Dot Physics

I already talked about increasing the temperature of a pool. My father commented that he thought the pool level rose by like half an inch when the temperature increased (by about 10 degrees F). So, this leaves the question: Is my father crazy, or is this possible? Or are both true?

Does water expand when it warms up? Yes, except when it melts. Why does this happen? Liquids are actually very complicated, but here is a basic answer. Take a look at this PhET gas simulator, I know it is for gases not liquids. I think we can make it behave like a liquid if you increase the gravity to the maximum (in the simulator) and cool it down some so that it looks like this:

i-d59c5f73d4f1ab609b71b113f75ab52c-cold-liquid.jpg

Now what happens if I increase the energy? Each particle has more energy and the result is an expansion.

i-f737040d6e8ed227c52beb9e340720ae-hot-liquid.jpg

I am really not sure if this is a valid comparison, but maybe it is enough for you to get the idea. The PhET simulator is awesome, by the way.

So, what about water? One way to model the expansion of water (or anything, really) is:

i-73ece96e6128bcd403a9c2fda68e645a-volume-expansion.jpg

Where ? is the expansion coefficient, which is typically not constant with temperature. For the range of water temperature in this case, I will use (from hypertextbook.com)

i-352b47b150c4b3dd0783a9cf9097989b-beta-water.jpg

Great. Now I can find the change in volume of the pool, but I want the change in the water level. Suppose the pool were a rectangular cube with a bottom of area A and depth of d. The initial volume of this pool will be:

i-f4ad5482e0fb74754a35a2440176285e-volume-1.jpg

And then when the volume increases, the final volume would be:

i-60bfe4a60a7c96be231b1761b8812c65-car-b-frame-v-2.jpg

So, now I can re-write the thermal expansion stuff:

i-b57298a0bbf2ecfbd51aa2bd2d2b17b1-delta-d.jpg

The best thing is that the bottom area of the pool cancels. I did make a guess that I can use the average depth of the pool, I am pretty sure this is ok (if not, the depth doesn’t change that much). At this point, I am ready to plug in some stuff. I will assume an average depth of 1.5 meters and a 7 C change in temperature, then:

i-39ca887532344431968fa8531a9a0a46-change-in-depth-answer.jpg

This is about 0.1 inches. So, yes. The water does expand when heated. 1/2 an inch change in the depth level seems a little high.