T8   The Transformations of  R · T iand  k · T

 
In section 6 we pointed out that the ideal gas law

holds (with all values primed) for a fast moving observer too. This equation is also satisfied if we set  P₂ jto the standard pressure  P_o j= 101'325 Paj, set V₂ jto n jtimes the standard molar volume of V_o j=j 0.022'414 m³/mol and set T₂ jto the standard temperature T_o j= j273.150 K .
We denote, as usual, the amount of substance in moles with jn = N / N_A . Reducing the constants on the right side we obtain

This are the definitions of the universal gas constant  R jand the Boltzmann constant k !

Rj is the product of a pressure and a volume, divided by a temperature. So R jhas to transform accordingly: Following 5 pressure is invariant, hence R and k jcan only be invariant when temperatures transform just as volumes do.

Using these shortcuts the following applies to a certain quantity of a gas in all states of equilibrium:

P · V n=  n · R · T n=  N · k · T

Even in this notation, the gas equation must be form-invariant. R jis only the abbreviation for P_o · V_o / T_o .
But we already know how to transform the left side of this equation for a fast-moving observer:

P' · V' n=  P · V · √

Since the particle number Nj, as well as the amount of substance n jare invariant it follows for the transformation of k jand T jthat

k' · T' n=   k · T · √

The product of k jand T jas well as the product of R jand T jare transformed through multiplication with the root.

Definition of Temperature The Second Law