Temperature — Question 6

Question 6

One indicator value can hide two temperatures

An ideal sensor has response $R(T)=100\,\Omega+(0.010\,\Omega\,\mathrm{K^{-2}})(T-300\,\mathrm K)^2$ over $280\le T\le320\,\mathrm K$. A measurement gives $R=101\,\Omega$ exactly.

1. Find every compatible temperature. Can this sensor alone rank unknown baths over the full domain?

2. Give a restricted operating range on which inversion is unique. Find the local response slope at each candidate and explain the ambiguity.

Question 6 - Solution

One indicator value can hide two temperatures

(a) Method and calculation

$$\frac{R-100\,\Omega}{0.010\,\Omega\,\mathrm{K^{-2}}}=100\,\mathrm{K^2}.$$$$T-300\,\mathrm K=\pm10\,\mathrm K,\qquad T=290\text{ or }310\,\mathrm K.$$Equal responses do not establish equal temperature when the response is not one-to-one.

(b) Interpretation and further calculation

Either $[280,300]$ K or $[300,320]$ K selects one branch.$$\frac{dR}{dT}=0.020(T-300\,\mathrm K)\,\Omega\,\mathrm{K^{-2}}.$$$$R_T(290\,\mathrm K)=-.200\,\Omega/\mathrm K,\quad R_T(310\,\mathrm K)=+.200\,\Omega/\mathrm K.$$The slope changes sign at the minimum; an independent branch identification is essential.

Physical check

Reproducibility is necessary but not sufficient for an unambiguous thermometer.