Decomposition of `NH_(4)NO_(2)(aq` into `N_(2)(g)` and `2H_(2)O(l)` is first order reaction.

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Answer

To analyze the decomposition of NH₄NO₂ into N₂ and H₂O as a first-order reaction, we can follow these steps:

### Step 1: Write the Reaction
The decomposition reaction can be written as:
\[ \text{NH}_4\text{NO}_2 (aq) \rightarrow \text{N}_2 (g) + 2\text{H}_2\text{O} (l) \]

### Step 2: Identify the Order of the Reaction
The problem states that this reaction is a first-order reaction. In first-order reactions, the rate of reaction is directly proportional to the concentration of one reactant.

### Step 3: Write the Integrated Rate Law for a First-Order Reaction
For a first-order reaction, the integrated rate law is given by:
\[ \ln[A] = \ln[A_0] - kt \]
Where:
- \([A]\) is the concentration of the reactant at time \(t\).
- \([A_0]\) is the initial concentration of the reactant.
- \(k\) is the rate constant.
- \(t\) is the time.

### Step 4: Rearranging the Equation
We can rearrange the equation to express it in the form of \(y = mx + b\):
\[ \ln[A] = -kt + \ln[A_0] \]
This indicates that if we plot \(\ln[A]\) (y-axis) against time \(t\) (x-axis), we will get a straight line with a slope of \(-k\) and a y-intercept of \(\ln[A_0]\).

### Step 5: Graphical Representation
In a first-order reaction, the graph of \(\ln[A]\) versus time will be a straight line that slopes downwards, indicating that as time increases, the concentration of the reactant decreases exponentially.

### Step 6: Conclusion
Based on the analysis, we can conclude that the correct graph representing the first-order decomposition of NH₄NO₂ is the one that shows a downward slope when plotting \(\ln[A]\) against time.