## 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 *P _{2}* jto the standard pressure

*P*j= 101'325 Paj, set

_{o}*V*jto

_{2}*n*jtimes the standard molar volume of

*V*j=j 0.022'414 m

_{o}^{3}/mol jand set

*T*jto the standard temperature

_{2}*T*j= j273.150 K .

_{o}We denote, as usual, the amount of substance in moles with j

*n*=

*N*/

*N*. Reducing the constants on the right side we obtain

_{A}This are the *definitions* of the universal gas constant *R* jand the Boltzmann constant *k* !

*R*j 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*. But we already know how to transform the left side of this equation for a fast-moving observer:

_{o}*P' · V'* n= *P · V *· √

Since the particle number *N*j, 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.