Thermodynamic temperature derivation

[kelvin490]

In deriving thermodynamic temperature scale, it uses the concept that heat transfer between two reservoirs is the function of the reservoirs' temperatures. i.e. Q₁/Q₂=Φ(T₁,T₂). And then further express that Φ(T₁,T₂)=ψ(T₁)/ψ(T₂).

I have two questions, 1. Is it a hidden assumption that the function Φ doesn't change its form for different temperatures? i.e. for different reservoirs we just plug in different temperatures Φ(T₁,T₂), Φ(T₃,T₄) but it can never be some other functions for different temperatures. Why?

2. Why Φ(T₁,T₂) can be expressed as ψ(T₁)/ψ(T₂)? Any underlying assumptions?

 

[MisterX]

I have two questions, 1. Is it a hidden assumption that the function Φ doesn't change its form for different temperatures? i.e. for different reservoirs we just plug in different temperatures Φ(T₁,T₂), Φ(T₃,T₄) but it can never be some other functions for different temperatures. Why?

Your question doesn't make any sense. The fact that it depends on temperatures means that Φ can change its value for different temperatures. It can.

2. Why Φ(T₁,T₂) can be expressed as ψ(T₁)/ψ(T₂)? Any underlying assumptions?

My understanding is this is an underlying assumption, at least in this presentation of the material.

 

[Chestermiller]

In deriving thermodynamic temperature scale, it uses the concept that heat transfer between two reservoirs is the function of the reservoirs' temperatures. i.e. Q₁/Q₂=Φ(T₁,T₂). And then further express that Φ(T₁,T₂)=ψ(T₁)/ψ(T₂).

I have two questions, 1. Is it a hidden assumption that the function Φ doesn't change its form for different temperatures? i.e. for different reservoirs we just plug in different temperatures Φ(T₁,T₂), Φ(T₃,T₄) but it can never be some other functions for different temperatures. Why?

2. Why Φ(T₁,T₂) can be expressed as ψ(T₁)/ψ(T₂)? Any underlying assumptions?

See Section 5.3 in Introduction to Chemical Engineering Thermodynamics by Smith and Van Ness.

Chet