Question

Short answer

The number of air molecules is \(1.119 \times {10^{27}}\;{\rm{molecules}}\), and the number of moles is \(1858.8\;{\rm{moles}}\).

Step-by-step solution

Given data

The length of the room is\(L = 6\;{\rm{m}}\).

The width of the room is\(w = 3\;{\rm{m}}\).

The height of the room is\(h = 2.5\;{\rm{m}}\).

The temperature is \(T = 22^\circ {\rm{C}}\).

Understanding the number of moles

In order to calculate the number of moles in a certain gas, divide the value of the number of molecules by Avogadro’s number.

Calculation of the number of molecules

The relation from the ideal gas law can be written as:

\(\begin{aligned}{c}PV &= nkT\\n &= \frac{{PV}}{{kT}}\end{aligned}\)

Here, Pis the atmospheric pressure, Vis the volume, kis the Boltzmann constant, and n is the number of molecules.

On plugging the values in the above relation, you get:

\(\begin{aligned}{l}n &= \left( {\frac{{\left( {1\;{\rm{atm}} \times \frac{{1.013 \times {{10}^5}\;{\rm{Pa}}}}{{1\;{\rm{atm}}}}} \right)\left( {6\;{\rm{m}} \times 3\;{\rm{m}} \times 2.5\;{\rm{m}}} \right)}}{{\left( {1.38 \times {{10}^{ - 23}}\;{\rm{J/K}}} \right)\left( {22^\circ {\rm{C}} + {\rm{273}}} \right)\;{\rm{K}}}}} \right)\\n &= 1.119 \times {10^{27}}\;{\rm{molecules}}\end{aligned}\)

Thus, \(n = 1.119 \times {10^{27}}\;{\rm{molecules}}\) is the number of molecules.

Calculation of the number of moles

The relation to find the number of moles can be written as:

\(N = \left( {n \times \frac{{1\,{\rm{mole}}}}{{{N_{\rm{A}}}}}} \right)\)

Here,\({N_{\rm{A}}}\)is Avogadro’s number.

On plugging the values in the above relation, you get:

\(\begin{aligned}{l}N &= \left( {1.119 \times {{10}^{27}}\;{\rm{molecules}} \times \frac{{1\,{\rm{mole}}}}{{6.02 \times {{10}^{23}}\;{\rm{molecules}}}}} \right)\\N &= 1858.8\;{\rm{moles}}\end{aligned}\)

Thus, \(N = 1858.8\;{\rm{moles}}\) is the required number of moles.