## 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 *C _{v}*

However, following **24 **and** 26** , the total energy required is *more than* d*Q* : 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 *C _{v}* is given:

*C*= 1.5 ·

_{v}*R*≈ 12.472 J/(K · mol). From the ideal gas law we derive in addition

*C*= 2.5 ·

_{p}*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

*C*/

_{p}*C*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.

_{v}