For the first order reaction
`2N_(2)O_(5)(g) rarr 4NO_(2)(g) + O_(2)(g)`

Options

A. The concentration of the reactant decreases exponentially with time.
B. The half life of the reaction decreases with increaisng temperature.
C. The half life of the reaction depends on the initial concentration of the reactant.
D. The reaction proceeds to `99.6%` completion in `8` half. Life durations.

Explanation

To solve the question regarding the first-order reaction:

**Reaction:**
\[ 2N_{2}O_{5}(g) \rightarrow 4NO_{2}(g) + O_{2}(g) \]

We need to analyze the four statements provided in the question and determine which ones are correct.

### Step 1: Analyze the first statement
**Statement 1:** The concentration of the reactant decreases exponentially with time.

**Solution:**
For a first-order reaction, the concentration of the reactant \( [A] \) at time \( t \) can be expressed as:
\[ [A] = [A_0] e^{-kt} \]
where \( [A_0] \) is the initial concentration, \( k \) is the rate constant, and \( t \) is time. As time increases, \( e^{-kt} \) decreases, indicating that the concentration of the reactant decreases exponentially with time.

**Conclusion:** This statement is **true**.

### Step 2: Analyze the second statement
**Statement 2:** The half-life of the reaction decreases with an increase in temperature.

**Solution:**
The half-life \( t_{1/2} \) for a first-order reaction is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
As temperature increases, the rate constant \( k \) increases (according to the Arrhenius equation). Since \( t_{1/2} \) is inversely proportional to \( k \), an increase in temperature leads to a decrease in half-life.

**Conclusion:** This statement is **true**.

### Step 3: Analyze the third statement
**Statement 3:** The half-life of the reaction depends on the initial concentration of the reactant.

**Solution:**
From the half-life formula \( t_{1/2} = \frac{0.693}{k} \), we see that the half-life for a first-order reaction does not depend on the initial concentration \( [A_0] \). It only depends on the rate constant \( k \).

**Conclusion:** This statement is **false**.

### Step 4: Analyze the fourth statement
**Statement 4:** The reaction proceeds to 99.6% completion in 8 half-life time duration when \( t_{1/2} = 8 \) hours.

**Solution:**
The percentage of reactant remaining after \( n \) half-lives is given by:
\[ \text{Percentage remaining} = 100 \times \left(\frac{1}{2}\right)^n \]
For \( n = 8 \):
\[ \text{Percentage remaining} = 100 \times \left(\frac{1}{2}\right)^8 = 100 \times \frac{1}{256} \approx 0.3906\% \]
Thus, the percentage of reactant that has reacted is:
\[ 100\% - 0.3906\% \approx 99.6094\% \]
This indicates that the reaction indeed proceeds to approximately 99.6% completion.

**Conclusion:** This statement is **true**.

### Final Conclusion:
The correct statements are:
- Statement 1: True
- Statement 2: True
- Statement 3: False
- Statement 4: True

Thus, the correct statements are A, B, and D.

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