The heating of ${\mathrm{NH}}_4{\mathrm{NO}}_2$ at $60-70°C$ and ${\mathrm{NH}}_4{\mathrm{NO}}_3$ at $200-250°C$ is associated with the formation of nitrogen containing compounds $X$ and $Y$, respectively.

$X$ and $Y$, respectively, are

${\mathrm{}}_{}$

The correct option is:
$N_2$ and $N_{2}O$

Step-by-step Explanation:

1. Thermal decomposition of Ammonium Nitrite (${\mathrm{NH}}_4{\mathrm{NO}}_2$):
When ammonium nitrite (${\mathrm{NH}}_4{\mathrm{NO}}_2$) is heated to a temperature of about $60-70°C$, it undergoes thermal decomposition to produce nitrogen gas ($N_2$) and water vapor. The balanced chemical equation for this reaction is:
${\mathrm{NH}}_4{\mathrm{NO}}_2\to N_2+2H_{2}O$
Therefore, the nitrogen-containing compound $X$ is $N_2$ (nitrogen gas).

2. Thermal decomposition of Ammonium Nitrate (${\mathrm{NH}}_4{\mathrm{NO}}_3$):
When ammonium nitrate (${\mathrm{NH}}_4{\mathrm{NO}}_3$) is heated to a temperature range of $200-250°C$, it decomposes to yield nitrous oxide ($N_{2}O$, commonly known as laughing gas) and water vapor. The balanced chemical equation is:
${\mathrm{NH}}_4{\mathrm{NO}}_3\to N_{2}O+2H_{2}O$
Thus, the nitrogen-containing compound $Y$ is $N_{2}O$ (nitrous oxide).

Conclusion:
The compounds $X$ and $Y$ are $N_2$ and $N_{2}O$, respectively.