The activation energy of a reaction can be determined by

Options

A. plotting substrate-time graph
B. plotting rate-time graph
C. calculating rate constants at two different temperatures
D. None of these

Explanation

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 temperature (T) and 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
- \( A \) = Arrhenius constant (frequency factor)
- \( E_a \) = activation energy
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 2: Take Natural Logarithm of the Arrhenius Equation
Taking the natural logarithm of both sides gives:
\[ \ln k = \ln A - \frac{E_a}{RT} \]
This can be rearranged to:
\[ \ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A \]

### Step 3: Prepare Two Equations for Two Different Temperatures
If we have two rate constants \( k_1 \) and \( k_2 \) at two different temperatures \( T_1 \) and \( T_2 \), we can write:
1. For \( k_1 \):
\[ \ln k_1 = \ln A - \frac{E_a}{RT_1} \]
2. For \( k_2 \):
\[ \ln k_2 = \ln A - \frac{E_a}{RT_2} \]

### Step 4: Subtract the Two Equations
Subtract the first equation from the second:
\[ \ln k_2 - \ln k_1 = \left(-\frac{E_a}{RT_2}\right) - \left(-\frac{E_a}{RT_1}\right) \]
This simplifies to:
\[ \ln \frac{k_2}{k_1} = -\frac{E_a}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]

### Step 5: Solve for Activation Energy (Ea)
Rearranging the equation gives:
\[ E_a = -R \cdot \frac{\ln \frac{k_2}{k_1}}{\left(\frac{1}{T_2} - \frac{1}{T_1}\right)} \]
This equation allows us to calculate the activation energy if we know the rate constants \( k_1 \) and \( k_2 \) at temperatures \( T_1 \) and \( T_2 \).

### Step 6: Graphical Method (Optional)
Alternatively, if you plot \( \ln k \) against \( \frac{1}{T} \), the slope of the resulting straight line will be:
\[ \text{slope} = -\frac{E_a}{R} \]
From the slope, you can also determine the activation energy.

### Conclusion
The activation energy of a reaction can be determined using the Arrhenius equation by measuring the rate constants at two different temperatures or by plotting \( \ln k \) against \( \frac{1}{T} \).

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