Question

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 what will be the value of ${n_f} - {n_i}$ ?
(A) $- 2.5 \times {10^{25}}$
(B) $- 1.61 \times {10^{23}}$
(C) $1.38 \times {10^{23}}$
(D) $2.5 \times {10^{25}}$

Answer 1

Hint
The ideal gas law is helpful in telling us about various measurable properties of an ideal gas or a gas that almost behaves ideally. On rearranging the equation to get a constant term on one side, other properties of the gas can be found from provided data.
Formula used: $PV = nRT$ , where $P$ is the pressure of the gas, $V$ is the volume, $T$ is the temperature, $R$ is the ideal gas constant and $n$ is the amount of substance.
$\Rightarrow T{\text{K}} = T{}^ \circ {\text{C}} + 273$ where $T{}^ \circ {\text{C}}$ is temperature in degree Celsius and $T{\text{K}}$ is temperature in Kelvin.

Complete step by step answer
In this question, we have room that increases in temperature. We are required to find the increase in substance in this volume when pressure and volume remains constant. We are provided with the following data:
Volume of the room is $V = 30{m^3}$ .
Temperature before heating ${T_1} = {17^0 }C = 17 + 273K = 290K$ . [We need to convert to Kelvins]
Temperature after heating ${T_2} = {27^0 }C = 27 + 273K = 300K$ .
Pressure is $P = {10^5}$ Pa
Ideal gas constant is $R = 8.314J \cdot mo{l^{ - 1}} \cdot {K^{ - 1}}$
We know that the ideal gas law is given as:
$\Rightarrow PV = nRT$
Rewriting the ideal gas equation, we can find the number of molecules as:
$\Rightarrow n = \dfrac{{PV}}{{RT}}$
Now, for finding the difference, we use the above equation as:
$\Rightarrow {n_f} - {n_i} = \dfrac{{PV}}{R}\left( {\dfrac{1}{{{T_2}}} - \dfrac{1}{{{T_1}}}} \right)$ [ $P$ , $V$ and $R$ come outside as they are constant]
Now, substituting the values we get,
$\Rightarrow {n_f} - {n_i} = \dfrac{{{{10}^5} \times 30}}{{8.314}}\left( {\dfrac{1}{{300}} - \dfrac{1}{{290}}} \right)$
Taking the LCM and solving further we get,
$\Rightarrow {n_f} - {n_i} = \dfrac{{{{10}^5} \times 30}}{{8.314}}\left( {\dfrac{{29 - 30}}{{30 \times 29 \times 10}}} \right)$
$\Rightarrow {n_f} - {n_i} = - 41.47$
As this is the difference in the amount of substance, to find the number of molecules, we multiply it by the Avogadro number.
$\Rightarrow - 41.47 \times 6.022 \times {10^{23}} = - 2.497 \times {10^{25}} \approx - 2.5 \times {10^{25}}$
Hence, the difference in the number of molecules is $- 2.5 \times {10^{25}}$ .
Therefore, the correct option is (A).

Note
The negative sign implies that the number of molecules decreased as the temperature increased. The Avogadro’s constant is an important point to remember while solving these kinds of questions because the gas law has the parameter of amount of substance, and not the molecules.