Question

Rate constants of a reaction at 500 K and 700 K are \(0.04 \, \text{s}^{-1}\) and \(0.14 \, \text{s}^{-1}\), respectively. Then, the activation energy of the reaction is:
{Given:} \[ \log 3.5 = 0.5441, \, R = 8.31 \, \text{J K}^{-1} \, \text{mol}^{-1} \]

• A. 182310 J
• B. 18500 J
• C. 18219 J
• D. 18030 J

Answer 1

The Correct Option is C

The Arrhenius equation, specifically its two-point form, can be utilized for this calculation:

\[ \ln{\frac{k_2}{k_1}} = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]

Key variables are defined as:

• $k_1$ and $k_2$: Rate constants at temperatures $T_1$ and $T_2$, respectively.
• $E_a$: Activation energy.
• R: Ideal gas constant.

Provided data includes:

• $k_1 = 0.04$ s$^{-1}$
• $k_2 = 0.14$ s$^{-1}$
• $T_1 = 500$ K
• $T_2 = 700$ K
• R = 8.31 J K$^{-1}$mol$^{-1}$
• $\log 3.5 = 0.5441$, which implies $\ln 3.5 = 2.303 \times 0.5441 = 1.253$

Substituting the given values into the Arrhenius equation yields:

\(\ln{\frac{0.14}{0.04}} = \frac{E_a}{8.31} \left( \frac{1}{500} - \frac{1}{700} \right)\)

\(\ln{3.5} = \frac{E_a}{8.31} \left( \frac{700 - 500}{500 \times 700} \right)\)

\(1.253 = \frac{E_a}{8.31} \left( \frac{200}{350000} \right)\)

Rearranging to solve for $E_a$:

\(1.253 \times 8.31 \times 350000 = E_a\)

\(E_a = \frac{1.253 \times 8.31 \times 350000}{200}\)

The calculated activation energy is:

\({E_a} = 18219 \text{J}\)