Calculate E G and K for The Following Reactions

This guide explains how to calculate activation energy (E), Gibbs free energy (G), and equilibrium constant (K) for chemical reactions. We'll cover the formulas, assumptions, and practical applications of these key thermodynamic and kinetic parameters.

What Are E, G, and K?

In chemical kinetics and thermodynamics, E, G, and K are fundamental parameters that describe different aspects of chemical reactions:

Activation Energy (E): The minimum energy required for a chemical reaction to occur. Measured in joules (J) or kilojoules per mole (kJ/mol).
Gibbs Free Energy (G): A measure of the energy available to do work in a system at constant temperature and pressure. Measured in joules (J) or kilojoules per mole (kJ/mol).
Equilibrium Constant (K): A measure of the ratio of concentrations of products to reactants at equilibrium. Dimensionless value.

These parameters help chemists understand reaction rates, spontaneity, and the position of equilibrium.

How to Calculate Activation Energy (E)

The activation energy (E) can be calculated using the Arrhenius equation:

Arrhenius Equation:

k = A × e^-E/RT

Where:

k = reaction rate constant
A = pre-exponential factor (frequency factor)
E = activation energy (J/mol)
R = universal gas constant (8.314 J/mol·K)
T = temperature (K)

To solve for E, rearrange the equation:

ln(k) = ln(A) - (E/RT)

E = -R × (d(ln k)/d(1/T))

This shows that plotting ln(k) vs. 1/T gives a straight line with slope -E/R.

How to Calculate Gibbs Free Energy (G)

The Gibbs free energy change (ΔG) for a reaction can be calculated using the standard Gibbs free energies of formation:

ΔG = ΣΔG_f (products) - ΣΔG_f (reactants)

Where ΔG_f is the standard Gibbs free energy of formation for each compound.

For temperature dependence, use:

ΔG = ΔH - TΔS

Where:

ΔH = enthalpy change
ΔS = entropy change

The sign of ΔG determines reaction spontaneity: negative ΔG means spontaneous under standard conditions.

How to Calculate Equilibrium Constant (K)

The equilibrium constant (K) can be calculated from the standard Gibbs free energy change:

ΔG° = -RT ln(K)

Where:

ΔG° = standard Gibbs free energy change
R = universal gas constant (8.314 J/mol·K)
T = temperature (K)
K = equilibrium constant

Rearranged to solve for K:

K = e^-ΔG°/RT

For reactions involving gases, the equilibrium constant can also be expressed in terms of partial pressures.

Example Calculation

Let's calculate E, G, and K for the reaction:

2H₂ + O₂ → 2H₂O

Step 1: Calculate Activation Energy (E)

From experimental data, the reaction rate constant (k) at different temperatures:

Temperature (K)	Rate Constant (k)
500	0.012
600	0.036
700	0.084

Plotting ln(k) vs. 1/T gives a slope of -12,000. Using the Arrhenius equation:

E = -R × slope = -8.314 × (-12,000) = 99,768 J/mol ≈ 100 kJ/mol

Step 2: Calculate Gibbs Free Energy (G)

Using standard Gibbs free energies of formation:

Compound	ΔG_f (kJ/mol)
H₂	0
O₂	0
H₂O	-237.1

For the reaction: 2H₂ + O₂ → 2H₂O

ΔG = [2 × (-237.1)] - [2 × 0 + 0] = -474.2 kJ/mol

Step 3: Calculate Equilibrium Constant (K)

Using ΔG° = -474.2 kJ/mol at 298 K:

ΔG° = -474,200 J/mol

K = e^-ΔG°/RT = e^{-(-474,200)/(8.314×298)} ≈ e^20.3 ≈ 5.1 × 10⁸

FAQ

What is the difference between activation energy and Gibbs free energy?

Activation energy (E) is the minimum energy needed to start a reaction, while Gibbs free energy (G) measures the energy available to do work. E is a kinetic parameter, while G is a thermodynamic parameter.

How does temperature affect the equilibrium constant?

The equilibrium constant is temperature-dependent. According to the van't Hoff equation, the natural logarithm of K is inversely proportional to temperature. Increasing temperature generally shifts equilibrium toward products that absorb heat.

What does a negative Gibbs free energy mean?

A negative ΔG indicates the reaction is spontaneous under standard conditions. The more negative ΔG is, the more favorable the reaction.