## 36 Summary for the Sections 1 to 35

If one defines temperature as a fundamental unit of physics, as in section **15** of this treatise, then the following values are relativistically invariant:

- temperature
*T*n( and temperature changes ) - pressure
*P*n( and pressure changes ) - particle number
*N*j and amount of substance*n*j=j*N*/*N*_{A} - thermodynamic efficiency
_{} - heat capacity ratio kappa =
*C*/_{P}*C*_{V}

The following values are transformed by multiplying by the root factor:

- volume
*V*n( and volume changes ) - the Boltzmann constant
*k*nand the universal gas constant*R* - average velocity of the particles relative to the center of mass
- heat quantity
*Q*n( and their changes ) - internal energy
*U*n( and their changes ) - volume work
*P · V*n( and their changes ) - enthalpy
*H*n( and their changes ) - entropy
*S*( and their changes ) - molar heat capacity
*C*and_{V}*C*_{P} - molrelated melting- and evaporation energies

The following values are transformed by dividing by the root factor:

- total energy
*E*jof the gas - particle density

There is no simple transformation in the sense of section **2** for the following process variable:

- peformed work j
*W*

Taking into account these transformations, **all** the essential relationships of thermodynamics persist !

The following relationships are invariant:

- the first law of thermodynamics, thus mmmjmm ∆
*E*j=*j*∆*Q**j*+ ∆*W* - the second law of thermodynamics, thus mnni ∆S j≥ j0 in a closed system
- for ideal gases mmmm
*C*j–_{P}*C*j=j_{V}*R* - for ideal gases mmm m
*P · V*j=*n · R · T*j=*jN · k · T* - for the entropy
*mmmmmm dS · T*j=j*dQ mm*and*mm S*j=*j k · ln(Ω)*

For non-relativistic ideal gases we further have the invariant relationships

It should be noted that we have **not** assumed the validity of these relationships for a fast moving observer in order to derive the transformation rules ! We only assumed that such a transformation exists for *T* , and in addition we have chosen a definition of temperature within the limiting conditions of *k' · T'* n= *k · T* · √ so that temperature is relativistically invariant. Most of the results were established even before this decision ( see section **33** ).

Planck, Einstein, Hasenöhrl and von Mosengeil made no errors (unlike many others) in the derivation of their results. They just decided that j*k' *= *k* should apply, and then they derived the other results in a logically correct way.

Avramov, however, made the better choice !