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

Answer

To solve the problem regarding the reaction \( A \rightarrow B \) with a zero-order reaction with respect to \( A \), we can follow these steps:

### Step 1: Understand Zero-Order Reactions
In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant. Therefore, the rate law can be expressed as:
\[ \text{Rate} = k[A]^0 = k \]
where \( k \) is the rate constant.

### Step 2: Write the Rate of Change of Concentration
Since the concentration of \( A \) is decreasing as the reaction proceeds, we can express the rate of change of concentration of \( A \) as:
\[ -\frac{d[A]}{dt} = k \]

### Step 3: Rearrange and Integrate
Rearranging the equation gives:
\[ d[A] = -k \, dt \]
Now, we will integrate both sides. The left side will be integrated from the initial concentration \( [A]_0 \) to \( [A] \) at time \( t \), and the right side will be integrated from \( 0 \) to \( t \):
\[ \int_{[A]_0}^{[A]} d[A] = -k \int_{0}^{t} dt \]

### Step 4: Perform the Integration
The integration results in:
\[ [A] - [A]_0 = -kt \]
Rearranging gives:
\[ [A] = [A]_0 - kt \]

### Step 5: Determine the Expression for Half-Life
For a zero-order reaction, the half-life \( t_{1/2} \) can be derived from the equation:
\[ t_{1/2} = \frac{[A]_0}{2k} \]
This shows that the half-life is directly proportional to the initial concentration of \( A \).

### Step 6: Identify the Correct Expression
From the derived equations, we can conclude that the expression that describes the reaction is:
\[ [A] = [A]_0 - kt \]
This indicates that the concentration of \( A \) decreases linearly over time.

### Final Answer
The correct expression that describes the zero-order reaction \( A \rightarrow B \) is:
\[ [A] = [A]_0 - kt \]