Expansion of mercury in a thermometer

[bacon]

I think that I’m doing this problem correctly, but the answer seems a bit unreasonable. Can someone else check my work?

A thermometer has a quartz body within which is sealed a total volume of 0.400[itex]cm^{3}[/itex] of mercury. The stem contains a cylindrical hole with a bore diameter of 0.10mm. How far does the mercury column extend in the process of rising from 10deg C to 90deg C? Neglect any change in the volume of the quartz.

I first found the change in volume of the mercury.
[itex]\Delta V=\beta V_{0}\Delta T[/itex]
[itex]\beta = 182x10^{-6}K^{-1}[/itex] from a table in the book.
[itex]V_{0}=.400cm^{3}[/itex]
[itex]\Delta T=90-10=80 deg[/itex]
[itex]\Delta V=.005824cm^{3}=5.824mm^{3}[/itex]

This change in volume will fill a portion of the slender cylindrical hole in the quartz.
The volume of a cylinder is [itex]V=L\pi \frac{D}{4}^{2}[/itex] Where L is the length of the cylinder and D is its diameter.
[itex]L =\frac{4V}{\pi D^{2}}[/itex]
V is the [itex]\Delta V[/itex] calculated above and D is given as 0.10mm.
L=741mm

This seems like a very long thermometer.

Thanks in advance for any responses.

 

[technician]

Delete
I thought you had made a mistake in your calculation but I have used your numbers and got the same as you

 

[technician]

In addition!...0.4 seems like a large volume and 0.1 mm seems like a very fine bore

 

[bacon]

Thanks for your help.

 

[asteriatic]

Your calculations appear to be correct. The expansion of mercury in a thermometer is a well-studied phenomenon and follows the equation you used, \Delta V=\beta V_{0}\Delta T. The length of the thermometer will indeed increase significantly as the temperature rises. This is why thermometers are often designed with a coiled or zigzag shape to fit into a smaller space. Additionally, the length of the thermometer is not necessarily an issue as long as it is able to accurately measure the temperature. However, if you are concerned about the length, you may want to consider using a different substance with a lower coefficient of expansion, such as alcohol or gas.