33   The Transformation of Molar Heat Capacities

 
In a isochoric heating process we have

From  ∆Q'  =  ∆Q · √  and  ∆T'  =  ∆T  we get for the transformation of  Cv

However, following 24 and 26 , the total energy required is more than dQ : Every increase of total energy goes along with an increase of kinetic energy.
So the total amount of energy required for this heating is

The portion

specifies the increase in kinetic energy ( cf. section 26 ).

The situation is completely analogous in the case of an isobaric heating process. The Definition

implies the transformation formula

Here again, the total energy required is more than dQ , namely dQ / √ , to provide the heat dQ' = dQ · √  and the kinetic energy.

Obviously, the heat capacity ratio  kappa  is relativistically invariant.

For a non-relativistic ideal gas ( section 30 ) we further have the invariant relationship

So the value of Cv  is given:  Cv  = 1.5 · R  ≈ 12.472 J/(K · mol). From the ideal gas law we derive in addition  Cp  =  2.5 · R  ≈  20.786 J/(K · mol). These numbers are in very good conformance to those listed for the noble gases. For the heat capacity ratio  Cp / Cv  of an ideal gas we get the number  5/3 . Looking up the values listed for the noble gases you will find numbers very close to 1.67.