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

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the decomposition of ammonium nitrite (NH₄NO₂) into nitrogen gas (N₂) and water (H₂O), we need to analyze the reaction and its characteristics as a first-order reaction. Here’s a step-by-step solution:

### Step 1: Write the Reaction
The decomposition reaction can be represented 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 is a first-order reaction. In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant.

### Step 3: Write the Rate Law
For a first-order reaction, the rate law can be expressed as:
\[ \text{Rate} = -\frac{d[\text{NH}_4\text{NO}_2]}{dt} = k[\text{NH}_4\text{NO}_2] \]
where \( k \) is the rate constant.

### Step 4: Integrate the Rate Law
To find the relationship between concentration and time, we need to integrate the rate law. Rearranging gives:
\[ \frac{d[\text{NH}_4\text{NO}_2]}{[\text{NH}_4\text{NO}_2]} = -k dt \]

Integrating both sides:
\[ \int \frac{d[\text{NH}_4\text{NO}_2]}{[\text{NH}_4\text{NO}_2]} = -k \int dt \]

This results in:
\[ \ln[\text{NH}_4\text{NO}_2] = -kt + C \]

Where \( C \) is the integration constant. At \( t = 0 \), let \( [\text{NH}_4\text{NO}_2] = [\text{NH}_4\text{NO}_2]_0 \):
\[ \ln[\text{NH}_4\text{NO}_2]_0 = C \]

Substituting back gives:
\[ \ln[\text{NH}_4\text{NO}_2] = -kt + \ln[\text{NH}_4\text{NO}_2]_0 \]

### Step 5: Exponential Form
Exponentiating both sides to eliminate the natural logarithm yields:
\[ [\text{NH}_4\text{NO}_2] = [\text{NH}_4\text{NO}_2]_0 e^{-kt} \]

### Step 6: Graphical Representation
The concentration of NH₄NO₂ decreases exponentially over time. The graph of concentration vs. time for a first-order reaction will show a downward curve, indicating that as time increases, the concentration of NH₄NO₂ decreases.

### Step 7: Identify the Correct Graph
Based on the characteristics of a first-order reaction, the graph will be an exponential decay curve. Among the options provided, the correct graph will be the one that shows a decreasing exponential trend.

### Conclusion
The correct answer is the graph that represents an exponential decrease in concentration over time.

---