Activation Energy Calculator
Calculate Ea using the Two-Point Arrhenius Equation
K
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°F
K
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Activation Energy (Ea):
0.00 kJ/mol
Activation Energy (Joules):
0.00 J/mol
Natural Log Ratio ln(k₂/k₁):
0.00
Temperature Difference (1/T₂ – 1/T₁):
0.00 K⁻¹
How to Calculate Activation Energy from Rate Constants
In physical chemistry, calculating the Activation Energy (Ea) of a reaction is crucial for understanding its kinetics and sensitivity to temperature changes. This calculator uses the two-point form of the Arrhenius Equation to determine the energy barrier required for a reaction to occur based on rate constants measured at two different temperatures.
The Arrhenius Equation Logic
The relationship between the rate constant ($k$), absolute temperature ($T$), and activation energy ($E_a$) is given by Svante Arrhenius as:
k = A × e^(-Ea / RT)
Where:
- k = Rate constant of the reaction
- A = Frequency factor (pre-exponential factor)
- Ea = Activation Energy (Joules/mol)
- R = Universal Gas Constant (8.314 J·mol⁻¹·K⁻¹)
- T = Temperature in Kelvin
Formula for Two Data Points
Often in laboratory settings, you determine the rate constant at two specific temperatures. By comparing these two points, you can eliminate the Frequency Factor ($A$) and solve directly for Activation Energy using the logarithmic form:
ln(k₂/k₁) = -(Ea/R) × (1/T₂ – 1/T₁)
Rearranging this to solve for Ea:
Ea = -R × ln(k₂/k₁) / (1/T₂ – 1/T₁)
Step-by-Step Calculation Example
Let's assume a chemical reaction has the following observed data:
- At T₁ = 298 K (25°C): Rate constant k₁ = 2.5 × 10⁻⁴ s⁻¹
- At T₂ = 308 K (35°C): Rate constant k₂ = 7.5 × 10⁻⁴ s⁻¹
1. Calculate the Natural Log of the Rate Ratio
Ratio = k₂ / k₁ = (7.5 × 10⁻⁴) / (2.5 × 10⁻⁴) = 3.0
ln(3.0) ≈ 1.0986
2. Calculate the Inverse Temperature Difference
1/T₂ = 1/308 ≈ 0.0032468
1/T₁ = 1/298 ≈ 0.0033557
Difference = 0.0032468 – 0.0033557 = -0.0001089 K⁻¹
3. Solve for Ea
Ea = -(8.314 J/mol·K) × (1.0986) / (-0.0001089)
Ea ≈ -9.133 / -0.0001089 ≈ 83,865 J/mol
Convert to kJ/mol: 83.9 kJ/mol
Why is Activation Energy Important?
Activation energy represents the minimum energy threshold required for reactants to transform into products.
- High Ea: The reaction is very sensitive to temperature. A small increase in temperature will significantly increase the rate.
- Low Ea: The reaction rate is less dependent on temperature changes.
Common Units and Conversions
| Variable | Standard Scientific Unit | Common Alternatives |
|---|---|---|
| Temperature (T) | Kelvin (K) | Celsius (°C), Fahrenheit (°F) |
| Activation Energy (Ea) | Joules per mole (J/mol) | Kilojoules per mole (kJ/mol) |
| Gas Constant (R) | 8.314 J·K⁻¹·mol⁻¹ | 0.008314 kJ·K⁻¹·mol⁻¹ |