# How did Kelvin make this fascinating calculation?

I was just reading Lord Kelvin's "The Sorting Demon Of Maxwell" where I found this quote concerning what Maxwell's Demon can do:

(He) can direct the energy of the moving molecules of a basin of water to throw the water up to a height and leave it there proportionately cooled (1 deg. Fahrenheit for 772 ft. of ascent)

Given this was written 135 years ago, I was wondering how in the world Lord Kelvin could possibly have calculated the kinetic energy of individual water molecules at different temperatures. Surely sophisticated statistical-mechanical theories of liquids weren't developed by 1878. What back-of-the-envelope calculation could he possibly have done?

• Is Lord Kelvin actually interested in the details, or is he just making a point about this being impossible without an external energy source? – CuriousOne Oct 14 '14 at 1:05
• @CuriousOne - the context was about Maxwell's Demon, a thought-experiment in which a tiny intelligent being is able to lower entropy just by cleverly making use of molecules' natural movements (opening and closing a tiny door when a molecule happens to approach from one direction but not the other, for example) rather than imparting any energy to them. Since here the increase in potential energy is exactly balanced by a decrease in molecular kinetic energy (which is why the temperature goes down), I presume Kelvin was thinking in these terms. – Hypnosifl Oct 14 '14 at 17:07
• @Hypnosifl: I know what a Maxwell Demon is. My question is about the context in which Lord Kelvin's statement rests, since I don't have the source. – CuriousOne Oct 15 '14 at 0:26
• @CuriousOne - OK, without having the source one can't be sure of what he meant, but since the whole idea of Maxwell's Demon is that it causes molecules to gather in particular preferred spatial locations (when they were previously spread out more randomly) without imparting energy to them, and we know Kelvin's comment was from the paper "The Sorting Demon of Maxwell", I think by far the most likely inference is that he was talking about the demon causing the atoms to gather at a greater height than most of them had been earlier using these same non-energy imparting methods. – Hypnosifl Oct 15 '14 at 1:38

How about lifting a litre or kilogram of water?

Lifting a kilogram of water $235.3\text{ m}$ (772 feet in sensible units) involves a potential energy change:$$PE=mgh=1\times9.8\times235=2.31 \text{ kJ}$$

Warming a kilogram of water, (specific heat $4.179\text{ kJ/kg/K}$), through a temperature change of $0.556\text{ K}$ (1 degree Farenheit) takes:$$E=mC\Delta T=1\times 4.179 \times0.556=2.32\text{ kJ}$$

No special knowledge of molecules needed...

The only trick is knowing the mechanical equivalent of heat, first "discovered" by Count Rumford (while boring out cannon barrels!) and then measured accurately by Joule (1845).

• Makes sense, and it might have also helped conceptually that they already had an idea that the equipartition theorem indicated temperature was proportional to the internal kinetic energy of the molecules (see the "history" section of the equipartition wiki article), and the constant of proportionality is just the heat capacity (equal to mass * specific heat) which can be determined empirically. So that makes it easy to see that raising potential energy at the expense of kinetic energy should lower temperature by dE/(heat capacity). – Hypnosifl Oct 14 '14 at 0:11
• Thanks, that's beautiful in the terribly-obvious-when-you-see-it kind of way :) – tom Oct 14 '14 at 1:07
• This is such a beautiful answer, I wish that we could star answers as opposed to questions. – dotancohen Oct 14 '14 at 15:42
• Note that Joule published a paper in 1850 titled, "On the Mechanical Equivalent of Heat". See dx.doi.org/10.1098/rstl.1850.0004 – mankoff Oct 14 '14 at 15:53