Measuring Temperature — Question 4

Question 4

Two calibration anchors cannot rule out curvature

An indicator is calibrated at $300$ and $400\,\mathrm K$. Two possible response laws, in numerical kelvin coordinates $t=T/(1\,\mathrm K)$, are $x_1=t$ and $x_2=t+.002(t-300)(t-400)$. Both are monotone on this range. An operator treats $x$ directly as a kelvin reading.

1. Compare both readings at $T=350\,\mathrm K$. Verify that both models fit the calibration endpoints.

2. Find the maximum magnitude of their disagreement within the calibrated range and propose one additional reference temperature that best distinguishes them.

Question 4 - Solution

Two calibration anchors cannot rule out curvature

(a) Method and calculation

$$x_1(300)=x_2(300)=300,\qquad x_1(400)=x_2(400)=400.$$$$x_1(350)=350,\qquad x_2(350)=350+.002(50)(-50)=345.$$Both fit the endpoints but disagree by $5$ indicated kelvins in the middle.

(b) Interpretation and further calculation

Complete the square:$$x_2-x_1=.002[(t-350)^2-2500].$$$$\max_{300\le t\le400}|x_2-x_1|=5\quad\text{at }t=350.$$A traceable $350$ K reference is most discriminating for these particular models. It does not exclude every other possible curvature.

Physical check

The curved model has $dx_2/dt=1+.002(2t-700)$ between $.8$ and $1.2$, so nonmonotonicity is not the cause.