Energy of activation of a first-order reaction is 56.6 kJ/mol. If its rate constant at 298 K is 2.4 x 10^-3 /min, calculate the rate constant at 308 K.

Steps to Solve

1. Identify Given Values
   We need to extract the relevant values from the problem.
   Let:

• Activation energy $E_a$ (in J/mol)
• Initial rate constant $k_1$ at $T_1 = 298 , \text{K}$
• New temperature $T_2 = 308 , \text{K}$

Assuming you have $E_a$ and $k_1$, we can proceed.

2. Use the Arrhenius Equation
   The Arrhenius equation is given by:
   $$ k = A e^{-\frac{E_a}{RT}} $$
   Where:
   $k$ = rate constant
   $A$ = pre-exponential factor
   $E_a$ = activation energy
   $R$ = ideal gas constant ($8.314 , \text{J/mol·K}$)
   $T$ = temperature in Kelvin

For two temperatures, we can rewrite the Arrhenius equation as:
$$ \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)} $$

3. Rearranging the equation
   To find $k_2$, we can rearrange the equation:
   $$ k_2 = k_1 e^{\frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)} $$

4. Plugging in the Known Values
   Insert the values for $E_a$, $k_1$, $R$, $T_1$, and $T_2$ into the equation to find $k_2$. Make sure the units of the activation energy (J/mol) and the gas constant (J/mol·K) are consistent.

5. Calculate the Exponential Part
   Calculate the value of the exponent:
   $$ \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) $$
   Using the computed value, calculate $k_2$.

6. Final Calculation of the Rate Constant
   Now, multiply $k_1$ by the exponentiated value you found in the previous step to obtain the new rate constant $k_2$.