The Zeroth Law of Thermodynamics — Question 9

Question 9

A finite-time null result depends on thermal contact strength

A passive contact test gives $|P|<.030\,\mathrm W$. Its independently verified conductance is in $[.10,.30]\,\mathrm{W/K}$, and $P=G(T_A-T_B)$ applies at the observation time. No other energy paths are present.

1. Find the strongest guaranteed upper bound on $|T_A-T_B|$. Why must the smallest possible conductance be used?

2. A second test constrains the conductance only to $0<G\le.30\,\mathrm{W/K}$. Is any finite temperature-difference bound guaranteed by the same null result?

Question 9 - Solution

A finite-time null result depends on thermal contact strength

(a) Method and calculation

$$|\Delta T|=\frac{|P|}{G}<\frac{.030}{.10}=.30\,\mathrm K.$$Using $G=.30$ would give $.10$ K, but$$G=.10,\quad |\Delta T|=.20\,\mathrm K\ \Longrightarrow\ |P|=.020\,\mathrm W,$$which is allowed and refutes that stronger bound.

(b) Interpretation and further calculation

For any proposed finite bound $M>0$, choose $|\Delta T|>M$ and sufficiently small positive $G$:$$0<G<\min\!\left(.30\,\mathrm{W/K},\frac{.030\,\mathrm W}{|\Delta T|}\right),$$so that $|P|<.030\,\mathrm W$ despite the large temperature difference. No finite guarantee follows without a positive lower conductance bound.

Physical check

Failure to resolve transfer is evidence about temperature only when the comparison mechanism has established sensitivity.