The following data were obtained at a certain temperature for the decomposition of `NH_3` in contact with catalyst.
`{:(P(mn), 25, 50),(t_(1//2),1.82,0.91):}`
The order of reaction is

Options

A. 2
B. 3
C. 1
D. 4

Answer

To determine the order of the reaction for the decomposition of ammonia (NH₃) based on the given data, we can follow these steps:

### Step 1: Understand the relationship between half-life and pressure
The half-life (t₁/₂) of a reaction is related to the initial concentration (or pressure in this case) of the reactants. For a reaction of order n, the relationship can be expressed as:

\[ t_{1/2} \propto \frac{1}{P_0^{n-1}} \]

Where:
- \( t_{1/2} \) is the half-life,
- \( P_0 \) is the initial pressure,
- \( n \) is the order of the reaction.

### Step 2: Analyze the given data
From the problem, we have the following data:

- For \( P_0 = 25 \) mm, \( t_{1/2} = 1.82 \) minutes.
- For \( P_0 = 50 \) mm, \( t_{1/2} = 0.91 \) minutes.

### Step 3: Compare the half-lives
Notice that when the pressure doubles from 25 mm to 50 mm, the half-life decreases from 1.82 minutes to 0.91 minutes. This indicates an inverse relationship between half-life and pressure.

### Step 4: Establish the relationship
Using the relationship established in Step 1, we can set up the following equations based on the two sets of data:

1. For \( P_0 = 25 \) mm:
\[ t_{1/2} \propto \frac{1}{25^{n-1}} \quad \text{(1)} \]

2. For \( P_0 = 50 \) mm:
\[ t_{1/2} \propto \frac{1}{50^{n-1}} \quad \text{(2)} \]

### Step 5: Set up the ratio
Since the half-lives are inversely proportional to the pressures raised to the power of \( n-1 \), we can set up the ratio of the two half-lives:

\[ \frac{t_{1/2}(25)}{t_{1/2}(50)} = \frac{50^{n-1}}{25^{n-1}} \]

Substituting the half-life values:

\[ \frac{1.82}{0.91} = \frac{50^{n-1}}{25^{n-1}} \]

### Step 6: Simplify the equation
Calculating the left side:

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

### Step 7: Solve for n
Taking logarithm or equating the powers:

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

This implies:

\[ n - 1 = 1 \quad \Rightarrow \quad n = 2 \]

### Conclusion
The order of the reaction is **2**.

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