For a certain reaction of order 'n' the time for half change `t_(1//2)` is given by : `t_(1//2)=(2-sqrt(2))/KxxC_(0)^(1//2)` where `K` is rate constant and `C_(0)` is the initial concentration. The value of `n` is:

Options

A. 1
B. 2
C. 0
D. 0.5

Answer

To determine the order of the reaction given the half-life equation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The half-life \( t_{1/2} \) is given by:
\[ t_{1/2} = \frac{2 - \sqrt{2}}{K \cdot C_0^{1/2}} \]
Here, \( K \) is the rate constant and \( C_0 \) is the initial concentration.

2. **Rearranging the Equation**:
We can express this equation in a form that highlights the relationship between \( t_{1/2} \) and \( C_0 \):
\[ t_{1/2} = k' \cdot C_0^{1/2} \]
where \( k' = \frac{2 - \sqrt{2}}{K} \) is a constant.

3. **Identify the Relationship for nth Order Reactions**:
For a reaction of order \( n \), the half-life is related to the initial concentration \( C_0 \) as follows:
\[ t_{1/2} \propto C_0^{\frac{1}{2}} \]
This implies that:
\[ t_{1/2} \propto C_0^{1-n} \]

4. **Comparing the Exponents**:
From the two proportionality relationships, we can equate the exponents:
\[ \frac{1}{2} = 1 - n \]

5. **Solving for n**:
Rearranging the equation gives:
\[ n = 1 - \frac{1}{2} = \frac{1}{2} \]

6. **Conclusion**:
Therefore, the order of the reaction \( n \) is \( 0.5 \).

### Final Answer:
The value of \( n \) is \( 0.5 \).

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