The activation energy of a reaction can be determined by

Options

A. Plotting a substrate-time graph
B. Plotting a rate time graph
C. Calculating rate constants at two different temperatures
D. None of these

Answer

To determine the activation energy (Ea) of a 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:

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

### Step 2: Take the Natural Logarithm
Taking the natural logarithm of both sides gives:
\[ \ln k = \ln A - \frac{E_a}{RT} \]
This can be rearranged to:
\[ \ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A \]
This equation is in the form of \( y = mx + b \), where:
- \( y = \ln k \)
- \( m = -\frac{E_a}{R} \)
- \( x = \frac{1}{T} \)
- \( b = \ln A \)

### Step 3: Plotting the Graph
If we plot \( \ln k \) against \( \frac{1}{T} \), we will get a straight line. The slope of this line will be equal to \(-\frac{E_a}{R}\).

### Step 4: Calculate Activation Energy
To find the activation energy, we can calculate the slope from the graph:
\[ \text{slope} = -\frac{E_a}{R} \]
From the slope, we can rearrange to find \( E_a \):
\[ E_a = -\text{slope} \cdot R \]

### Step 5: Using Rate Constants at Two Different Temperatures
Alternatively, we can determine \( E_a \) using the rate constants at two different temperatures \( T_1 \) and \( T_2 \):
1. Calculate \( k_1 \) at \( T_1 \) and \( k_2 \) at \( T_2 \).
2. Use the equation:
\[ \ln \frac{k_2}{k_1} = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
3. Rearranging gives:
\[ E_a = -\frac{R \ln \frac{k_2}{k_1}}{\left( \frac{1}{T_2} - \frac{1}{T_1} \right)} \]

### Conclusion
Thus, the activation energy of a reaction can be determined by calculating the rate constants at two different temperatures and using the Arrhenius equation.