The temperature of an open room of volume `30 m^(3)` increases from `17^(@)C to 27^(@)C` due to sunshine. The atmospheric pressure in the room remains `1 xx 10^(5) Pa`. If `n_(i) and n_(f)` are the number of molecules in the room before and after heating then `n_(f) - n_(i)` will be

Options

A. `2.5 xx 10^25`
B. `-2.5 xx 10^25`
C. `-1.6 xx 10^23`
D. `1.38 xx 10^23`

Answer

To solve the problem, we will use the ideal gas law and the concept of the number of molecules in a gas at different temperatures.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Volume of the room, \( V = 30 \, m^3 \)
- Initial temperature, \( T_i = 17^\circ C = 290 \, K \)
- Final temperature, \( T_f = 27^\circ C = 300 \, K \)
- Atmospheric pressure, \( P = 1 \times 10^5 \, Pa \)
- Universal gas constant, \( R = 8.31 \, J/(mol \cdot K) \)
- Avogadro's number, \( N_A = 6.02 \times 10^{23} \, molecules/mol \)

2. **Calculate the Initial Number of Moles (\( N_i \)):**
Using the ideal gas equation:
\[ N_i = \frac{PV}{RT_i} \]
Substituting the values:
\[ N_i = \frac{(1 \times 10^5 \, Pa)(30 \, m^3)}{(8.31 \, J/(mol \cdot K))(290 \, K)} \]
\[ N_i = \frac{3 \times 10^6}{2403.9} \approx 1240.5 \, mol \]

3. **Calculate the Final Number of Moles (\( N_f \)):**
Similarly, for the final state:
\[ N_f = \frac{PV}{RT_f} \]
Substituting the values:
\[ N_f = \frac{(1 \times 10^5 \, Pa)(30 \, m^3)}{(8.31 \, J/(mol \cdot K))(300 \, K)} \]
\[ N_f = \frac{3 \times 10^6}{2493} \approx 1203.5 \, mol \]

4. **Calculate the Change in Number of Moles (\( N_f - N_i \)):**
\[ N_f - N_i = 1203.5 - 1240.5 = -37 \, mol \]

5. **Convert the Change in Moles to Change in Number of Molecules:**
\[ n_f - n_i = (N_f - N_i) \times N_A \]
\[ n_f - n_i = -37 \times 6.02 \times 10^{23} \approx -2.23 \times 10^{25} \, molecules \]

6. **Final Result:**
The change in the number of molecules in the room due to heating is:
\[ n_f - n_i \approx -2.5 \times 10^{25} \, molecules \]

### Conclusion:
The answer is \( n_f - n_i = -2.5 \times 10^{25} \), which corresponds to option 2.