Among the following graphs showing variation of rate (k) with temperature (T) for a recation, the one that exhibits arrhenius behaviour over the entire temperature range is

Options

A.
B.
C.
D.

Explanation

To determine which graph exhibits Arrhenius behavior over the entire temperature range, we can follow these steps:

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

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

### Step 3: Identify the Linear Relationship
From the equation, we can see that if we plot \( \ln k \) (y-axis) against \( \frac{1}{T} \) (x-axis), we should get a straight line with:
- Slope = \(-\frac{E_a}{R}\) (which is negative),
- Y-intercept = \(\ln A\).

### Step 4: Analyze the Graphs
Now, we need to analyze the provided graphs to find which one shows a straight line with a negative slope over the entire temperature range.

1. **Graph A**: Check if it is linear and has a negative slope.
2. **Graph B**: Check if it is linear and has a negative slope.
3. **Graph C**: Check if it is linear and has a negative slope.
4. **Graph D**: Check if it is linear and has a negative slope.

### Step 5: Determine the Correct Graph
The graph that maintains a straight line with a consistent negative slope across the entire temperature range represents the Arrhenius behavior.

### Conclusion
After analyzing the graphs, you will identify the one that fits the criteria of exhibiting Arrhenius behavior over the entire temperature range.