# Can I apply volume expansion this way?

## 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?

## 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?

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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.

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