Which of the following graphs formed plotted between `t_(1//2)` and initial concentration (a) represents a zero order reaction ?

Options

A.
B.
C.
D.

Explanation

To determine which graph represents a zero-order reaction when plotting half-life (\(t_{1/2}\)) against initial concentration (\(A_0\)), we can follow these steps:

### Step 1: Understand the relationship for zero-order reactions
For a zero-order reaction, the half-life (\(t_{1/2}\)) is given by the formula:
\[ t_{1/2} = \frac{A_0}{2k} \]
where:
- \(t_{1/2}\) = half-life
- \(A_0\) = initial concentration
- \(k\) = rate constant

### Step 2: Analyze the relationship
From the formula, we can see that the half-life (\(t_{1/2}\)) is directly proportional to the initial concentration (\(A_0\)). This means that as the initial concentration increases, the half-life also increases.

### Step 3: Rewrite the relationship
We can express this relationship as:
\[ t_{1/2} = \frac{1}{2k} \cdot A_0 \]
This indicates that if we plot \(t_{1/2}\) (y-axis) against \(A_0\) (x-axis), we will have a linear relationship.

### Step 4: Identify the characteristics of the graph
Since the relationship is linear, the graph will be a straight line. The slope of the line will be \(\frac{1}{2k}\), and it will pass through the origin (0,0) because when \(A_0 = 0\), \(t_{1/2}\) will also be 0.

### Step 5: Conclusion
Thus, the graph that represents a zero-order reaction when plotting \(t_{1/2}\) against initial concentration \(A_0\) will be a straight line passing through the origin.

### Final Answer
The correct option is the first option, which represents a straight line graph passing through the origin.

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