The Zeroth Law of Thermodynamics — Question 5

Question 5

A connected passive network has one equilibrium temperature

Three unchanged equilibrium bodies at temperatures $T_A,T_B,T_C$ are connected in a chain. The supplied link laws are $P_{A\to B}=G_1(T_A-T_B)$ and $P_{B\to C}=G_2(T_B-T_C)$, with $G_1,G_2>0$. There are no external supplies. At global thermal equilibrium neither endpoint gains or loses energy.

1. Use the endpoint conditions to derive the relation among all three temperatures. Does unequal conductance permit unequal equilibrium temperatures?

2. For $G_1=.50$ and $G_2=.20\,\mathrm{W/K}$, endpoint rate bounds are $.010\,\mathrm W$ each. Find the largest allowed $|T_A-T_C|$.

Question 5 - Solution

A connected passive network has one equilibrium temperature

(a) Method and calculation

Endpoint A requires$$G_1(T_A-T_B)=0\ \Longrightarrow\ T_A=T_B.$$Endpoint C similarly requires$$G_2(T_B-T_C)=0\ \Longrightarrow\ T_B=T_C.$$Thus $T_A=T_B=T_C$; conductance controls rates, not the equilibrium equality.

(b) Interpretation and further calculation

$$|T_A-T_B|\le .010/.50=.020\,\mathrm K,\quad |T_B-T_C|\le .010/.20=.050\,\mathrm K.$$$$|T_A-T_C|\le.020+.050=.070\,\mathrm K.$$The maximum is attainable when both differences have the same sign.

Physical check

Positive conductances and no external supplies are essential. One disconnected edge removes the equality constraint across that edge.