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:
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