For the first order reaction given below select the set having correct statements `2N_(2)O_(5)(g) rarr 4NO_(2)(g) + O_(2)(g)`
(1) The concentration of the reactant decreases exponentially with time
(2) the reaction proceeds to 99.6 % completion in eight half life duration
(3) the half life of the reaction depends on the intial concentration of the reactant
(4) the half life of the reaction decrease with increasing temperature

Options

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

Answer

To solve the question regarding the first-order reaction \(2N_{2}O_{5}(g) \rightarrow 4NO_{2}(g) + O_{2}(g)\), we will analyze each statement provided and determine their correctness step by step.

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

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. This equation shows that the concentration decreases exponentially over time.

**Conclusion:** Statement 1 is **correct**.

### Step 2: Analyze the second statement
**Statement 2:** The reaction proceeds to 99.6% completion in eight half-life durations.

The fraction of reactant remaining after \( n \) half-lives is given by:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^n \]
For \( n = 8 \):
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^8 = \frac{1}{256} \approx 0.0039 \]
This means that approximately 99.6% of the reactant has reacted.

**Conclusion:** Statement 2 is **correct**.

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

For a first-order reaction, the half-life \( t_{1/2} \) is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
This shows that the half-life is independent of the initial concentration \( [A_0] \).

**Conclusion:** Statement 3 is **incorrect**.

### Step 4: Analyze the fourth statement
**Statement 4:** The half-life of the reaction decreases with increasing temperature.

As temperature increases, the rate constant \( k \) generally increases (according to the Arrhenius equation). Since the half-life is inversely proportional to \( k \):
\[ t_{1/2} \propto \frac{1}{k} \]
This means that as \( k \) increases with temperature, \( t_{1/2} \) decreases.

**Conclusion:** Statement 4 is **correct**.

### Final Conclusion
The correct statements are:
1. The concentration of the reactant decreases exponentially with time.
2. The reaction proceeds to 99.6% completion in eight half-life durations.
4. The half-life of the reaction decreases with increasing temperature.

Thus, the correct set of statements is **1, 2, and 4**.