Measuring Temperature — Question 10

Question 10

A fixed Boltzmann constant does not make a realization exact

The SI kelvin fixes $k_B=1.380649\times10^{-23}\,\mathrm{J/K}$ exactly. An ideal primary thermometer measures an energy scale $E=k_BT$ and reports $E=(4.141947\pm.001380649)\times10^{-21}\,\mathrm J$ as a hard interval. Other errors are included in this bound. The triple point of water is a practical reference, not the current defining constant.

1. Infer the central temperature and full interval. Does the exact value of $k_B$ eliminate the measurement bound?

2. A practical reference has assigned $T_r=273.1600\pm.0002\,\mathrm K$, and an exact measured ratio $T_x/T_r=1.1000$. Infer $T_x$ and its uncertainty. Distinguish a scale assignment from an exact physical realization.

Question 10 - Solution

A fixed Boltzmann constant does not make a realization exact

(a) Method and calculation

$$T_0=\frac{4.141947\times10^{-21}}{1.380649\times10^{-23}}=300.000\,\mathrm K.$$$$\delta T=\frac{.001380649\times10^{-21}}{1.380649\times10^{-23}}=.100\,\mathrm K,\quad T\in[299.9,300.1]\,\mathrm K.$$The constant contributes no uncertainty; the measured energy still does.

(b) Interpretation and further calculation

$$T_x=1.1000(273.1600)=300.4760\,\mathrm K.$$$$\delta T_x=1.1000(.0002)=.00022\,\mathrm K.$$An assigned reference-scale value can be exact by convention while realizing that physical state remains uncertain. The supplied reference uncertainty must be propagated.

Physical check

The modern definition is consistent with BIPM: the fixed constant is $k_B$, while realizations and thermodynamic fixed-point temperatures are experimental quantities.