Finding coefficients of volumetric expandtion from a know coefficient of linear expandtion Starting from a homework problem:

An aluminum cup of $100 cm^3$ capacity is completely filled with glycerin at $22°C$. How much glycerin, if any, will spill out of the cup if the temperature of both the cup and the glycerin is increased to $28°C$? (The coefficient of volume expansion of glycerin is $5.1x10^4/C°$.)

I find that I have the for efficient of linear expansion for aluminum, but I need to know how the volume of the cup changes. Worse, I don't know the dimensions of the cup.
I think I use the linear expansion equation for metal rod $\Delta L = L \alpha \Delta T$ to find how much taller the cup is after the temperature changed and the volume expansion equation for a solid of liquid $\Delta V = V \beta \Delta T$ but not knowing any of the dimension of the cup I do not see how to determine this?
 A: To leading order in $\alpha$, the volume expansion of your container does not depend on its shape, and is equal to $\Delta V = 3V\alpha\Delta T$. It is fairly simple to verify this for a cylinder, cube or sphere. Also the expansion coefficient for aluminum is going to be quite a bit smaller than glycerol, so you may be intended to simply neglect the expansion of the container.
A: A few things to ask yourself and keep in mind RE Aluminium:
- Does the proportionality of the change in Volume matter in this context?
- If you know the linear expansion (Remember this is the change in length/area/volume (all scaler units) VS the change in Temp), then you can work out what the final volume is by simply multiplying the initial volume by (expansion co-efficient * dT)
- If this has brought you to a point of uncertainty "50/50", do your unit check:
Where C - Temperature in Celsius
Where d - Delta (Change in Unit)
Where V - Volume
(V)cm^3 * (Coefficient)1/C * (dT)C. This will leave you with dV in cm^3
Volume capacity change of Aluminium Cup:
100 * (23e-6 * 6) = X
Volume change of Glycerin:
100 * (5.1e-4 * 6) = Y
Spilt amount (Z) = Y - X
You can do the math.
All the best,
Blake
