In a thermometer manufacturing plant, a type of mercury thermometer is built
at room temperature \(\left(20^{\circ} \mathrm{C}\right)\) to measure
temperatures in the \(20^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) range,
with \(\mathrm{a}\) \(1-\mathrm{cm}^{3}\) spherical reservoir at the bottom and a
\(0.5-\mathrm{mm}\) inner diameter expansion tube. The wall thickness of the
reservoir and tube is negligible, and the \(20^{\circ} \mathrm{C}\) mark is at
the junction between the spherical reservoir and the tube. The tubes and
reservoirs are made of fused silica, a transparent glass form of
\(\mathrm{SiO}_{2}\) that has a very low linear expansion coefficient \((\alpha=\)
\(\left.0.4 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .\) By mistake, the
material used for one batch of thermometers was quartz, a transparent
crystalline form of \(\mathrm{SiO}_{2}\) with a much higher linear expansion
coefficient \(\left(\alpha=12.3 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)
.\) Will the manufacturer have to scrap the batch, or will the thermometers
work fine, within the expected uncertainty of \(5 \%\) in reading the
temperature? The volume expansion coefficient of mercury is \(\beta=181 \cdot
10^{-6}{ }^{\circ} \mathrm{C}^{-1}\).
