31. Boltzmann constant

The equation \((3-11)\) includes the ratio of the universal gas constant \(R\) to the Avogadro number \(N_A\). This relation is identical for all substances and is named Boltzmann constant in honor of Austrian scientist Ludwig Boltzmann, one of founders of the molecular-kinetic theory.

Boltzmann constant is equal to

\( k ~= ~\frac{R}{N_A} ~= ~1.38 \times 10^{-23} ~J/K \)

Equation \((3-11)\) can now be written in this way

\(\overline{E} ~= ~\frac{3}{2}kT \)(3-12)

Inserting \((3-12)\) in \((3-8)\), we find the equation for gas pressure through the concentration of molecules and temperature

\( p {~=~} nkT \)

From this, in particular, it follows that at the same pressures and temperatures the concentration of molecules is the same for all gases.
At \({~}p {~=~} 1 {~atm~}\) and \({~}T {~=~} 273^0K {~}(0^0C)\), \({~}n {~=~} 2.69 \times 10^{25} {~m^3~} (per{~}cubic{~}metre)\). This number is called the Loschmidt constant or Loschmidt's number.

Historically, temperature was introduced for the first time as a thermodynamic value and the unit of measurement was set to a degree (see \(\S3\)). After establishing the relationship between temperature and the average kinetic energy of molecules, it became clear that temperature can be defined as the average kinetic energy of molecules and measured in joules, i.e. instead of the value of \(T\), enter the value of \(T^{\star}\) so that

\(T^{\star} {~=~} \frac{m\overline\upsilon{^2}}{2} \)

Defined temperature is related to the temperature measured in degrees

\(T^{\star} {~=~} \frac{3}{2}kT \)

Therefore, Boltzmann constant can be considered as a value linking temperature in energy units with temperature expressed in degrees. However, temperature measurement in degrees is still generally accepted.