Question

The ratio of the activation energy of a chemical reaction to the universal gas constant is \( 1000 \, \text{K} \). The temperature dependence of the reaction rate constant follows the collision theory. The ratio of the rate constant at \( 600 \, \text{K} \) to that at \( 400 \, \text{K} \) is:

• A. 2.818
• B. 4.323
• C. 1.502
• D. 1.000

Hint:

For temperature dependence of reaction rates, use the Arrhenius equation and carefully calculate the exponential factor for the given temperatures.

Answer 1

The Correct Option is A

Step 1: Use the Arrhenius equation for the rate constant. The Arrhenius equation is given by: \[ k = A e^{-\frac{E_a}{RT}}, \] where: - \( k \) is the rate constant, - \( A \) is the pre-exponential factor, - \( E_a \) is the activation energy, - \( R \) is the universal gas constant, - \( T \) is the temperature. The ratio of rate constants at two temperatures \( T_1 \) and \( T_2 \) is: \[ \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)}. \] Step 2: Substitute the given values. The ratio \( \frac{E_a}{R} = 1000 \, \text{K} \), \( T_1 = 400 \, \text{K} \), and \( T_2 = 600 \, \text{K} \): \[ \frac{k_2}{k_1} = e^{1000 \left(\frac{1}{400} - \frac{1}{600}\right)}. \] Step 3: Simplify the exponent. Calculate \( \frac{1}{400} - \frac{1}{600} \): \[ \frac{1}{400} - \frac{1}{600} = \frac{3 - 2}{1200} = \frac{1}{1200}. \] Thus: \[ \frac{k_2}{k_1} = e^{\frac{1000}{1200}} = e^{0.8333}. \] Step 4: Calculate the exponential term. Using \( e^{0.8333} \): \[ \frac{k_2}{k_1} \approx 2.818. \] Step 5: Conclusion. The ratio of the rate constant at \( 600 \, \text{K} \) to that at \( 400 \, \text{K} \) is \( 2.818 \).