Questions: The rate constant of a certain reaction is known to obey the Arrhenius equation, and to have an activation energy Eα=67.0 kJ / mol. If the rate constant of this reaction is 7.6 M^-1 s^-1 at 158.0°C, what will the rate constant be at 201.0°C? Round your answer to 2 significant digits.

Solution Steps

First, we need to convert the given temperatures from Celsius to Kelvin.

\[ T_1 = 158.0^\circ \text{C} + 273.15 = 431.15 \text{ K} \] \[ T_2 = 201.0^\circ \text{C} + 273.15 = 474.15 \text{ K} \]

The Arrhenius equation is given by:

\[ k = A e^{-\frac{E_a}{RT}} \]

We can use the ratio form of the Arrhenius equation to find the new rate constant \( k_2 \):

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

Given:

• \( E_a = 67.0 \text{ kJ/mol} = 67000 \text{ J/mol} \)
• \( R = 8.314 \text{ J/mol·K} \)
• \( k_1 = 7.6 \text{ M}^{-1}\text{s}^{-1} \)
• \( T_1 = 431.15 \text{ K} \)
• \( T_2 = 474.15 \text{ K} \)

\[ \frac{k_2}{k_1} = e^{\left(\frac{67000}{8.314} \left(\frac{1}{431.15} - \frac{1}{474.15}\right)\right)} \]

First, calculate the difference in the reciprocals of the temperatures:

\[ \frac{1}{431.15} - \frac{1}{474.15} = 0.002319 - 0.002109 = 0.000210 \]

Next, calculate the exponent:

\[ \frac{67000}{8.314} \times 0.000210 = 16.88 \]

Then, calculate the exponential:

\[ e^{16.88} \approx 2.06 \times 10^7 \]

Now, use the ratio to find \( k_2 \):

\[ k_2 = k_1 \times e^{16.88} = 7.6 \times 2.06 \times 10^7 \]

\[ k_2 \approx 1.57 \times 10^8 \text{ M}^{-1}\text{s}^{-1} \]

Final Answer