A substance decomposes according to zero order kinetics. If the rate constant is k and initial concentration is ‘a’ then half life period of the reaction will be:

Options

A. `1//ak`
B. `0.693//k`
C. `k//2a`
D. `a//2k`

Answer

To find the half-life period of a substance decomposing according to zero-order kinetics, we can follow these steps:

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

### Step 2: Write the Integrated Rate Equation
For a zero-order reaction, the integrated rate equation is given by:
\[ [A] = [A_0] - kt \]
where:
- \([A]\) is the concentration at time \( t \),
- \([A_0]\) is the initial concentration,
- \( k \) is the rate constant,
- \( t \) is the time.

### Step 3: Define Half-Life
The half-life (\( t_{1/2} \)) is the time required for the concentration of the reactant to decrease to half of its initial value. Therefore, we set:
\[ [A] = \frac{[A_0]}{2} \]

### Step 4: Substitute into the Integrated Rate Equation
Substituting \([A] = \frac{[A_0]}{2}\) into the integrated rate equation:
\[ \frac{[A_0]}{2} = [A_0] - kt_{1/2} \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ kt_{1/2} = [A_0] - \frac{[A_0]}{2} \]
\[ kt_{1/2} = \frac{[A_0]}{2} \]

### Step 6: Solve for Half-Life
Now, we can solve for \( t_{1/2} \):
\[ t_{1/2} = \frac{[A_0]}{2k} \]

### Conclusion
Thus, the half-life period of the reaction is:
\[ t_{1/2} = \frac{[A_0]}{2k} \]

### Final Answer
The half-life of a zero-order reaction is given by:
\[ t_{1/2} = \frac{a}{2k} \]
where \( a \) is the initial concentration.

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