8 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
Po j= 101'325 Paj, set V2 jto n jtimes the standard molar volume of Vo j=j 0.022'414 m3/mol jand set T2 jto the standard temperature To j= j273.150 K .
We denote, as usual, the amount of substance in moles with jn = N / NA . 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 Po · Vo / To . 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.