Benene diaxonium chloride in aqeous solution decomposes as : `C_6H_5-N=N^(+)Cl_(aq)^(-) +H_2O_(aq)rarrC_6H_5OH_(aq)+N_2(g)+HCl_(aq)` The reaction follows first order kinetics. If `P_t` is the pressure of `N_2` at constant volume and temperature corresponding to different intervals of time `t and p_f` that after completion of the reaction, then which of the following graphs conforms to the kinetics of the reaction ?

Options

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Answer

To solve the problem, we need to analyze the decomposition of benzene diazonium chloride and understand how it relates to first-order kinetics. The reaction is given as:

\[ C_6H_5-N=N^+Cl^- + H_2O \rightarrow C_6H_5OH + N_2 + HCl \]

### Step-by-Step Solution:

1. **Understanding the Reaction**:
- The reaction involves the decomposition of benzene diazonium chloride in aqueous solution, producing phenol, nitrogen gas, and hydrochloric acid.
- The key point is that this reaction follows first-order kinetics, which means the rate of reaction is proportional to the concentration of one reactant.

2. **Identifying Variables**:
- Let \( P_t \) be the pressure of nitrogen gas (\( N_2 \)) at time \( t \).
- Let \( P_f \) be the pressure of nitrogen gas after the completion of the reaction.

3. **Relation of Pressure to Concentration**:
- In a first-order reaction, the concentration of the reactant decreases exponentially over time. This can be expressed as:
\[ \ln \left( \frac{P_f - P_t}{P_f} \right) = -kt \]
- Here, \( P_f \) is the final pressure of nitrogen after the reaction is complete, and \( P_t \) is the pressure at time \( t \).

4. **Graphing the Data**:
- For first-order kinetics, if we plot \( \ln(P_f - P_t) \) versus time \( t \), we should get a straight line.
- The slope of this line will be equal to \(-k\), where \( k \) is the rate constant.

5. **Choosing the Correct Graph**:
- Among the provided graphs, we need to look for one that shows a linear relationship when plotting \( \ln(P_f - P_t) \) against time \( t \).
- The graph should have a negative slope, indicating that as time increases, the logarithm of the pressure difference decreases, consistent with first-order kinetics.

### Conclusion:
The correct graph will be the one that shows a straight line when plotting \( \ln(P_f - P_t) \) against time \( t \).