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 \( 2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g) \), we will analyze each statement provided in the question step by step.

### Step-by-Step Solution:

1. **Statement 1: The concentration of the reactant decreases exponentially with time.**
- For a first-order reaction, the concentration of the reactant decreases according to the equation:
\[ [A] = [A_0] e^{-kt} \]
where \([A_0]\) is the initial concentration, \([A]\) is the concentration at time \(t\), \(k\) is the rate constant, and \(t\) is time. This indicates that the concentration decreases exponentially over time.
- **Conclusion**: This statement is **true**.

2. **Statement 2: The reaction proceeds to 99.6% completion in eight half-life durations.**
- The relationship between the completion of a reaction and half-lives can be expressed as:
\[ \text{Completion} = 1 - \frac{1}{2^n} \]
where \(n\) is the number of half-lives. For \(n = 8\):
\[ \text{Completion} = 1 - \frac{1}{2^8} = 1 - \frac{1}{256} \approx 0.996 \text{ or } 99.6\% \]
- **Conclusion**: This statement is **true**.

3. **Statement 3: The half-life of the reaction depends on the initial concentration of the reactant.**
- For a first-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
This indicates that the half-life is independent of the initial concentration \([A_0]\). Therefore, this statement is incorrect.
- **Conclusion**: This statement is **false**.

4. **Statement 4: The half-life of the reaction decreases with increasing temperature.**
- The half-life for a first-order reaction is inversely proportional to the rate constant \(k\):
\[ t_{1/2} = \frac{0.693}{k} \]
According to the Arrhenius equation, as temperature increases, the rate constant \(k\) increases, leading to a decrease in half-life \(t_{1/2}\).
- **Conclusion**: This statement is **true**.

### Summary of Statements:
- Statement 1: True
- Statement 2: True
- Statement 3: False
- Statement 4: True

### Final Answer:
The correct statements are **1, 2, and 4**.

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