Question

Calculate the number of molecules/m³in an ideal gas at STP.

Short answer

There are \(2.69 \times {10^{25}}\;{\rm{molecules/}}{{\rm{m}}^3}\)in an ideal gas at STP.

Step-by-step solution

Step 1:Statement of the ideal gas law in terms of molecules 

The ideal gas law gives the relation between the pressure (P), volume (V), and temperature (T) of n moles of the ideal gas by the following expression:

\(PV = nRT\)

Here,R is the universal gas constant whose value is\(8.14\;{\rm{J/mol}} \cdot {\rm{K}}\).

In terms of the number of molecules (N) of the gas, the ideal gas law is written as

\(PV = \frac{N}{{{N_{\rm{A}}}}}RT\).

Here, \({N_{\rm{A}}}\)is the number of molecules in one mole of asubstance and is known as Avogadro’s number. Its value is\(6.02 \times {10^{23}}\).

Determination of the number of molecules in one cubic meter ofan ideal gas at STP

The volume of 1 mole of an ideal gas at STPis \(V = 22.4\;{\rm{L}} = 22.4 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{3}}}\).

The value of the Avogadro number is\({N_{\rm{A}}} = 6.02 \times {10^{23}}\).

Avogadro’s number is the number of molecules in volume \(V\;{{\rm{m}}^{\rm{3}}}\). Thus, the number of molecules in \(1\;{{\rm{m}}^{\rm{3}}}\) is

\(\begin{array}{c}\frac{{{N_{\rm{A}}}}}{V} = \frac{{6.02 \times {{10}^{23}}\;{\rm{molecules}}}}{{22.4 \times {{10}^{ - 3}}\;{{\rm{m}}^{\rm{3}}}}}\\ = 0.269 \times {10^{26}}\;{\rm{molecules/}}{{\rm{m}}^3}\\ = 2.69 \times {10^{25}}\;{\rm{molecules/}}{{\rm{m}}^3}.\end{array}\)

Thus, there are \(2.69 \times {10^{25}}\;{\rm{molecules/}}{{\rm{m}}^3}\)in an ideal gas at STP.