The rate constant & for a certain reaction is measured at two different temperatures: temperature k 286.0 °C 1.2×1011 393.0 °C 1.4×10¹1 Assuming the rate constant obeys the Arrhenius equation, calculate the activation energy E for this reaction Round your answer to 2 significant digits. £.-0-0/0 E mol x10
The rate constant & for a certain reaction is measured at two different temperatures: temperature k 286.0 °C 1.2×1011 393.0 °C 1.4×10¹1 Assuming the rate constant obeys the Arrhenius equation, calculate the activation energy E for this reaction Round your answer to 2 significant digits. £.-0-0/0 E mol x10
Chemistry
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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
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![### Activation Energy Calculation Using the Arrhenius Equation
The rate constant \( k \) for a certain reaction is measured at two different temperatures:
| Temperature (°C) | \( k \) \\
|------------------|----------|
| 286.0 °C | \( 1.2 \times 10^{11} \) |
| 393.0 °C | \( 1.4 \times 10^{11} \) |
Assuming the rate constant obeys the Arrhenius equation, calculate the activation energy \( E_a \) for this reaction. Round your answer to 2 significant digits.
\[ E_a = \boxed{ \ \ \ \ \ \ } \ \text{kJ/mol} \]
### Explanation of Concepts
**Arrhenius Equation:**
The Arrhenius equation is given as:
\[ k = A \cdot e^{ -E_a / (RT) } \]
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.
**Conversion of Temperature:**
- Convert the temperature from Celsius to Kelvin:
\[ T_1 = 286.0^\circ C + 273.15 = 559.15 \, K \]
\[ T_2 = 393.0^\circ C + 273.15 = 666.15 \, K \]
**Calculation Steps:**
1. **Apply the Arrhenius Equation to both temperatures:**
\[ \ln k_1 = \ln A - \frac{E_a}{RT_1} \]
\[ \ln k_2 = \ln A - \frac{E_a}{RT_2} \]
2. **Form the difference of the two equations to eliminate \( A \):**
\[ \ln \left( \frac{k_2}{k_1} \right) = -E_a \left( \frac{1}{RT_2} - \frac{1}{RT_1} \right) \]
3. **Insert the known values:**
\[ \ln \left( \frac{1.4 \times 10^{11}}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20b515c9-5e3a-448e-90a2-ac3b562e1523%2F4c6dea7f-7b96-4063-9f9b-297968d00b85%2F3fbzwa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Activation Energy Calculation Using the Arrhenius Equation
The rate constant \( k \) for a certain reaction is measured at two different temperatures:
| Temperature (°C) | \( k \) \\
|------------------|----------|
| 286.0 °C | \( 1.2 \times 10^{11} \) |
| 393.0 °C | \( 1.4 \times 10^{11} \) |
Assuming the rate constant obeys the Arrhenius equation, calculate the activation energy \( E_a \) for this reaction. Round your answer to 2 significant digits.
\[ E_a = \boxed{ \ \ \ \ \ \ } \ \text{kJ/mol} \]
### Explanation of Concepts
**Arrhenius Equation:**
The Arrhenius equation is given as:
\[ k = A \cdot e^{ -E_a / (RT) } \]
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.
**Conversion of Temperature:**
- Convert the temperature from Celsius to Kelvin:
\[ T_1 = 286.0^\circ C + 273.15 = 559.15 \, K \]
\[ T_2 = 393.0^\circ C + 273.15 = 666.15 \, K \]
**Calculation Steps:**
1. **Apply the Arrhenius Equation to both temperatures:**
\[ \ln k_1 = \ln A - \frac{E_a}{RT_1} \]
\[ \ln k_2 = \ln A - \frac{E_a}{RT_2} \]
2. **Form the difference of the two equations to eliminate \( A \):**
\[ \ln \left( \frac{k_2}{k_1} \right) = -E_a \left( \frac{1}{RT_2} - \frac{1}{RT_1} \right) \]
3. **Insert the known values:**
\[ \ln \left( \frac{1.4 \times 10^{11}}
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