Activation energy of a chemical reaction can be determined by:

Options

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

Answer

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. Here’s a step-by-step solution to the question:

### Step 1: Understand the Arrhenius Equation
The Arrhenius equation is given by:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = Arrhenius factor (frequency factor)
- \( E_a \) = activation energy
- \( R \) = gas constant (8.314 J/mol·K)
- \( T \) = temperature in Kelvin

### Step 2: Set Up the Equations for Two Different Temperatures
To find the activation energy, we can measure the rate constants at two different temperatures:
1. At temperature \( T_1 \), the rate constant is \( k_1 \):
\[ k_1 = A e^{-\frac{E_a}{RT_1}} \]
2. At temperature \( T_2 \), the rate constant is \( k_2 \):
\[ k_2 = A e^{-\frac{E_a}{RT_2}} \]

### Step 3: Divide the Two Equations
By dividing the two equations, we can eliminate the Arrhenius factor \( A \):
\[ \frac{k_2}{k_1} = \frac{A e^{-\frac{E_a}{RT_2}}}{A e^{-\frac{E_a}{RT_1}}} \]
This simplifies to:
\[ \frac{k_2}{k_1} = e^{-\frac{E_a}{RT_2} + \frac{E_a}{RT_1}} = e^{\frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)} \]

### Step 4: Take the Logarithm
Taking the natural logarithm (ln) of both sides gives:
\[ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]

### Step 5: Solve for Activation Energy
Rearranging the equation to solve for \( E_a \):
\[ E_a = R \cdot \ln\left(\frac{k_2}{k_1}\right) \cdot \frac{1}{\left(\frac{1}{T_1} - \frac{1}{T_2}\right)} \]

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