The half-life of `2` sample are `0.1` and `0.4` seconds. Their respctive concentration are `200` and `50` respectively. What is the order of the reaction

Options

A. 0
B. 2
C. 1
D. 4

Explanation

To determine the order of the reaction based on the given half-lives and concentrations, we can follow these steps:

### Step 1: Write the relationship for half-life of nth order reactions.
The half-life (\(t_{1/2}\)) for a reaction of order \(n\) is given by the formula:
\[ t_{1/2} = \frac{1}{k} (n - 1) \left( \frac{A_0}{n - 1} \right)^{n - 1} \]
where \(A_0\) is the initial concentration.

### Step 2: Establish the relationship between the half-lives and concentrations.
From the relationship, we can derive that:
\[ \frac{t_{1/2,1}}{t_{1/2,2}} = \left( \frac{A_2}{A_1} \right)^{n - 1} \]
where:
- \(t_{1/2,1} = 0.1\) seconds (for the first sample)
- \(t_{1/2,2} = 0.4\) seconds (for the second sample)
- \(A_1 = 200\) (concentration of the first sample)
- \(A_2 = 50\) (concentration of the second sample)

### Step 3: Substitute the values into the equation.
Substituting the values we have:
\[ \frac{0.1}{0.4} = \left( \frac{50}{200} \right)^{n - 1} \]

### Step 4: Simplify the equation.
This simplifies to:
\[ \frac{1}{4} = \left( \frac{1}{4} \right)^{n - 1} \]

### Step 5: Equate the powers.
Since the bases are the same, we can equate the exponents:
\[ 1 = n - 1 \]

### Step 6: Solve for \(n\).
Solving for \(n\):
\[ n = 2 \]

### Conclusion
The order of the reaction is \(2\).