The number density of an ideal gas at STP is called the Loschmidt number. Calculate the Loschmidt number.

As Loschmidt's number, the Loschmidt constant or Loschmidt's number refers to the number of ideal gas particles in a given volume at standard temperatures and pressures.

This is the number of atoms or molecules per cubic meter in a system, which indicates how tightly the molecules are bonded in the system.

It can be calculated by equation (18.2) using the form:

$numberdensity=\frac{N}{V}\to \left(1\right)$

An ideal gas law shows the relation between the four state variables, which are the volume $V$, pressure $p$, temperature $T$, and number of molecules, or moles $n$, in the container of the ideal gas.

Ideal gas law is as follows:

$pV=N{k}_{B}T$

localid="1648287708748" $\frac{N}{V}=\frac{p}{k_{\mathrm{B}}T}\to \left(2\right)$

$k_{\mathrm{B}}$is Boltzmann's constant and in $\mathrm{SI}$unit its value is,

$k_{\mathrm{B}}=1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$

At STP, the pressure is $p=1.013\times {10}^5\mathrm{Pa}$

Temperature is $T=273K$

In order to get number density or Loschmidt number, put it in the above equation (2):

$Loschmidtnumber=\frac{p}{k_{B}T}$

$=\frac{1.013\times {10}^5\mathrm{Pa}}{\left(1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}\right)(273\mathrm{K})}$

$=2.7\times {10}^{25}{\mathrm{m}}^{-3}$

Hence, the Loschmidt number$=2.7\times {10}^{25}{\mathrm{m}}^{-3}.$