Question

Given, \[ k = Ae^{\frac{2800}{T}} \] Find activation energy.

• A. 15.89 kJ/mol
• B. 56 kcal/mol
• C. 232.8 kJ/mol
• D. 5600 kcal/mol

Answer 1

The Correct Option is A

The Arrhenius equation is: \[ k = A e^{\frac{-E_a}{RT}} \] Where:
- \( k \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant (8.314 J/mol·K),
- \( T \) is the temperature in Kelvin.
The equation can be rearranged to calculate the activation energy: \[ \ln(k) = \ln(A) - \frac{E_a}{RT} \] To find the activation energy \( E_a \), we can use the following form of the equation for two different temperature points: \[ \ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \] Step-by-Step Calculation: Step 1: Calculate the natural logarithm of the ratio of rate constants: \[ \ln \left( \frac{k_2}{k_1} \right) = \ln \left( \frac{2.5}{1.0} \right) = \ln(2.5) \approx 0.9163 \] Step 2: Substitute the values into the equation: \[ 0.9163 = \frac{E_a}{8.314} \left( \frac{1}{300} - \frac{1}{350} \right) \] Step 3: Calculate the difference in the reciprocals of the temperatures: \[ \frac{1}{300} - \frac{1}{350} = 0.0033333 - 0.0028571 = 0.0004762 \] Step 4: Substitute into the equation: \[ 0.9163 = \frac{E_a}{8.314} \times 0.0004762 \] Step 5: Solve for \( E_a \): \[ E_a = \frac{0.9163 \times 8.314}{0.0004762} \approx 15.89 \, \text{kJ/mol} \]