Can I apply volume expansion this way?

[1MileCrash]

Homework Statement

An aluminum cup of 120 cm3 capacity is filled with glycerin at 15°C. How much glycerin, if any, will spill out of the cup if the temperature of the cup and glycerin is raised to 34°;C?

Homework Equations

The Attempt at a Solution

I think I can solve this no problem, but I have a question.

Does the capacity of the aluminum cup increase just like aluminum does? IE, would 120 cm3 of aluminum expand just like a 120 cm3 empty space within aluminum?

 

[ehild]

Does the capacity of the aluminum cup increase just like aluminum does? IE, would 120 cm3 of aluminum expand just like a 120 cm3 empty space within aluminum?

Yes.
Imagine you fill the empty volume of the cup (120 cm³) with the same aluminium, and say, the volume of the wall is 50 cm³. You have then Vo=Vfilling +Vwall=170 cm³ of aluminium altogether which volume will increase by a factor of k=(1+βΔT).
So the new volume of the whole piece of aluminium is V=kVo=170k=(120+50)k=120k +50k.

(1+βΔT)Vo=(1+βΔT)(Vfilling) +(1+βΔT)(Vwall) The volume of the filling is the same as the volume of the empty place inside the cup, so they increase by the same amount.

ehild