Question

Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity \[\mathop {25.0m}\nolimits^2 \] at a temperature of \[27^\circ {\text{ }}C\] and 1 atm pressure.

Answer 1

Hint: The ideal gas is the gas that is hypothetical gas. This gas contains particles that are in random motion. No gas is perfectly ideal when we observe. These gases are not subjected to interparticle interaction. The ideal gas equation is given by \[PV{\text{ }} = {\text{ }}nRT\] ( where P is pressure , T is temperature , V is volume , n is number of moles and R is gas constant.)

Complete step by step answer:
We are provided here with
\[Pressure{\text{ }}\left( P \right){\text{ }} = {\text{ }}1{\text{ }}atm{\text{ }} = {\text{ }}1.013{\text{ }} \times {\text{ }}{10^5}Pascal\] (it is in standard form)
\[Temperature{\text{ }}\left( T \right){\text{ }} = {\text{ }}27^\circ C{\text{ }} = {\text{ }}27{\text{ }} + {\text{ }}273K{\text{ }} = {\text{ }}300K\]
\[Volume{\text{ }}\left( V \right){\text{ }} = {\text{ }}25.0{\text{ }}{m^3}\]
Now as we have to find out molecules of air ,
We will relate ideal gas equation according to number of molecules: -
\[PV{\text{ }} = {\text{ }}{K_B}NT\;\]Where KB = Boltzmann constant and its value is = \[1.38{\text{ }} \times {\text{ }}{10^{ - 23}}m^2kgs^ - 2K^ - 1.\]
N is number of molecules (that is unknown)
Now substituting values in ideal gas equation \[PV{\text{ }} = {\text{ }}{K_B}NT\;\]
\[1.013{\text{ }} \times {\text{ }}{10^5} \times {\text{ }}25{\text{ }} = \;1.38{\text{ }} \times {\text{ }}{10^{ - 23}}N{\text{ }} \times {\text{ }}300\]
\[N{\text{ }} = {\text{ }}6.11{\text{ }} \times {\text{ }}{10^{26.}}\]

Note:
When we are provided with number of moles or have to find we use ideal gas equation \[PV{\text{ }} = {\text{ }}nRT\] but when number of molecules are given or we have to find number of molecule we use ideal gas equation \[PV{\text{ }} = {\text{ }}{K_B}NT\;\]. An ideal gas follows gas law at all temperature and pressure.