The half-life period of a substance is 50 minutes at a certain concentration. When the concentration is reduced to one half of the initial concentration, the half-life period is 25 minutes. Calculate order of the reaction.

Explanation

To determine the order of the reaction based on the given half-life periods at different concentrations, we can follow these steps:

### Step 1: Understand the relationship between half-life and concentration
The half-life \( T_{1/2} \) of a reaction is related to the initial concentration \( [A]_0 \) and the order of the reaction \( n \) by the formula:

\[ T_{1/2} \propto [A]_0^{(1-n)} \]

This means that the half-life is directly proportional to the initial concentration raised to the power of \( (1-n) \).

### Step 2: Set up the equation using the given data
We have two scenarios:
1. When the concentration is \( [A]_0 \), the half-life \( T_{1/2} = 50 \) minutes.
2. When the concentration is reduced to \( \frac{1}{2}[A]_0 \), the half-life \( T_{1/2} = 25 \) minutes.

Using the proportionality, we can write:

\[ \frac{T_{1/2,1}}{T_{1/2,2}} = \left( \frac{[A]_0}{\frac{1}{2}[A]_0} \right)^{(1-n)} \]

### Step 3: Substitute the known values
Substituting the values we have:

\[ \frac{50}{25} = \left( \frac{[A]_0}{\frac{1}{2}[A]_0} \right)^{(1-n)} \]

This simplifies to:

\[ 2 = \left( \frac{1}{\frac{1}{2}} \right)^{(1-n)} \]

### Step 4: Simplify the equation
The fraction simplifies as follows:

\[ 2 = (2)^{(1-n)} \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:

\[ 1 = 1 - n \]

### Step 6: Solve for n
Rearranging the equation gives:

\[ n = 0 \]

### Conclusion
The order of the reaction is \( n = 0 \).

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