How to Calculate Activation Energy from Temperature and Rate Constant

Calculate activation energy (Ea) using the two-point Arrhenius equation form.

Condition 1

Kelvin (K) Celsius (°C)

Condition 2

Kelvin (K) Celsius (°C)

How to Calculate Activation Energy from Temperature and Rate Constant

In chemical kinetics, the Activation Energy ($E_a$) represents the minimum amount of energy required for a chemical reaction to occur. One of the most common methods to determine this value experimentally is by measuring the rate constant ($k$) of a reaction at two different temperatures ($T$) and applying the Arrhenius equation.

The Arrhenius Equation (Two-Point Form)

While the standard Arrhenius equation is $k = Ae^{-E_a/RT}$, calculating $E_a$ without knowing the frequency factor ($A$) requires comparing two different conditions. The logarithmic two-point form is derived as follows:

ln(k2 / k1) = (Ea / R) * ((1 / T1) – (1 / T2))

Rearranging this to solve for Activation Energy ($E_a$) gives the formula used by this calculator:

Ea = (R * ln(k2 / k1)) / ((1 / T1) – (1 / T2))

Where:

• Ea = Activation Energy (Joules per mole)
• R = Universal Gas Constant (8.314 J/mol·K)
• T1, T2 = Absolute temperatures in Kelvin
• k1, k2 = Rate constants at T1 and T2 respectively

Step-by-Step Calculation Guide

1. Identify your data points: You need two pairs of data: (Temp 1, Rate 1) and (Temp 2, Rate 2).
2. Convert Temperature: If your temperatures are in Celsius, convert them to Kelvin by adding 273.15.
   Example: 25°C + 273.15 = 298.15 K
3. Check Rate Constants: Ensure $k_1$ and $k_2$ use the same units (e.g., $s^{-1}$ or $M^{-1}s^{-1}$).
4. Apply the Formula: Calculate the natural log of the ratio of rate constants, and divide by the difference of the inverse temperatures. Multiply by the Gas Constant ($R$).

Example Calculation

Let's say a decomposition reaction has a rate constant of 2.5 × 10^-4 s^-1 at 300 K ($T_1$) and increases to 8.0 × 10^-4 s^-1 at 320 K ($T_2$).

1. Calculate the log ratio of rates:
$ln(8.0/2.5) = ln(3.2) \approx 1.163$

2. Calculate the difference in inverse temperatures:
$(1/300) – (1/320) = 0.003333 – 0.003125 = 0.0002083\ K^{-1}$

3. Solve for Ea:
$E_a = \frac{8.314 \times 1.163}{0.0002083} \approx 46,415\ \text{J/mol}$

Result: The Activation Energy is approximately 46.4 kJ/mol.

Why is Kelvin Required?

Thermodynamic equations involving the Gas Constant ($R$) rely on absolute temperature. Celsius is a relative scale where 0 is arbitrarily set at the freezing point of water. Kelvin is an absolute scale starting at absolute zero. Using Celsius directly in the denominator of the Arrhenius equation would yield incorrect mathematical results and undefined values at 0°C.