The Zeroth Law of Thermodynamics — Question 8

Question 8

Equilibrium classes do not choose a unique numerical scale

Temperature labels $\theta$ on one scale are positive. Another laboratory assigns $\psi=2\theta+10$ and a third assigns $\chi=\theta^3$. These are numerical labels only. The zeroth-law relation is represented by equality of labels.

1. For $\theta_A=3$ and $\theta_B=5$, find both transformed pairs and verify that ordering is preserved.

2. Show that both transformations preserve equilibrium classes but give different ratios and interval ratios. Can the zeroth law alone select kelvin as the absolute scale?

Question 8 - Solution

Equilibrium classes do not choose a unique numerical scale

(a) Method and calculation

$$(\psi_A,\psi_B)=(16,20),\qquad(\chi_A,\chi_B)=(27,125).$$$$3<5\ \Longrightarrow\ 16<20\ \text{and}\ 27<125.$$Both maps are strictly increasing in the specified positive domain.

(b) Interpretation and further calculation

$$\theta_A=\theta_B\ \Longleftrightarrow\ \psi_A=\psi_B\ \Longleftrightarrow\ \chi_A=\chi_B.$$$$\theta_B/\theta_A=5/3,\quad\psi_B/\psi_A=5/4,\quad\chi_B/\chi_A=125/27.$$Also $\Delta\psi=2\Delta\theta$ while $\Delta\chi=(\theta_B^2+\theta_A\theta_B+\theta_A^2)\Delta\theta$. Equality classes alone do not determine physical interval or ratio calibration.

Physical check

The modern kelvin requires additional physical structure and an SI definition; it is not fixed by transitivity alone.