How Does Temperature Affect the Movement of Mercury in a Thermometer?

[Kingrly]

Homework Statement

(a) The tube of a mercury thermometer (made of Pyrex glass) has an inside diameter of 0.140 mm. The bulb has a volume of 0.275 cm³. How far will the thread of mercury move when the temperature changes from 10.5 degrees Celsius to 33.0 degrees Celsius?

(b) Determine a formula for the change in length of the mercury column in terms of relevant variables. Ignore tube volume compared to bulb volume.

α_PYREX = 3 × 10^-6 (Degrees Celsius)^-1
β_PYREX = 9 × 10^-6 (Degrees Celsius)^-1
β_MERCURY = 180 × 10^-6 (Degrees Celsius)^-1

Homework Equations

(I) ∆V = β(V₀)∆T
(II) ∆L = α(L₀)∆T

The Attempt at a Solution

I calculated the change in the volume of the bulb via equation I, but I am unsure where to head from here. Without knowing the height of the mercury within the thermometer (save for the temperature intervals), I am unsure of how to find the change in volume of the tube or the change in volume of the mercury itself. Any hints?

 

[Nate810]

For (b), I was thinking of an equation in the form of:∆L = mα(L0)∆T + nβ(V0)∆TWhere m and n are coefficients to be determined. Is this a valid approach?

 

[thomate1]

Your approach so far is correct. To find the change in volume of the tube, you can use the fact that the volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height. In this case, you know the inside diameter of the tube (0.140 mm) and the change in temperature (∆T = 33.0 - 10.5 = 22.5 degrees Celsius). You can use equation II to find the change in length of the tube, and then plug that value into the volume equation to find the change in volume.

To find the change in volume of the mercury, you can use the fact that the density of mercury is constant, so the change in volume will be directly proportional to the change in length. You can use the βMERCURY value to find the change in length of the mercury column, and then use that value to find the change in volume.

Once you have the change in volumes of the tube and the mercury, you can add them together to find the total change in volume. From there, you can use equation I to find the change in length of the thread of mercury. Just remember to use the appropriate β value for each component (βPYREX for the tube and βMERCURY for the mercury).