Question

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

Short answer

The Loschmidt number is$2.7\times {10}^{25}{\mathrm{m}}^{-3}.$

Step-by-step solution

Concept Introduction (Part a)

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.

Explanation (Part a)

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):

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

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

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

Final Answer

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