Question

The rate of a particular reaction quadruples when the temperature increases from 300 K to 320 K. Calculate the energy of activation \( (E_a) \) of the reaction assuming that it does not change with temperature.
(Given : \( \log 4 = 0 \cdot 60, R = 8 \cdot 314 \text{ JK}^{-1} \text{ mol}^{-1} \))

Hint:

Ensure temperatures are in Kelvin.
Check the units of \( R \); using 8.314 gives \( E_a \) in Joules.
Most activation energies are expressed in kJ/mol.

Answer 1

Concept:

• The temperature dependence of the rate constant is given by the Arrhenius Equation.

• For two different temperatures, the integrated form of the Arrhenius equation is used.
Step 1: Identify the variables and the formula
Given:
\( T_1 = 300 \text{ K}, T_2 = 320 \text{ K} \).
Since the rate quadruples, \( k_2/k_1 = 4 \).
\( R = 8.314 \text{ JK}^{-1} \text{ mol}^{-1} \).
The formula is:
\[ \log \left( \frac{k_2}{k_1} \right) = \frac{E_a}{2.303R} \left( \frac{T_2 - T_1}{T_1 T_2} \right) \]

Step 2: Substitute values into the equation
\[ \log(4) = \frac{E_a}{2.303 \times 8.314} \left( \frac{320 - 300}{320 \times 300} \right) \]
\[ 0.60 = \frac{E_a}{19.147} \left( \frac{20}{96000} \right) \]
\[ 0.60 = \frac{E_a}{19.147} \left( \frac{1}{4800} \right) \]

Step 3: Solve for \( E_a \)
\[ E_a = 0.60 \times 19.147 \times 4800 \]
\[ E_a = 11.4882 \times 4800 = 55143.36 \text{ J/mol} \]
To convert to kJ/mol: \( E_a = 55.14 \text{ kJ/mol} \). Final Answer: The energy of activation for the reaction is 55.14 kJ/mol.