Q:

I have read that the expansion ration of water at 212F is 1700 to 1 and at 1,000F it's 4,200 to 1. My question is how and why this happens. What causes it? Is it temperature alone or pressure or both? I'm very interested to find the answer to this question. I've been trying to find answers but have failed.

- Kyle Simmons (age 24)

Houston Tx USA

- Kyle Simmons (age 24)

Houston Tx USA

A:

Those expansion ratios of water vapor compared to liquid water are all taken at *fixed* pressure. If you don't specify the pressure, you don't know how much the expansion is. So the expansion you describe is purely due to how as the temperature goes up, while the pressure is held fixed, the volume goes up.

There are many ways to understand the effect. The difference between 212°F and 1000°F (a volume ratio of 4200/1700=2.5, according to your numbers) is the easiest to understand. Water vapor at atmospheric pressure in this temperature range is close to being an ideal gas, for which V=NkT/p, where N is the number of molecules, k is a constant, T is the absolute temperature, and p is the pressure. So let's convert the T's to absolute tempratures in Kelvin: 373 K and 811 K. We expect a V ratio of 811/373= 2.2. That's not far from what you gave.

Why does the pressure go up about proportional to the absolute T? T is, in this range, a measure of how much translational kinetic energy (mv^{2}/2) the molecules have. The average force on a wall (pressure times area) is proportional to how often molecules hit it, which goes as their speed v. It's also proportional to how much momentum they transfer on each hit, which is proportional to mv. So the pressure goes as mv^{2}, as does T. There are more fundamental and elegant ways to derive this, but I think this way is good for building physical intuition.

Ok, now for the big question. Why does the volume go up times 1700 when the liquid water boils to vapor at 212°F? We basically understand the gas volume by our simple pV=NkT argument. So the question becomes why is there a liquid and why is it so small but not smaller? A quick answer is that the molecules attract each other. When they aren't too hot, they tend to stick to each other. It's much easier to stick to a blob of other molecules than to stick to just one or two, so the choice is all or nothing: either the molecules form a big liquid blob or a free gas, not something in between. At very short range, the molecules repel each other- you can think of them as "touching". So the blob takes on a volume about as small as it can get without the molecules smashing into each other. That's the liquid.

Mike W.

*(published on 09/08/2015)*