3. A reaction has a rate constant of 0.0117 M/s at 400.0K and 0.689 M/s at 450.0K. What is the activation energy of the reaction?

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3)can someone explain the question
### Chemical Kinetics Problem

**Problem Statement:**

"A reaction has a rate constant of 0.0117 M/s at 400.0 K and 0.689 M/s at 450.0 K. What is the activation energy of the reaction?"

**Graphical Explanation (if applicable):**
To calculate the activation energy, you can use the Arrhenius equation:

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

where:
- \( k \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant (8.314 J/(mol·K)),
- \( T \) is the temperature in Kelvin.

Taking the natural logarithm of both sides gives:

\[ \ln(k) = \ln(A) - \frac{E_a}{RT} \]

For two different temperatures, the equation becomes:

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

Given:
- \( k_{1} = 0.0117 \, \text{M/s} \) at \( T_{1} = 400.0 \, \text{K} \)
- \( k_{2} = 0.689 \, \text{M/s} \) at \( T_{2} = 450.0 \, \text{K} \)

We can solve for \( E_a \):

\[ \ln\left(\frac{0.689}{0.0117}\right) = -\frac{E_a}{8.314} \left( \frac{1}{450.0} - \frac{1}{400.0} \right) \]

Calculating the left side:

\[ \ln\left(\frac{0.689}{0.0117}\right) = \ln(58.8034) \approx 4.0732 \]

Calculating the right side:

\[ \frac{1}{450.0} - \frac{1}{400.0} = 0.002222 - 0.0025 = -0.000278 \]

Then:

\[
Transcribed Image Text:### Chemical Kinetics Problem **Problem Statement:** "A reaction has a rate constant of 0.0117 M/s at 400.0 K and 0.689 M/s at 450.0 K. What is the activation energy of the reaction?" **Graphical Explanation (if applicable):** To calculate the activation energy, you can use the Arrhenius equation: \[ k = A e^{-\frac{E_a}{RT}} \] where: - \( k \) is the rate constant, - \( A \) is the pre-exponential factor, - \( E_a \) is the activation energy, - \( R \) is the universal gas constant (8.314 J/(mol·K)), - \( T \) is the temperature in Kelvin. Taking the natural logarithm of both sides gives: \[ \ln(k) = \ln(A) - \frac{E_a}{RT} \] For two different temperatures, the equation becomes: \[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \] Given: - \( k_{1} = 0.0117 \, \text{M/s} \) at \( T_{1} = 400.0 \, \text{K} \) - \( k_{2} = 0.689 \, \text{M/s} \) at \( T_{2} = 450.0 \, \text{K} \) We can solve for \( E_a \): \[ \ln\left(\frac{0.689}{0.0117}\right) = -\frac{E_a}{8.314} \left( \frac{1}{450.0} - \frac{1}{400.0} \right) \] Calculating the left side: \[ \ln\left(\frac{0.689}{0.0117}\right) = \ln(58.8034) \approx 4.0732 \] Calculating the right side: \[ \frac{1}{450.0} - \frac{1}{400.0} = 0.002222 - 0.0025 = -0.000278 \] Then: \[
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