# Thermodynamic temperature derivation

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## Main Question or Discussion Point

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?

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## Answers and Replies

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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
Mentor
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