equilibrium Constant from ΔG

The calculation of the equilibrium constant from Gibbs free energy change (ΔG) is fundamental in thermodynamics. This conversion links thermodynamic spontaneity with reaction equilibrium quantitatively.

This article explores the detailed mathematical framework, common values, and real-world applications of the equation ΔG = -RT ln K. You will gain expert-level insights into its calculation and interpretation.

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Calculate the equilibrium constant K for a reaction with ΔG = -10 kJ/mol at 298 K.
Determine ΔG given K = 5.0 × 10³ at 310 K.
Find K at 350 K if ΔG = 15 kJ/mol for a biochemical reaction.
Explain how temperature affects K when ΔG = -25 kJ/mol.

Comprehensive Table of Common ΔG and Corresponding Equilibrium Constants (K)

ΔG (kJ/mol)	Temperature (K)	R (J/mol·K)	Calculated ln K	Equilibrium Constant (K)	Reaction Tendency
-40	298	8.314	16.13	1.0 × 10⁷	Strongly product-favored
-25	298	8.314	10.07	2.3 × 10⁴	Product-favored
-10	298	8.314	4.03	56.7	Moderately product-favored
0	298	8.314	0	1	Equilibrium
10	298	8.314	-4.03	0.018	Reactant-favored
25	298	8.314	-10.07	4.4 × 10^-5	Strongly reactant-favored
40	298	8.314	-16.13	1.0 × 10^-7	Very strongly reactant-favored
-15	310	8.314	5.83	340	Product-favored
-5	310	8.314	1.94	6.96	Slightly product-favored
5	310	8.314	-1.94	0.14	Slightly reactant-favored
20	350	8.314	-6.90	0.001	Strongly reactant-favored
-20	350	8.314	6.90	1000	Strongly product-favored

Mathematical Framework and Variable Definitions

The fundamental equation connecting Gibbs free energy change (ΔG) and the equilibrium constant (K) is expressed as:

ΔG = -RT ln K

Where:

ΔG (Gibbs free energy change): The change in Gibbs free energy for the reaction under standard or specified conditions, typically expressed in joules per mole (J/mol) or kilojoules per mole (kJ/mol). Negative ΔG indicates spontaneous reactions.
R (Universal gas constant): A physical constant that relates energy scale to temperature, with a value of 8.314 J/(mol·K).
T (Temperature): Absolute temperature in kelvin (K). It is critical to use the absolute scale to maintain consistency in thermodynamic calculations.
ln K (Natural logarithm of the equilibrium constant): The natural logarithm of the equilibrium constant K, which is dimensionless.
K (Equilibrium constant): A dimensionless number representing the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their stoichiometric coefficients.

Rearranging the equation to solve for K yields:

K = exp(-ΔG / RT)

Where exp denotes the exponential function.

Additional Relevant Formulas

In many practical scenarios, the standard Gibbs free energy change (ΔG°) is used, which corresponds to standard conditions (1 bar, 1 M concentrations). The relationship is:

ΔG = ΔG° + RT ln Q

Where:

Q is the reaction quotient, representing the ratio of product and reactant concentrations at any point in time.

At equilibrium, ΔG = 0 and Q = K, so:

0 = ΔG° + RT ln K  →  ΔG° = -RT ln K

This confirms the primary equation used for calculating K from ΔG°.

Common Values and Their Significance

R = 8.314 J/(mol·K): This constant is universal and does not change with reaction or conditions.
T: Typically ranges from 273 K (0 °C) to 373 K (100 °C) in laboratory conditions, but can extend to higher temperatures in industrial processes.
ΔG: Values can range widely; negative values indicate spontaneous reactions, positive values indicate non-spontaneous reactions under standard conditions.
K: Values much greater than 1 indicate product-favored equilibria; values much less than 1 indicate reactant-favored equilibria.

Real-World Application Examples

Example 1: Estimating the Equilibrium Constant for the Haber Process

The Haber process synthesizes ammonia (NH₃) from nitrogen and hydrogen gases:

N₂(g) + 3H₂(g) ⇌ 2NH₃(g)

At 298 K, the standard Gibbs free energy change ΔG° for this reaction is approximately -33.0 kJ/mol.

Calculate the equilibrium constant K at 298 K.

Step 1: Convert ΔG° to joules per mole:

ΔG° = -33.0 kJ/mol × 1000 = -33000 J/mol

Step 2: Use the formula:

K = exp(-ΔG° / RT)

Step 3: Substitute values:

R = 8.314 J/(mol·K), T = 298 K

K = exp(-(-33000) / (8.314 × 298)) = exp(33000 / 2477.6) = exp(13.31) ≈ 6.0 × 10⁵

Interpretation: The large K value indicates the reaction strongly favors ammonia formation at room temperature.

Example 2: Calculating ΔG from Known Equilibrium Constant in Enzyme Kinetics

Consider an enzymatic reaction with an equilibrium constant K = 50 at 310 K (human body temperature). Calculate the standard Gibbs free energy change ΔG°.

Step 1: Use the rearranged formula:

ΔG° = -RT ln K

Step 2: Calculate ln K:

ln 50 ≈ 3.912

Step 3: Substitute values:

ΔG° = – (8.314 J/mol·K)(310 K)(3.912) = – (8.314)(310)(3.912) = -10090 J/mol ≈ -10.1 kJ/mol

Interpretation: The negative ΔG° indicates the enzymatic reaction is spontaneous under physiological conditions.

Factors Influencing the Calculation and Interpretation

Temperature Dependence: Since ΔG and K depend on temperature, accurate temperature measurement is critical. Increasing temperature can shift equilibrium constants, especially for endothermic or exothermic reactions.
Pressure and Concentration: Standard ΔG° assumes 1 bar pressure and 1 M concentrations. Deviations require corrections using reaction quotient Q.
Units Consistency: Ensure ΔG is in joules per mole when using R in J/(mol·K). Mixing units leads to incorrect K values.
Non-ideal Behavior: Real systems may deviate from ideality, requiring activity coefficients for precise calculations.

Advanced Considerations and Extensions

For reactions involving gases, the equilibrium constant can be expressed in terms of partial pressures (K_p) or concentrations (K_c). The relationship between K_p and K_c is:

Kp = Kc (RT)Δn

Where Δn is the change in moles of gas (moles of gaseous products minus moles of gaseous reactants).

In such cases, ΔG° is related to K_p by the same fundamental equation:

ΔG° = -RT ln Kp

For biochemical reactions, standard Gibbs free energy changes are often reported under biochemical standard conditions (pH 7, 1 M concentrations of reactants except protons). Adjustments must be made to convert between biochemical and chemical standard states.

Summary of Key Points for Expert Application

Always confirm the temperature and units before calculation.
Use the natural logarithm (ln) for K in the equation ΔG = -RT ln K.
Interpret K values in the context of reaction spontaneity and equilibrium position.
Apply corrections for non-standard conditions using reaction quotient Q and ΔG = ΔG° + RT ln Q.
Consider the physical state and phase of reactants/products when relating K_p and K_c.

Calculation of the Equilibrium Constant from ΔG (ΔG = -RT ln K)