Temperature — Question 5

Question 5

Overlapping temperature bounds do not prove equality

Three equilibrium samples have independently allowed hard temperature intervals: A is $300.0\pm0.4\,\mathrm K$, B is $300.5\pm0.3\,\mathrm K$, and C is $301.2\pm0.2\,\mathrm K$. Every allowed combination is possible.

1. Find the full ranges of $T_B-T_A$ and $T_C-T_A$. Which temperature ordering is certain?

2. Is one common temperature compatible with all three readings? Distinguish compatibility for A and B from proof of equality.

Question 5 - Solution

Overlapping temperature bounds do not prove equality

(a) Method and calculation

$$T_B-T_A\in[300.2-300.4,300.8-299.6]=[-.2,1.2]\,\mathrm K.$$$$T_C-T_A\in[301.0-300.4,301.4-299.6]=[.6,1.8]\,\mathrm K.$$B versus A is undecided; C is certainly hotter than A.

(b) Interpretation and further calculation

$$I_A\cap I_B=[300.2,300.4]\,\mathrm K,\qquad I_A\cap I_C=\varnothing.$$$$I_A\cap I_B\cap I_C=\varnothing.$$A and B can have the same temperature, but many unequal pairs also fit. All three cannot share one value under these bounds.

Physical check

Hard endpoint bounds require extreme differences; a root-sum-square statistical rule would answer a different question.