Which of the following option is valid for zero reaction?

Options

A. `t_(1//2)=(3)/(2)t_(1//4)`
B. `t_(1//2)=(4)/(3)t_(1//4)`
C. `t_(1//2)=2t_(1//4)`
D. `t_(1//4)=(t_(1//2))^(2)`

Explanation

To solve the question regarding the valid option for a zero-order reaction, we need to derive the expressions for the half-life (t₁/₂) and the quarter-life (t₁/₄) of a zero-order reaction and then find their ratio.

### Step-by-Step Solution:

1. **Understanding Zero-Order Reaction**:
- In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant.
- The general form of the rate equation for a zero-order reaction is:
\[ \text{Rate} = k \]
- The concentration of the reactant (A) at time t can be expressed as:
\[ [A] = [A_0] - kt \]
where [A₀] is the initial concentration and k is the rate constant.

2. **Deriving the Half-Life (t₁/₂)**:
- The half-life is the time required for the concentration of the reactant to reduce to half of its initial concentration.
- Setting [A] = [A₀]/2 in the equation:
\[ \frac{[A_0]}{2} = [A_0] - kt_{1/2} \]
- Rearranging gives:
\[ kt_{1/2} = [A_0] - \frac{[A_0]}{2} = \frac{[A_0]}{2} \]
- Therefore, we can express t₁/₂ as:
\[ t_{1/2} = \frac{[A_0]}{2k} \]

3. **Deriving the Quarter-Life (t₁/₄)**:
- The quarter-life is the time required for the concentration of the reactant to reduce to one-fourth of its initial concentration.
- Setting [A] = [A₀]/4 in the equation:
\[ \frac{[A_0]}{4} = [A_0] - kt_{1/4} \]
- Rearranging gives:
\[ kt_{1/4} = [A_0] - \frac{[A_0]}{4} = \frac{3[A_0]}{4} \]
- Therefore, we can express t₁/₄ as:
\[ t_{1/4} = \frac{3[A_0]}{4k} \]

4. **Finding the Ratio of t₁/₂ to t₁/₄**:
- Now we can find the ratio of t₁/₂ to t₁/₄:
\[ \frac{t_{1/2}}{t_{1/4}} = \frac{\frac{[A_0]}{2k}}{\frac{3[A_0]}{4k}} = \frac{[A_0]}{2k} \cdot \frac{4k}{3[A_0]} = \frac{4}{6} = \frac{2}{3} \]

5. **Conclusion**:
- The ratio of the half-life to the quarter-life for a zero-order reaction is \( \frac{2}{3} \).
- Therefore, we check the options provided in the question to find the one that corresponds to this ratio.

### Final Answer:
- The correct option is **C**, as it represents the valid ratio for a zero-order reaction.