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

Options

A.
B.
C.
D.

Explanation

To determine the correct graph for a first-order reaction, we need to analyze the characteristics of first-order kinetics and how they relate to graphical representations.

### Step-by-Step Solution:

1. **Understanding the Kinetic Equation**:
The kinetic equation for a first-order reaction is given by:
\[ k = \frac{2.303}{t} \log \left( \frac{[A_0]}{[A]} \right) \]
where:
- \( k \) is the rate constant,
- \( [A_0] \) is the initial concentration,
- \( [A] \) is the concentration at time \( t \).

2. **Rearranging the Equation**:
We can rearrange the equation to express it in the form of a straight line:
\[ \log [A_0] - \log [A] = \frac{k}{2.303} t \]
This can be rewritten as:
\[ \log [A] = \log [A_0] - \frac{k}{2.303} t \]
This is in the form \( y = mx + b \), where:
- \( y = \log [A] \),
- \( m = -\frac{k}{2.303} \) (the slope),
- \( x = t \),
- \( b = \log [A_0] \) (the y-intercept).

3. **Graph Characteristics**:
The graph of a first-order reaction will be a straight line when plotting \( \log [A] \) (y-axis) against time \( t \) (x-axis). The slope of this line will be negative, indicating that as time increases, the logarithm of the concentration decreases.

4. **Identifying the Correct Graph**:
Based on the analysis, we are looking for a graph that shows a straight line with a negative slope when \( \log [A] \) is plotted against time \( t \).

5. **Half-Life of First-Order Reactions**:
The half-life \( t_{1/2} \) for a first-order reaction is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
This indicates that the half-life is constant and does not depend on the initial concentration. Therefore, a graph showing half-life against concentration will be a horizontal line.

6. **Conclusion**:
The correct graphs for a first-order reaction are:
- A graph of \( \log [A] \) vs. time (which is a straight line).
- A graph showing that half-life is constant regardless of concentration.

### Final Answer:
The correct options for the graphs representing a first-order reaction are **Option 1** (for half-life) and **Option 4** (for concentration vs. time).