## 25   The First Law of Thermodynamics

We suppose that conservation of energy holds for the 'fast moving observer':

ΔE'  =  ΔQ'  +  ΔW'

In the system at rest this reads as "the total increase of energy is the sum of the added heat energy and the performed work". The fast moving observer has a slightly different reading: "The total increase of energy is the sum of the increase in heat and the increase in mechanical and kinetic energy". He may extend the above equation to

ΔE'  =  ΔQ'  +  ΔW'   =  ΔH'  +  ΔEkin'

In section 24 we have made clear that there is no addition of energy to a fast system without an increase of kinetic energy.

But how to calculate  ΔW'  ? Pauli gives us [3, p.694, formula 365] the following answer:

ΔW'   =  v · Δp  –  P' · ΔV'

In STR,  v  is constant. However, this does not mean the first term to disappear. Adding energy inevitably increases the momentum p . Let us check Pauli's formula looking at the following Epstein diagram:

We have

ΔE'  =  red line  +  blue line  =  c · Δp · sin(φ)  –  P · ΔV · cos(φ)  =

=  c · Δp · v / c   –  P · ΔV · √    =   v · Δp   –   P' · ΔV'

The first term  v · Δp  contributes to the kinetic part of E' , while   – P' · ΔV'  jgives us the increase of enthalpy  H'  .

For a fast moving gas we get the following equation

ΔE'  =  ΔQ'  +  v · Δp  –  P' · ΔV'

In the system at rest this reduces to the well-known expression

ΔE  =  ΔQ  –  P · ΔV   =  ΔH