How can the activation enrgy for a reaction be determined graphically?

Options

A. Plot `k` versue `T`, the slope of the line will be equal to `E_(a)`
B. Plot `1[A]_(t)`, versus `t`, the slope of the line will be equal to `E_(a)`
C. Plot In `[A]_(t)` versus `t`, the slope of the line be equal to `-E_(a)`.
D. Plot `Ink` versus `1//T`, the slope of the line will be equal to `-E_(a)//R`.

Explanation

To determine the activation energy for a reaction graphically, we can use the Arrhenius equation, which relates the rate constant (k) to temperature (T) and activation energy (Ea). 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
To linearize the equation, take the natural logarithm of both sides:
\[ \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \]

### Step 3: Rearrange to Match the Linear Form
This can be rearranged to fit the linear equation format \( y = mx + c \):
\[ \ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A \]
Here, we can identify:
- \( y = \ln k \)
- \( m = -\frac{E_a}{R} \) (slope)
- \( x = \frac{1}{T} \)
- \( c = \ln A \) (y-intercept)

### Step 4: Plot the Graph
To determine the activation energy graphically:
1. Plot \( \ln k \) (y-axis) against \( \frac{1}{T} \) (x-axis).
2. The resulting graph will be a straight line.

### Step 5: Determine the Slope
The slope of the line obtained from the graph will be:
\[ \text{slope} = -\frac{E_a}{R} \]
From this slope, you can calculate the activation energy \( E_a \):
\[ E_a = -\text{slope} \cdot R \]

### Conclusion
Thus, the activation energy can be determined graphically by plotting \( \ln k \) versus \( \frac{1}{T} \) and calculating the slope of the resulting line.

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