Why Does Turpentine Not Overflow When Heated in an Aluminum Cylinder?

[Feodalherren]

Homework Statement

A hollow aluminum cylinder 2cm deep has an internal capacity of 2L at 20C. It is completely filled with turpentine and then slowly warmed to 80C.
a) How much turpentine overflows?
b)If the cylinder is then cooled back to 20 C how far below the cylinder's rim does the turpentine recede?

Homework Equations

Vnew=Vold(1+αΔT)^3

The Attempt at a Solution

I'm so confused. I just plugged and chugged into the formula.

It seems that the solutions manual uses the formula for the Al container, getting 2.0087L, which I follow. Then for the turpentine, they just do
2L(1+αΔT)=2.108L
So they do NOT cube it. What am I missing here? Why aren't they cubing the turpentine?

 

[Simon Bridge]

Did they use the volume or the linear expansion coefficient?

 

[Feodalherren]

Ahh they used linear for the Al and volume for the turpentine. Thanks a bunch!

 

[Chestermiller]

Ahh they used linear for the Al and volume for the turpentine. Thanks a bunch!

They should have cubed for both, unless the used the volumetric coefficient of expansion for aluminum.

Chet

 

[HalfLostAndSearching]

It appears that the solutions manual is using the formula for volume expansion of a liquid, Vnew = Vold(1+βΔT), where β is the coefficient of volume expansion for the liquid. This formula does not involve cubing the temperature change, as it is only meant to account for the expansion of the liquid itself, not the container. In this case, the turpentine expands from 2L to 2.108L due to the increase in temperature, so the excess volume is 0.108L.

To answer the first question, since the cylinder has a capacity of 2L, all of the turpentine will fit inside without any overflow.

For the second question, when the cylinder is cooled back to 20C, the turpentine will contract, but it will not recede below the cylinder's rim. This is because the original volume of the turpentine at 20C is 2L, which is the same as the capacity of the cylinder. Therefore, there will still be some space left in the cylinder even after the turpentine contracts due to the decrease in temperature.

I hope this helps clarify the solution for you. Keep in mind that it is important to use the correct formula for the specific scenario in order to get an accurate answer.