The Zeroth Law of Thermodynamics — Question 4

Question 4

A differential thermometer can cancel a common offset

At equilibrium an instrument indicates $r=T+b$, where $b$ is unknown but constant across a comparison. Its readings for A and B are $306.4$ and $306.4\,\mathrm K$. Each indicated reading has an independent hard error no larger than $.10\,\mathrm K$.

1. Write the true temperature-difference interval. Can the unknown fixed offset prevent this comparison?

2. If the offset can change by any amount in $[-.30,.30]\,\mathrm K$ between readings, find the new difference interval. Does repeated equality of the displayed numbers restore the earlier bound?

Question 4 - Solution

A differential thermometer can cancel a common offset

(a) Method and calculation

Let $r_X=T_X+b+e_X$. Then$$T_B-T_A=(r_B-r_A)-(e_B-e_A).$$$$T_B-T_A\in[-.20,.20]\,\mathrm K.$$The common $b$ cancels even though the absolute temperature remains uncertain.

(b) Interpretation and further calculation

Allowing $\Delta b=b_B-b_A$,$$T_B-T_A=\Delta r-\Delta b-\Delta e.$$$$|T_B-T_A|\le.30+.10+.10=.50\,\mathrm K.$$Repeated displays do not exclude a correlated offset drift. Additional reference control is needed to justify a smaller interval.

Physical check

This is an operational finite-bound comparison, not an exact proof of the zeroth-law premise.