Let's analyze each compound:
\( \text{Ba(N}_3\text{)}_2 (s) \rightarrow \text{Ba} (s) + 3\text{N}_2 (g) \)
\( \text{NH}_4\text{NO}_2 (s) \rightarrow \text{N}_2 (g) + 2\text{H}_2\text{O} (g) \)
\( \text{NH}_4\text{NO}_3 (s) \rightarrow \text{N}_2\text{O} (g) + 2\text{H}_2\text{O} (g) \)
\( (\text{NH}_4)_2 \text{Cr}_2\text{O}_7 (s) \rightarrow \text{Cr}_2\text{O}_3 (s) + \text{N}_2 (g) + 4\text{H}_2\text{O} (g) \)
Conclusion: The correct answer is (C) \( \text{NH}_4\text{NO}_3 \) because its primary decomposition reaction produces dinitrogen monoxide (\( \text{N}_2\text{O} \)), **not** dinitrogen (\( \text{N}_2 \)).
The reaction for the compounds listed is as follows:
1. Ba(N\(_3\))\(_2\): When heated, barium azide decomposes to form barium and nitrogen gas. The decomposition reaction is: \[ \text{Ba(N}_3\text{)}_2 \xrightarrow{\text{heat}} \text{Ba} + \text{N}_2 \] So, Ba(N\(_3\))\(_2\) does produce nitrogen gas, but it does not release dinitrogen (N\(_2\)) directly in the form of a stable dimer. Hence, it is not considered for this particular reaction.
2. NH\(_4\)NO\(_2\): Ammonium nitrite decomposes on heating to form nitrogen gas and water: \[ \text{NH}_4\text{NO}_2 \xrightarrow{\text{heat}} \text{N}_2 + 2\text{H}_2\text{O} \]
3. NH\(_4\)NO\(_3\): Ammonium nitrate decomposes upon heating to release nitrogen gas, water, and oxygen: \[ \text{NH}_4\text{NO}_3 \xrightarrow{\text{heat}} \text{N}_2 + 2\text{H}_2\text{O} + \text{O}_2 \]
4. (NH\(_4\))\(_2\) Cr\(_2\)O\(_7\): Ammonium dichromate decomposes on heating, releasing nitrogen gas: \[ \text{(NH}_4\text{)}_2\text{Cr}_2\text{O}_7 \xrightarrow{\text{heat}} \text{N}_2 + \text{Cr}_2\text{O}_3 + 4\text{H}_2\text{O} \]
Thus, the only compound listed that does not produce nitrogen gas directly on heating is Ba(N\(_3\))\(_2\).