Which of the following is correct for zero and first order reactions respectively, where 'a' is initial concentration of the reactant ?

Options

A. `t_(1//2)propa,t_(1//2)prop1/a^2`
B. `t_(1//2)propa,t_(1//2)prop1/a`
C. `t_(1//2)propa,t_(1//2)propa^0`
D. `t_(1//2)propa^0,t_(1//2)propa`

Answer

To solve the problem regarding the relationship between half-life periods and initial concentration for zero and first order reactions, we will follow these steps:

### Step 1: Understand 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: Zero Order Reaction
For a zero-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{[A]_0}{2k} \]
Where:
- \( [A]_0 \) is the initial concentration of the reactant.
- \( k \) is the rate constant.

From this formula, we can see that the half-life is directly proportional to the initial concentration \( [A]_0 \):
\[ t_{1/2} \propto [A]_0 \]

### Step 3: First Order Reaction
For a first-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
In this case, the half-life is independent of the initial concentration. Therefore, we can express it as:
\[ t_{1/2} \propto [A]_0^0 \]
This indicates that the half-life does not change with varying initial concentrations.

### Step 4: Combine Results
From the above steps, we can summarize:
- For zero-order reactions: \( t_{1/2} \propto [A]_0^1 \)
- For first-order reactions: \( t_{1/2} \propto [A]_0^0 \)

### Step 5: Identify the Correct Option
Now, we need to match these results with the options given in the question. The correct expressions for zero and first order reactions respectively are:
- Zero order: \( [A]_0^1 \)
- First order: \( [A]_0^0 \)

Thus, the correct answer is the option that states \( [A]_0^1 \) for zero order and \( [A]_0^0 \) for first order.

### Conclusion
The correct option is the one that states:
- For zero order: \( [A]_0^1 \)
- For first order: \( [A]_0^0 \)