In a first order reaction , on plotting a graph between `t_(1//2)` and concentration of reactant

Options

A. A curve is obtained
B. A straight line is obtained having slope `45^@`
C. A straight line is parallel to time axis
D. A straight line is parallel to the concentration axis

Answer

To solve the question regarding the relationship between the half-life (\(t_{1/2}\)) and the concentration of reactant in a first-order reaction, we can follow these steps:

### Step 1: Understand the relationship between half-life and concentration
In a first-order reaction, the half-life (\(t_{1/2}\)) is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
where \(k\) is the rate constant. Importantly, for a first-order reaction, the half-life is independent of the initial concentration of the reactant.

### Step 2: Analyze the dependence of half-life on concentration
Since the half-life does not depend on the concentration of the reactant, we can conclude that regardless of the initial concentration (\(A_0\)), the half-life remains constant. This means that as we change the concentration, the value of \(t_{1/2}\) does not change.

### Step 3: Plotting the graph
When we plot a graph of \(t_{1/2}\) (on the y-axis) against the initial concentration of the reactant \(A_0\) (on the x-axis), we will observe that:
- Since \(t_{1/2}\) is constant and does not vary with \(A_0\), the graph will be a horizontal line.
- This line is parallel to the concentration axis (x-axis).

### Step 4: Conclusion
From the analysis, we can conclude that the graph of \(t_{1/2}\) versus the concentration of the reactant in a first-order reaction results in a straight line that is parallel to the concentration axis.

### Final Answer
The correct option is:
- **Option 4**: A straight line is parallel to the concentration axis.

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