Measuring Temperature — Question 8

Question 8

Bulb expansion changes a gas-thermometer ratio

A sealed ideal gas thermometer has reference state $T_r=300\,\mathrm K$, $p_r=100\,\mathrm{kPa}$, volume $V_r$. In the unknown bath its absolute pressure is $120\,\mathrm{kPa}$ and its bulb volume is independently known to be $1.002V_r$. Use $pV/T=\text{constant}$.

1. Infer the corrected temperature and the error from assuming constant volume.

2. If the volume ratio lies in $[1.0015,1.0025]$ while pressures are exact, give the full temperature interval and half-width.

Question 8 - Solution

Bulb expansion changes a gas-thermometer ratio

(a) Method and calculation

$$T=T_r\frac p{p_r}\frac V{V_r}=300(1.20)(1.002)=360.72\,\mathrm K.$$$$T_{\mathrm{fixed\ volume}}-T=360-360.72=-.72\,\mathrm K.$$

(b) Interpretation and further calculation

The conversion is monotone in the volume ratio:$$T\in[360(1.0015),360(1.0025)]\,\mathrm K=[360.54,360.90]\,\mathrm K.$$$$\delta T=.18\,\mathrm K.$$A pressure measurement alone cannot correct an unknown volume change.

Physical check

At a given measured pressure, a larger volume requires a higher temperature for the same gas amount.