The Zeroth Law of Thermodynamics — Question 6

Question 6

A steady thermometer between baths is not in equilibrium with either

A probe at $T_p$ exchanges energy with two maintained baths: $T_A=320\,\mathrm K$ through $G_A=2\,\mathrm{W/K}$ and $T_B=280\,\mathrm K$ through $G_B=1\,\mathrm{W/K}$. At steady state, $G_A(T_A-T_p)+G_B(T_B-T_p)=0$.

1. Find $T_p$ and the magnitude of the energy transfer through it.

2. The probe reading stops changing. Is it in thermal equilibrium with A, B, both, or neither? Explain using each link separately.

Question 6 - Solution

A steady thermometer between baths is not in equilibrium with either

(a) Method and calculation

$$(G_A+G_B)T_p=G_AT_A+G_BT_B,\quad T_p=\frac{2(320)+280}{3}=306.667\,\mathrm K.$$$$P_{A\to p}=2(320-306.667)=26.667\,\mathrm W.$$The same amount leaves toward B.

(b) Interpretation and further calculation

$$T_A-T_p=13.333\,\mathrm K,\qquad T_p-T_B=26.667\,\mathrm K.$$$$P_{A\to p}=P_{p\to B}\ne0.$$The probe is in equilibrium with neither bath. Zero net accumulation results from cancellation of nonzero transfers, so the zeroth-law premises are absent.

Physical check

The weighted probe temperature lies between the bath temperatures; equal and opposite powers are not pairwise thermal equilibrium.