## 25 The First Law of Thermodynamics

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

Δ*E'* = Δ*Q'* + Δ*W' *

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

Δ*E'* = Δ*Q'* + Δ*W'* = Δ*H'* + Δ*E _{kin}'*

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 Δ*W'* ? Pauli gives us [3, p.694, formula 365] the following answer:

Δ*W'* = *v* · Δ*p* – *P'* · Δ*V'*

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

Δ*E'* = red line + blue line = *c* · Δ*p* · sin(*φ*) – *P* · Δ*V* · cos(*φ*) =

= *c* · Δ*p* · v / c – *P* · Δ*V* · √ = *v* · Δ*p* – *P'* · Δ*V'*

The first term *v* · Δ*p* contributes to the kinetic part of *E'* , while – *P'* · Δ*V'* jgives us the increase of enthalpy *H'* .

For a fast moving gas we get the following equation

Δ*E'* = Δ*Q'* + *v* · Δ*p* – *P'* · Δ*V'*

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

Δ*E* = Δ*Q* – *P* · Δ*V *= Δ*H*