Question

Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity \[25.0{ }{{m}^{3}}~\] at a temperature of \[27{{~}^{\circ }}C~\]and \[1{ }atm\] pressure.

Answer 1

Hint: The total number of air molecules of air can be calculated by using the ideal gas equation, as it establishes a relationship between the temperature at which the molecules are present, and the pressure as well as the volume occupied by the gas.
The volume occupied by the gas will be the same as the total volume of the room in this case, as air is a homogenous mixture of several gases and it tends to flow. So it will occupy the total space which is occupied by the room.

Complete step by step answer:
The ideal gas law, which is also termed as the general gas equation, is the equation of state of a ideal gas which is hypothetical in nature. It is a general approximation of the behaviour of various gases under several given conditions, although it has several limitations.
Let us write the value of each variable which is given in the question, along with the definitions,
Volume of the room, which is the total volume occupied by the room, is given as
\[V=25.0{ }{{m}^{3}}~\]
Then we write the given temperature of the room at which we will estimate the number of air molecules,
\[T=270{}^\circ C=300K\].
Here we convert the value of temperature from Celsius to kelvin, as it is the standard unit of temperature. Now we will consider the value of pressure in the room, which is given as
\[P=1atm=1\times 1.013\times {{10}^{5}}Pa\]
Now we know that the ideal gas equation which relates the values of pressure ($P$), Volume ($V$), as well as absolute temperature ($T$) can be mathematically represented by the following equation:
\[PV={{k}_{B}}NT\]
Where,
\[{{k}_{B}}\]​ represents the value of Boltzmann constant \[=1.38\times {{10}^{-23}}{{m}^{2}}kg{{s}^{-2}}{{K}^{-1}}\]
And \[N\] is the number of air molecules which are present in the room. Now we will simple write all the values of variables given and known to us, on one side of the equation, and the number of molecules which is unknown to us on the other side of the equation.
\[N=\dfrac{PV}{{{k}_{B}}T}\]
Now we will substitute the values of the pressure, volume, temperature and the Boltzmann constant which is known to us.
\[~=\dfrac{1.013\times {{10}^{5}}\times 25}{1.38\times {{10}^{-23}}\times 300}\]
Now after we calculate the following equation we get,
\[=6.11\times {{10}^{26}}~molecules\]

So the total number of air molecules in the given room is \[6.11\times {{10}^{26}}~molecules\]

Note:
The ideal gas equation is applicable only for the gases when they are kept in certain temperature and pressure, where they show ideal behaviour. No gas shows ideal behaviour in all temperature and pressure, the behaviour will deviate with change in the physical conditions.