T27   The Transformation of Heat in the General Case

 
Now we drop the condition of 'constant volume'. Heat is added and work is performed, compressing the volume of the gas. We start from the following relations:

• ΔE  =  ΔQ  +  ΔW  =  ΔQ  –  P · ΔV 
  
  
• ΔE'  =  ΔE  / √
  
  
• ΔE'  =  ΔQ'  +  ΔW'   =  ΔQ'  +  v · Δp  –  P' · ΔV'
  
  
• v   is constant
  
  
• Δp  =  ΔE' · v / c²
  
  

We repeat the calculation of section 26 :

ΔQ'   =  ΔE'  –  v · Δp  +  P' · ΔV'   =  ΔE'  –  ΔE' · v² / c²   +  P' · ΔV'   =  ΔE' · ( 1 – v² / c²  )  +  P' · ΔV'  =
         =  ( ΔE / √ ) · ( 1 – v² / c²  )  +  P' · ΔV'   =   ΔE · √  +  P · ΔV · √  =  ( ΔE  +  P · ΔV  ) · √   =   ΔQ · √

Heat energy is transformed by multiplication by the root term.

Q'  =  Q · √

In section 30 we will reach this result on a completely different path.

Transformation of ∆Q in a Isochoric Process The Transformations of U and H