For a reaction , `A rarrB` , it has been found that the order of the reaction , is zero with respect to A. Which of the following expressions correctly describes the reaction ?

Options

A. `K=2.303/tlog.([A0])/([A])`
B. `[A]_0-[A]=Kt`
C. `t_(1//2)=(0.693)/K`
D. `t_(1//2)prop1/([A]_0)`

Answer

To solve the problem regarding the zero-order reaction \( A \rightarrow B \), we will derive the expressions for the rate constant and half-life of the reaction.

### Step-by-Step Solution:

1. **Understanding Zero-Order Reaction**:
- A zero-order reaction means that the rate of the reaction is independent of the concentration of the reactant \( A \). Thus, the rate can be expressed as:
\[ \text{Rate} = k \]
where \( k \) is the rate constant.

2. **Writing the Rate Expression**:
- The rate of the reaction can also be expressed in terms of the change in concentration of \( A \):
\[ \text{Rate} = -\frac{d[A]}{dt} \]
- For a zero-order reaction, we can equate this to the rate constant:
\[ -\frac{d[A]}{dt} = k \]

3. **Integrating the Rate Equation**:
- Rearranging gives:
\[ d[A] = -k \, dt \]
- Integrating both sides from the initial concentration \( [A]_0 \) to \( [A] \) over the time from 0 to \( t \):
\[ \int_{[A]_0}^{[A]} d[A] = -k \int_{0}^{t} dt \]
- This results in:
\[ [A] - [A]_0 = -kt \]
- Rearranging gives:
\[ [A] = [A]_0 - kt \]

4. **Finding the Expression for the Rate Constant**:
- From the integrated rate law, we can express the change in concentration:
\[ [A]_0 - [A] = kt \]
- Thus, the expression for the rate constant \( k \) can be derived as:
\[ k = \frac{[A]_0 - [A]}{t} \]

5. **Calculating the Half-Life of the Reaction**:
- The half-life \( t_{1/2} \) for a zero-order reaction is defined as the time required for the concentration of \( A \) to decrease to half of its initial value:
\[ [A] = \frac{[A]_0}{2} \]
- Substituting this into the integrated rate equation:
\[ [A]_0 - \frac{[A]_0}{2} = kt_{1/2} \]
- This simplifies to:
\[ \frac{[A]_0}{2} = kt_{1/2} \]
- Rearranging gives:
\[ t_{1/2} = \frac{[A]_0}{2k} \]

6. **Conclusion**:
- The correct expression for the rate constant \( k \) is \( k = [A]_0 - [A] \) and the half-life \( t_{1/2} = \frac{[A]_0}{2k} \).

### Final Answer:
The correct expression that describes the reaction is option number 2, which corresponds to the derived expressions for the rate constant and half-life of a zero-order reaction.