Which of the following expressions is correct for zero order and first order respectively [Where a is initial concentration]?

Options

A. `t_(1//2)propa,t_(1//2)prop(1)/(a)`
B. `t_(1//2)propa,t_(1//2)propa^(@)`
C. `t_(1//2)propa^(@),t_(1//2)propa`
D. `t_(1//2)propa,t_(1//2)prop(1)/(a^(2))`

Explanation

To determine the correct expressions for the half-life of zero-order and first-order reactions in terms of the initial concentration (A), we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life (t_half) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration.

### Step 2: Write the expression for zero-order reactions
For a zero-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{[A]_0}{k} \]
where \([A]_0\) is the initial concentration and \(k\) is the rate constant.

### Step 3: Substitute the initial concentration
Since we are focusing on the relationship with the initial concentration \(A\), we can express it as:
\[ t_{1/2} \propto A \]
This indicates that the half-life is directly proportional to the initial concentration for zero-order reactions.

### Step 4: Write the expression for first-order reactions
For a first-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
This shows that the half-life is independent of the initial concentration.

### Step 5: Understand the relationship for first-order reactions
For first-order reactions, the concentration does not appear in the half-life expression, which means:
\[ t_{1/2} \propto A^0 \]
This indicates that the half-life is constant and does not depend on the initial concentration.

### Step 6: Summarize the expressions
Thus, the correct expressions for the half-lives of zero-order and first-order reactions in terms of the initial concentration \(A\) are:
- Zero-order: \( t_{1/2} \propto A \)
- First-order: \( t_{1/2} \propto A^0 \) (which simplifies to a constant)

### Conclusion
Based on the analysis, the correct answer is that the half-life for zero-order is proportional to the initial concentration \(A\) and for first-order is independent of \(A\).

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