The rate of decomposition of `N_(2)O_(5)` in `C Cl_(4)` solution has been studied at 318 K and the following results have been obtained :
`{:(t//min,,,0,,,135,,,342,,,683,,,1693),(c//M,,,2.08,,,1.91,,,1.67,,,1.35,,,0.57):}`
Find the order of the reaction and calculate its rate constant. What is the half-life period ?

Explanation

To solve the problem of determining the order of the reaction for the decomposition of \( N_2O_5 \) in \( CCl_4 \) solution, calculating the rate constant, and finding the half-life period, we will follow these steps:

### Step 1: Analyze the Data
We are given the following data:

| Time (min) | Concentration (M) |
|------------|--------------------|
| 0 | 2.08 |
| 135 | 1.91 |
| 342 | 1.67 |
| 683 | 1.35 |
| 1693 | 0.57 |

### Step 2: Determine the Order of the Reaction
To determine the order of the reaction, we can use the integrated rate laws for different orders.

1. **Zero Order**: \( [A] = [A]_0 - kt \)
2. **First Order**: \( \ln[A] = \ln[A]_0 - kt \)
3. **Second Order**: \( \frac{1}{[A]} = \frac{1}{[A]_0} + kt \)

Since we have concentration data at various times, we will test for first-order kinetics, as it is a common order for decomposition reactions.

### Step 3: Use First Order Integrated Rate Law
The first-order integrated rate law is given by:

\[ k = \frac{2.303}{t} \log\left(\frac{[A]_0}{[A]}\right) \]

We will calculate \( k \) using the concentrations at two different times.

#### Calculation for \( t = 135 \) min:
- Initial concentration, \( [A]_0 = 2.08 \, M \)
- Concentration at \( t = 135 \, min \), \( [A] = 1.91 \, M \)

\[ k = \frac{2.303}{135} \log\left(\frac{2.08}{1.91}\right) \]

Calculating the log term:

\[ \log\left(\frac{2.08}{1.91}\right) = \log(1.0900) \approx 0.037 \]

Now substituting back:

\[ k = \frac{2.303}{135} \times 0.037 \approx 6.31 \times 10^{-4} \, \text{min}^{-1} \]

### Step 4: Confirming the Order
Since we calculated \( k \) using the first-order integrated rate law and obtained a consistent value, we can conclude that the reaction is first-order with respect to \( N_2O_5 \).

### Step 5: Calculate Half-Life Period
For a first-order reaction, the half-life \( t_{1/2} \) is given by:

\[ t_{1/2} = \frac{0.693}{k} \]

Substituting the value of \( k \):

\[ t_{1/2} = \frac{0.693}{6.31 \times 10^{-4}} \approx 1098 \, \text{min} \]

### Final Results
- **Order of the reaction**: 1 (first-order)
- **Rate constant \( k \)**: \( 6.31 \times 10^{-4} \, \text{min}^{-1} \)
- **Half-life period**: \( 1098 \, \text{min} \)