Measuring Temperature — Question 2

Question 2

Extrapolating gas thermometry to vanishing density

At each gas filling, a thermometer forms the ratio $r=p_x/p_r$ at fixed bulb volume. Over the tested range, assume $r(p_r)=r_0+a p_r$ exactly. Measurements give $r=1.2050$ at $p_r=50.0\,\mathrm{kPa}$ and $r=1.2100$ at $100.0\,\mathrm{kPa}$. The reference temperature is $300.0\,\mathrm K$, and the dilute-gas result is $T_x/T_r=r_0$.

1. Infer $a,r_0,T_x$. What bias results if the higher-pressure ratio is used directly?

2. A third filling at $75.0\,\mathrm{kPa}$ gives $1.2080$. Test the exact linear model and explain whether the original extrapolation is now established by all three points.

Question 2 - Solution

Extrapolating gas thermometry to vanishing density

(a) Method and calculation

$$a=\frac{1.2100-1.2050}{100-50}=1.00\times10^{-4}\,\mathrm{kPa^{-1}}.$$$$r_0=1.2050-a(50)=1.2000,\qquad T_x=360.0\,\mathrm K.$$The high-pressure shortcut gives $363.0$ K, a $+3.0$ K bias.

(b) Interpretation and further calculation

$$r_{\mathrm{predicted}}(75)=1.2000+.0001(75)=1.2075.$$$$r_{\mathrm{observed}}-r_{\mathrm{predicted}}=.0005.$$The exact linear model cannot fit all three exact values. Real data would require uncertainty bounds or a justified nonlinear model before an improved extrapolation.

Physical check

Low-density extrapolation addresses gas nonideality; it cannot by itself remove gauge offsets or bulb-volume changes.