`(P)overset(Delta)(rarr)(Q)+(R )uarr+(S)uarr`
`(X)overset(Delta)(rarr)(Y)+(R )uarr+(S)uarr`
P and X are respectively

Options

A. `AgNO_3, LiNO_3`
B. `AgNO_3, Pb(NO_(3))_(2)`
C. `Hg_(2)(NO_(3))_(2),Ca(NO_3)_(2)`
D. `NaNO_(3), Zn(NO_3)_(2)`

Answer

To solve the problem, we need to identify the compounds P and X based on the information given about their thermal decomposition reactions.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
We know that both P and X decompose upon heating to produce gases. The products of both reactions include the same gases, which are NO2 and O2.

2. **Analyzing the Options**:
We have four pairs of compounds to consider:
- Silver Nitrate (AgNO3) and Lithium Nitrate (LiNO3)
- Silver Nitrate (AgNO3) and Lead Nitrate (Pb(NO3)2)
- Mercury Nitrate (Hg(NO3)2) and Calcium Nitrate (Ca(NO3)2)
- Sodium Nitrate (NaNO3) and Zinc Nitrate (Zn(NO3)2)

3. **Decomposition of Silver Nitrate**:
When Silver Nitrate is heated:
\[ 2 \text{AgNO}_3 \rightarrow 2 \text{Ag} + 2 \text{NO}_2 + \text{O}_2 \]
This reaction produces silver, nitrogen dioxide (NO2), and oxygen (O2).

4. **Decomposition of Lead Nitrate**:
When Lead Nitrate is heated:
\[ 2 \text{Pb(NO}_3)_2 \rightarrow 2 \text{PbO} + 4 \text{NO}_2 + \text{O}_2 \]
This reaction produces lead oxide, nitrogen dioxide (NO2), and oxygen (O2).

5. **Comparing Products**:
Both reactions produce NO2 and O2, but the first reaction produces silver, while the second produces lead oxide. Therefore, P could be Silver Nitrate and X could be Lead Nitrate.

6. **Decomposition of Lithium Nitrate**:
When Lithium Nitrate is heated:
\[ 2 \text{LiNO}_3 \rightarrow \text{Li}_2\text{O} + 2 \text{NO}_2 + \text{O}_2 \]
This reaction also produces NO2 and O2, but it produces lithium oxide instead of a metal.

7. **Decomposition of Mercury Nitrate**:
When Mercury Nitrate is heated:
\[ 2 \text{Hg(NO}_3)_2 \rightarrow 2 \text{HgO} + 4 \text{NO}_2 + \text{O}_2 \]
This reaction also produces NO2 and O2, but it produces mercury oxide.

8. **Decomposition of Calcium Nitrate**:
When Calcium Nitrate is heated:
\[ 2 \text{Ca(NO}_3)_2 \rightarrow 2 \text{CaO} + 4 \text{NO}_2 + \text{O}_2 \]
This reaction also produces NO2 and O2, but produces calcium oxide.

9. **Decomposition of Sodium Nitrate**:
When Sodium Nitrate is heated:
\[ 2 \text{NaNO}_3 \rightarrow 2 \text{NaNO}_2 + \text{O}_2 \]
This reaction produces only one gas (O2) and sodium nitrite.

10. **Conclusion**:
Based on the analysis, the pairs that produce the same gases (NO2 and O2) upon heating are:
- Silver Nitrate and Lead Nitrate
- Silver Nitrate and Lithium Nitrate
- Mercury Nitrate and Calcium Nitrate

Thus, the answer is that P and X are Silver Nitrate (AgNO3) and Lead Nitrate (Pb(NO3)2).