Measuring Temperature — Question 1

Question 1

Absolute pressure matters in a gas thermometer

A sealed constant-volume ideal gas thermometer obeys $p/T=\text{constant}$ with absolute pressure. At a reference temperature $300\,\mathrm K$, its gauge pressure is $20.0\,\mathrm{kPa}$ above ambient $100.0\,\mathrm{kPa}$. In an unknown bath, the gauge reads $44.0\,\mathrm{kPa}$ with unchanged ambient pressure.

1. Find the bath temperature. Quantify the result from incorrectly taking the ratio of gauge pressures.

2. If ambient pressure in the second measurement is instead $98.0\,\mathrm{kPa}$ with the same gauge reading, infer the corrected bath temperature.

Question 1 - Solution

Absolute pressure matters in a gas thermometer

(a) Method and calculation

$$p_r=120\,\mathrm{kPa},\quad p=144\,\mathrm{kPa},\quad T=300\frac{144}{120}=360\,\mathrm K.$$$$T_{\mathrm{gauge}}=300\frac{44}{20}=660\,\mathrm K.$$Gauge zero is an atmospheric reference, not zero molecular pressure.

(b) Interpretation and further calculation

$$p=98+44=142\,\mathrm{kPa},\qquad T=300\frac{142}{120}=355\,\mathrm K.$$Ignoring the ambient change produces$$T_{\mathrm{uncorrected}}-T=360-355=5\,\mathrm K.$$

Physical check

Ratios of absolute pressures remove the gas amount and fixed volume; ratios of shifted gauge readings do not.