Activation energy of a chemical reaction can be determined by

Options

A. evaluating rate constant at standard temperature
B. evaluating velocities of reaction at two different temperatures
C. evaluating rate constants at two different temperatures
D. changing concentration of reactants

Explanation

To determine the activation energy (Ea) of a chemical reaction, we can use the Arrhenius equation, which relates the rate constant (K) of a reaction to the temperature (T) and the activation energy. The Arrhenius equation is given by:

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

Where:
- \( K \) = rate constant
- \( A \) = Arrhenius constant (pre-exponential factor)
- \( E_a \) = activation energy (in Joules per mole)
- \( R \) = universal gas constant (8.314 J/mol·K)
- \( T \) = temperature in Kelvin

### Step-by-Step Solution:

1. **Understanding the Arrhenius Equation**:
- The Arrhenius equation shows that the rate constant \( K \) increases with an increase in temperature \( T \) and decreases with an increase in activation energy \( E_a \).

2. **Evaluating Rate Constants at Two Different Temperatures**:
- To determine \( E_a \), we can evaluate the rate constants at two different temperatures, \( T_1 \) and \( T_2 \).
- Let \( K_1 \) be the rate constant at temperature \( T_1 \) and \( K_2 \) be the rate constant at temperature \( T_2 \).

3. **Writing the Equations for Two Temperatures**:
- From the Arrhenius equation, we can write:
\[ K_1 = A e^{-\frac{E_a}{RT_1}} \]
\[ K_2 = A e^{-\frac{E_a}{RT_2}} \]

4. **Dividing the Two Equations**:
- Dividing the first equation by the second:
\[ \frac{K_1}{K_2} = \frac{A e^{-\frac{E_a}{RT_1}}}{A e^{-\frac{E_a}{RT_2}}} \]
- The constant \( A \) cancels out:
\[ \frac{K_1}{K_2} = e^{-\frac{E_a}{RT_1} + \frac{E_a}{RT_2}} = e^{-\frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)} \]

5. **Taking the Natural Logarithm**:
- Taking the natural logarithm of both sides:
\[ \ln\left(\frac{K_1}{K_2}\right) = -\frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]

6. **Rearranging to Solve for Activation Energy**:
- Rearranging the equation to solve for \( E_a \):
\[ E_a = -R \cdot \frac{\ln\left(\frac{K_1}{K_2}\right)}{\left(\frac{1}{T_1} - \frac{1}{T_2}\right)} \]
- Here, \( R \) is the universal gas constant (8.314 J/mol·K), and \( K_1 \), \( K_2 \), \( T_1 \), and \( T_2 \) are known values.

### Conclusion:
Thus, the activation energy of a chemical reaction can be determined by evaluating the rate constants at two different temperatures.