# Water evaporation problem. Exercise from Feynman's lectures

I read lectures of Richard Feynman and try to solve problems and I have some trouble with a problem on water evaporation.

A glass full of water is left standing on an average outdoor window in California.

1. How long do you think it would take to evaporate completely?
2. How many molecules $$~\text{cm}^{-2} ~\text{s}^{-1}$$ would be leaving the water glass at this rate?
3. Briefly discuss the connection, if any, between your answer to part 1. above and the average rainfall over the earth

I think calculation the rate of evaporation at first is the easiest way for solving this problem. So I break glass into layers, every layer as thick as one water molecule ($$2.8\cdot10^{-8}\,\text{cm}$$). I know the number of molecules in one cubic centimeter in the water ($$3\cdot10^{22}$$). The average diameter of glass is $$6.75\,\text{cm}$$ and I suppose that it is surface. Next I look at surface layer in 2 dimensional view. So each molecule moves in 4 main directions. I suppose that in $$\frac{1}{4}$$ of cases, the molecule goes into the air. I take normal humidity of air as 60%. But I don't know the velocity of this molecule that I need for calculating rate of evaporation. This velocity should be related to the temperature (because temperature is only movement of molecules).

Should I use the temperature in Joule?

Can someone help me with algorithm of solution? I want to understand analytical part of this problem. I am not sure that my method is right.

• I suspect this is one of Feynman's attempts to see how much physics you recognize goes into the answer. Is the water absorbing energy from sunlight? Do you know the partial pressure law + evaporation rate? and so on. Sep 7 '15 at 14:37
• I do not understand why this question has been put on hold. It seems to meet all the criteria for homework problems listed at the meta site. It does show some effort (perhaps misguided, but effort nonetheless) and it asks about a specific concept: how to find the velocity of water molecules at a given temperature. It references the source. It (now) uses the homework tag. So, can you please explain why it has been put on hold? Much thanks. Sep 9 '15 at 11:03
• "If this question can be reworded to fit the rules in the help center, please edit the question." Once again, I would like to ask ACuriousMind, John Rennie, John Duffield, and Qmechanic how this question fails to comply with the guidelines given at the meta-site, and ask their advice about how it should be edited. I believe that the person who posted this question is sincere about wanting to better understand physics, and should be aided, rather than hindered, in that. Sep 9 '15 at 23:20
• Related meta post: meta.physics.stackexchange.com/q/7042/2451 Sep 10 '15 at 11:00
• @MichaelA.Gottlieb - Just an aside, if you want a response from these gentlemen, just writing their names here won't suffice. They would have come across it in some review queue, took action and that was the end of it (i.e. they may never visit the post ever again). At least 3 out of these 4 can be reached in the Physics chatroom, The $\hbar$. You can ping their names, and they will be alerted, and then, most likely respond. Cheers :) Sep 10 '15 at 11:04

This is an order-of-magnitude problem. The idea is to avoid accounting for specific processes, since there are too many. Perhaps the bottom of the glass focuses the sunlight and the windowsill is darkly colored, causing it all to heat up. We don't know. But the point is, something like that might increase the evaporation rate 3x, but it won't make a 10x, order of magnitude, difference.

We can get a rough idea of global rainfall or the molecular process of evaporation without really trying very hard.

1. It's reasonable to say, "a few weeks," just based on experience with standing water. I'd say that water evaporates about 2 cm/week.

2. Water has a molecular mass of 18, and a density of 1 g/cm3, so NA molecules make a column 1 cm2 × 18 cm — coincidentally, as high as a tall glass. So, the rate is 6×1023/(cm2 × 9 weeks), or 1017 cm–2 sec–1. A centimeter and a second are both small, but that number is still unimaginably large.

3. What goes up must come down *, so the average rainfall on Earth should be about the same as the average evaporation from a typical glass. Google says that the average global rainfall is 1 m/year, which is about 2 cm/week. Yay!

(* Water empirically follows this rule, anyway, or the planet would be dessicated.)

• +1 for "what goes up must come down". That is indeed the aha moment about this entire question. No detailed physics needed. Jul 11 '16 at 19:02

These are parts (a) and (b) of Exercise 1.15 in "Exercises for The Feynman Lectures on Physics" (2014, New Millennium Edition). There is also a part (c):

(c) Briefly discuss the connection, if any, between your answer to part (a) above and the average rainfall over the earth.

For a glass not in direct sunlight filled with water to a depth of 6 cm the answer to part (a) is ~2 weeks.

• Thank you for the part C. What about answer ~2 weeks it is not enough for me, did you find it somewhere or solve yourself? Sep 8 '15 at 13:10
• This is the answer that has been used at Caltech since 1961, when I was one year old. I have seen an analytical solution that yields this answer. Of course it was based on many assumptions; the problem is not stated in a way that it can be solved analytically without making additional assumptions, and figuring out what kinds of assumptions to make was part of the problem. However, I think it is not unlikely that Rochus Vogt, who wrote the answer, found it by putting a glass of water on his window sill, which would, of course, be in the true spirit of Physics :-). Sep 8 '15 at 14:25
• I can put a glass outdoor and check how many days the water will evaporate. =) But I think that it don't speak a lot about this process. It is good experiment that I can create after theoretical solution. Like a checking myself. There is a velocity of water molecule in analytical solution that you have been seen? Sep 8 '15 at 14:39
• IMHO, the answer to part (a) is not as important as the answers to parts (b) and (c). That is why part (a) asks, "How long do you think it would take to evaporate completely?" It does not ask to calculate it analytically nor to find it experimentally: it's a question of opinion. Part (b) can be calculated from part (a) using the answer from another problem in this set. Bear in mind this problem was in the 1st homework assignment for freshman, given after hearing only two introductory non-technical lectures: they didn't know enough yet to calculate this, nor to follow a calculation of it. Sep 8 '15 at 14:42
• Yes, you are right about importance question (b) and (c), but you know without calculation I can do some mistakes. Calculation gives me more understanding I think. May be it is only checking my opinion that I created before calculation, why not. It show me the truth, what depends from what =) Sep 8 '15 at 14:50