At 300 K, a catalyst lowers the activation barrier from 20 kJ/mol to 10 kJ/mol. Which of the following statements is true regarding the rate constant for the catalyzed reaction compared to the uncatalyzed reaction? (R = 8.314 J mol⁻¹ K⁻¹)

• A. The rate constant for the catalyzed reaction is 10 times larger than the rate constant for the uncatalyzed reaction.
• B. The rate constant for the catalyzed reaction is 2 times larger than the rate constant for the uncatalyzed reaction.
• C. The rate constant for the catalyzed reaction is 100 times larger than the rate constant for the uncatalyzed reaction.
• D. The rate constant for the catalyzed reaction is 55.1 times larger than the rate constant for the uncatalyzed reaction.

Answer: (D)

Verified step by step guidance

1. Identify the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea): k = A * exp(-Ea / (R * T)), where A is the pre-exponential factor, R is the gas constant, and T is the temperature in Kelvin.
2. Recognize that the problem involves comparing the rate constants of catalyzed and uncatalyzed reactions. The activation energy for the uncatalyzed reaction is 20 kJ/mol, and for the catalyzed reaction, it is 10 kJ/mol.
3. Convert the activation energies from kJ/mol to J/mol for consistency with the gas constant R (8.314 J mol⁻¹ K⁻¹). Thus, 20 kJ/mol becomes 20000 J/mol and 10 kJ/mol becomes 10000 J/mol.
4. Use the Arrhenius equation to express the rate constants for both reactions: k_uncatalyzed = A * exp(-20000 / (8.314 * 300)) and k_catalyzed = A * exp(-10000 / (8.314 * 300)).
5. Calculate the ratio of the rate constants (k_catalyzed / k_uncatalyzed) to determine how many times larger the catalyzed rate constant is compared to the uncatalyzed one. This involves simplifying the expression: (exp(-10000 / (8.314 * 300))) / (exp(-20000 / (8.314 * 300))).