Which of the following graph is correct for a first order reaction ?

Options

A.
B.
C.
D.

Explanation

To determine which graphs are correct for a first-order reaction, we need to analyze the characteristics of first-order kinetics and the relevant equations.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions**:
A first-order reaction is characterized by a rate that is directly proportional to the concentration of one reactant. The rate law can be expressed as:
\[ \text{Rate} = k[A] \]
where \( k \) is the rate constant and \([A]\) is the concentration of the reactant.

2. **Half-Life of First-Order Reactions**:
The half-life (\( t_{1/2} \)) of a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
Importantly, this half-life is independent of the initial concentration of the reactant. This means that a graph plotting half-life against concentration will not show a relationship; hence, this graph is incorrect.

3. **Concentration vs. Time**:
For a first-order reaction, the relationship between concentration and time can be expressed as:
\[ \ln[A] = \ln[A_0] - kt \]
Rearranging gives:
\[ \ln[A] = -kt + \ln[A_0] \]
This is a linear equation in the form \( y = mx + b \), where:
- \( y = \ln[A] \)
- \( m = -k \)
- \( x = t \)
- \( b = \ln[A_0] \)

Therefore, a plot of \(\ln[A]\) versus time will yield a straight line with a slope of \(-k\).

4. **Logarithmic Form**:
Alternatively, using the logarithmic form, we can express the concentration in terms of its initial value:
\[ \log[A] = \log[A_0] - \frac{k}{2.303}t \]
This indicates that plotting \(\log[A]\) versus time will also yield a straight line.

5. **Identifying Correct Graphs**:
- **Graph of Half-Life vs. Concentration**: Incorrect, as half-life does not depend on concentration.
- **Graph of \(\ln[A]\) vs. Time**: Correct, as it shows a linear relationship.
- **Graph of \(\log[A]\) vs. Time**: Correct, as it also shows a linear relationship.

### Conclusion:
The correct graphs for a first-order reaction are:
- Graph A: Half-life vs. Concentration (Incorrect)
- Graph B: \(\ln[A]\) vs. Time (Correct)
- Graph C: \(\log[A]\) vs. Time (Correct)
- Graph D: Concentration vs. Time (Incorrect)

Thus, the correct options are B and C.