The rate constant $ k $ of a reaction doubles when the temperature is raised from 300 K to 310 K. Calculate the activation energy $ E_a $ of the reaction. (Use $ R = 8.314 \, J/mol\cdot K $)

Question

Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radii $r_A$ and $r_B$ respectively. On supplying an equal amount of heat to both the cylinders, their pressures remain constant and their pistons are displaced by 16 cm and 9 cm respectively. If the change in their internal energies is the same, then the ratio $r_A / r_B$ is:

Answer 1

The activation energy $E_a$ is determined using the Arrhenius equation in its logarithmic form: $\ln\left(\frac{k_2}{k_1}\right)=-\frac{E_a}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$.

Given that the rate constant $k$ doubles ($\frac{k_2}{k_1}=2$) as the temperature increases from $T_1=300\,K$ to $T_2=310\,K$, we substitute these values into the equation: $\ln(2)=-\frac{E_a}{8.314}\left(\frac{1}{310}-\frac{1}{300}\right)$.

The temperature difference is calculated as $\frac{1}{310}-\frac{1}{300}=\frac{300-310}{310\times300}=-\frac{10}{93,000}\approx-0.0001075$.

Substituting $\ln(2)\approx0.693$ yields:\[0.693=-\frac{E_a}{8.314}\times-0.0001075\].

Solving for $E_a$ gives:\[E_a=\frac{0.693\times8.314}{0.0001075}\approx53,000\ J/mol=53\ kJ/mol\].

Therefore, the activation energy $E_a$ for the reaction is ${53\ kJ/mol}$.

The rate constant $ k $ of a reaction doubles when the temperature is raised from 300 K to 310 K. Calculate the activation energy $ E_a $ of the reaction. (Use $ R = 8.314 \, J/mol\cdot K $)

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The Correct Option is C

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