Question

For a chemical reaction, the ratio of rate constant at 500K to that at 400K is 2.5. Given R = 8.314 J mol\(^{-1}\)K\(^{-1}\), the value of activation energy (in kJ/mol) is _______

• A. 10.5
• B. 12.0
• C. 15.2
• D. 18.4

Hint:

Always pay attention to units in activation energy calculations.
The gas constant $R = 8.314$ yields $E_a$ in J/mol, so don't forget to divide by 1000 to convert to kJ/mol as requested by the options.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to calculate the activation energy ($E_a$) of a chemical reaction using the ratio of its rate constants at two different temperatures.
The temperature dependence of chemical reaction rate constants is described by the Arrhenius equation.

Step 2: Key Formula or Approach:
The Arrhenius equation is given by: \[ k = k_0 e^{-E_a / RT} \] For two different temperatures $T_1$ and $T_2$, with corresponding rate constants $k_1$ and $k_2$, the ratio of the rate constants is expressed as: \[ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \] where:
$E_a$ is the activation energy (in J/mol).
$R$ is the universal gas constant = $8.314 \text{ J mol}^{-1}\text{K}^{-1}$.
$T_1, T_2$ are absolute temperatures in Kelvin.

Step 3: Detailed Explanation:
Let us list the given parameter values:
- $T_1 = 400$ K
- $T_2 = 500$ K
- Ratio of rate constants, $\frac{k_2}{k_1} = 2.5$
- $R = 8.314 \text{ J mol}^{-1}\text{K}^{-1}$
Substitute these values into the Arrhenius equation: \[ \ln(2.5) = \frac{E_a}{8.314} \left( \frac{1}{400} - \frac{1}{500} \right) \] First, compute the logarithmic term: \[ \ln(2.5) \approx 0.91629 \] Next, compute the temperature difference term: \[ \frac{1}{400} - \frac{1}{500} = \frac{500 - 400}{400 \times 500} = \frac{100}{200000} = 0.0005 \text{ K}^{-1} = 5 \times 10^{-4} \text{ K}^{-1} \] Now, substitute these back into the main equation: \[ 0.91629 = \frac{E_a}{8.314} \times 0.0005 \] Rearrange to solve for $E_a$: \[ E_a = \frac{0.91629 \times 8.314}{0.0005} \] \[ E_a = 0.91629 \times 8.314 \times 2000 \] \[ E_a = 15236.42 \text{ J/mol} \] To convert the activation energy from J/mol to kJ/mol, we divide by $1000$: \[ E_a = \frac{15236.42}{1000} \approx 15.24 \text{ kJ/mol} \] This value is closest to $15.2$ kJ/mol, which matches option (C).

Step 4: Final Answer
The activation energy is $15.2$ kJ/mol, corresponding to option (C).