If initial concentration is reduced to `1//4^(th)` in a zero order reaction , the time taken for half the reaction to complete

Options

A. Remains same
B. Becomes 4 time
C. Become one fourth
D. Doubles

Answer

To solve the problem, we need to determine the time taken for half the reaction to complete when the initial concentration of a zero-order reaction is reduced to \( \frac{1}{4} \) of its original value.

### Step-by-Step Solution:

1. **Understanding Zero-Order Reactions**:
In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactants. The half-life (\( t_{1/2} \)) for a zero-order reaction is given by the formula:
\[ t_{1/2} = \frac{[A_0]}{2k} \]
where \( [A_0] \) is the initial concentration and \( k \) is the rate constant.

2. **Define Initial Concentration**:
Let the initial concentration be \( [A_0] = a \).

3. **Calculate Half-Life with Original Concentration**:
Using the formula for half-life, we can express the half-life with the original concentration:
\[ t_{1/2} = \frac{a}{2k} \quad \text{(Equation 1)} \]

4. **Determine New Concentration**:
According to the problem, the initial concentration is reduced to \( \frac{1}{4} \) of its original value:
\[ [A_0] = \frac{1}{4}a \]

5. **Calculate New Half-Life**:
Now, we can calculate the new half-life (\( t'_{1/2} \)) using the reduced concentration:
\[ t'_{1/2} = \frac{\frac{1}{4}a}{2k} = \frac{a}{8k} \quad \text{(Equation 2)} \]

6. **Finding the Ratio of New Half-Life to Original Half-Life**:
To find how the new half-life compares to the original half-life, we can take the ratio of \( t'_{1/2} \) to \( t_{1/2} \):
\[ \frac{t'_{1/2}}{t_{1/2}} = \frac{\frac{a}{8k}}{\frac{a}{2k}} = \frac{1/8}{1/2} = \frac{1}{4} \]

7. **Conclusion**:
This means that the new half-life is \( \frac{1}{4} \) of the original half-life. Thus, the time taken for half the reaction to complete when the initial concentration is reduced to \( \frac{1}{4} \) is four times longer than the original half-life.

### Final Answer:
The time taken for half the reaction to complete when the initial concentration is reduced to \( \frac{1}{4} \) is **four times the original half-life**.

---