The temperature of an open room of volume $$30\,{m^3}$$  increases from $${17^ \circ }C$$  to $${27^ \circ }C$$  due to sunshine. The atmospheric pressure in the room remains $$1 \times {10^5}\,Pa.$$   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:

Solution :
Given : Temperature $${T_i} = 17 + 273 = 290\,K$$
Temperature $${T_f} = 27 + 273 = 300\,K$$
Atmospheric pressure, $${P_0} = 1 \times {10^5}\,Pa$$
Volume of room, $${V_0} = 30\,{m^3}$$
Difference in number of molecules, $${N_f} - {N_i} = ?$$
The number of molecules
$$\eqalign{ & \Rightarrow \,\,N = \frac{{PV}}{{RT}}\left( {{N_0}} \right) \cr & \therefore \,\,{N_f} - {N_i} = \frac{{{P_0}{V_0}}}{R}\left( {\frac{1}{{{T_f}}} - \frac{1}{{{T_i}}}} \right){N_0} \cr & = \frac{{1 \times {{10}^5} \times 30}}{{8.314}} \times 6.023 \times {10^{23}}\left( {\frac{1}{{300}} - \frac{1}{{290}}} \right) \cr & = - 2.5 \times {10^{25}} \cr} $$