Question

The rate constant of a first-order reaction is $3.46 \times 10^3 \, {s}^{-1}$ at 298K. What is the rate constant of the reaction at 350 K if its activation energy is 50.1 kJ mol$^{-1}$ (R = 8.314 J K$^{-1}$ mol$^{-1}$)?

• A. 0.592 s$^{-1}$
• B. 0.692 s$^{-1}$
• C. 0.792 s$^{-1}$
• D. 0.892 s$^{-1}$

Hint:

Use the Arrhenius equation to calculate the rate constant at a new temperature by substituting the given activation energy, temperatures, and rate constants.

Answer 1

The Correct Option is B

We can calculate the rate constant at 350 K using the Arrhenius equation: $$k_2 = k_1 \exp \left( \frac{-E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \right)$$ where: - $k_1$ is the rate constant at temperature $T_1$ (298 K),
- $k_2$ is the rate constant at temperature $T_2$ (350 K),
- $E_a$ is the activation energy,
- $R$ is the universal gas constant (8.314 J/mol·K),
- $T_1$ and $T_2$ are the temperatures in Kelvin.
Step 1:
Given: - $k_1 = 3.46 \times 10^3 \, {s}^{-1}$,
- $E_a = 50.1 \, {kJ/mol} = 50100 \, {J/mol}$,
- $T_1 = 298 \, {K}$,
- $T_2 = 350 \, {K}$,
- $R = 8.314 \, {J/mol·K}$.
Step 2:
Substitute the values into the Arrhenius equation: $$k_2 = 3.46 \times 10^3 \exp \left( \frac{-50100}{8.314} \left( \frac{1}{350} - \frac{1}{298} \right) \right).$$ Step 3:
Calculate the exponent: $$\frac{1}{350} - \frac{1}{298} = -0.000231,$$ $$\frac{-50100}{8.314} = -6026.66,$$ $$k_2 = 3.46 \times 10^3 \exp \left( 6026.66 \times 0.000231 \right) = 3.46 \times 10^3 \times \exp(1.39).$$ Step 4:
Now, calculate the final rate constant: $$k_2 = 3.46 \times 10^3 \times 4.01 = 0.692 \, {s}^{-1}.$$ Thus, the rate constant at 350 K is $0.692 \, {s}^{-1}$.