Feynman feels the pressure

 

Richard Feynman famously said “Science is the belief in the ignorance of experts”. What he meant thereby, was presumably that unlike other subjects where people tend to blindly accept the ignorant opinions of experts, in science people are more sceptical of experts’ opinions. Whether Feynman’s claim has much validity, I know not, nonetheless I thought it might be interesting to point out an example of where Feynman himself had shown ignorance of the physical world in ‘The Feynman Lectures on Physics’.

The errata page of the website contains the following:

In his preface to The Definitive Edition of The Feynman Lectures on Physics, Professor Kip Thorne writes:

“The errata corrected in this edition come from three sources: about 80 per cent are from Michael Gottlieb; most of the rest are from a long list by an anonymous reader, submitted to Feynman in the early 1970s via the publisher; and the remainder are from scattered short lists provided to Feynman or us by various readers.

The corrected errata are mainly of three types: (i) typographical errors in the prose; (ii) roughly 150 typographical and mathematical errors in equations, tables, and figures—sign errors, incorrect numbers (e.g., a 5 that should be a 4), and missing subscripts, summation signs, parentheses and terms in equations; (iii) roughly 50 incorrect cross references to chapters, tables, and figures. These kinds of errors, though not terribly serious to a mature physicist, can be frustrating and confusing to students, the audience Feynman was trying to reach.

It is remarkable that the errata included only two inadvertent errors in physics: Volume I, page 45-4 now says “When a rubber band is stretched its temperature rises,” not “falls” as claimed in previous editions; and Volume II, page 5-9 now says “…no static distribution of charges inside a closed grounded conductor can produce any [electric] fields outside” (the word grounded was omitted in previous editions). This second error was pointed out to Feynman by a number of readers, including Beulah Elizabeth Cox, a student at The College of William and Mary, who had relied on Feynman’s erroneous passage in an exam. To Ms. Cox, Feynman wrote in 1975,[1] “Your instructor was right not to give you any points, for your answer was wrong, as he demonstrated using Gauss’s law. You should, in science, believe logic and arguments, carefully drawn, and not authorities. You also read the book correctly and understood it. I made a mistake, so the book is wrong. I probably was thinking of a grounded conducting sphere, or else of the fact that moving the charges around in different places inside does not affect things on the outside. I am not sure how I did it, but I goofed. And you goofed, too, for believing me.”

Since the lectures had clearly been subjected to extensive scrutiny, with so few errors of physics apparently found, you might think that finding fault would be difficult, nonetheless I found a major error in the very first lecture.

The first lecture is beautifully written, and should make perfect sense even to somebody with little knowledge of physics, so you might like to read the lecture before checking out my criticism.

Feynman’s Lecture 1

This is what Feynman wrote:

“Now imagine this great drop of water with all of these jiggling particles stuck together and tagging along with each other. The water keeps its volume; it does not fall apart, because of the attraction of the molecules for each other. If the drop is on a slope, where it can move from one place to another, the water will flow, but it does not just disappear—things do not just fly apart—because of the molecular attraction. Now the jiggling motion is what we represent as heat: when we increase the temperature, we increase the motion. If we heat the water, the jiggling increases and the volume between the atoms increases, and if the heating continues there comes a time when the pull between the molecules is not enough to hold them together and they do fly apart and become separated from one another. Of course, this is how we manufacture steam out of water—by increasing the temperature; the particles fly apart because of the increased motion.”

Feynman seems to be making it very clear that the existence of a drop of water, is a kind of battle between the jiggling of molecules trying to force them apart, and the molecular attraction of the molecules trying to hold them together. That sounded plausible. Then I recalled that you cannot boil an egg, or make a proper cup of tea, on Mount Everest, because water boils at 72 degrees Celsius. So clearly pressure must have something important to say about the existence of liquid water.

Then I recalled the classic experiment whereby water can be made to boil at room temperature, by putting it in vessel and pumping out most of the air. So I began to wonder whether water can ever actually exist without any pressure at all. So I looked at a phase diagram, of the type that I think we were shown at school as part of a basic physics course:

phase

And lo and behold, it shows that liquid water can never exist when the pressure is less that 0.6 % of normal atmospheric pressure (see the triple point of water). Since liquid water requires pressure for its very existence, Feynman’s characterisation of it merely as a battle between molecular forces and the jiggling of heat, is severely lacking.

Of course the phase diagram does not tell the whole story; because if you go to the middle of the liquid area, you will still have water-vapour molecules flying around above the liquid. Also the phase diagram indicates that you cannot even have solid ice without pressure, even though clearly solid ice lasts a lot longer in the absence of pressure than liquid water; for instance comets partly composed of ice last for ages.

What the phase diagram does tell us, is that if we were to put a lump of ice in a test tube, pump the air out, seal the tube, and then let the ice melt and reach room temperature; then the pressure in the tube due to the water vapour would be about 2% of an atmosphere. If we increased the temperature to 100 degrees, then the vapour pressure would be about 1 atmosphere. As we continued to heat the water, the pressure in the tube would continue to increase, and there would continue to be waterline until we reached the critical temperature 374 degrees, at which point the waterline would disappear and we would just be left with highly compressed water-vapour, or perhaps highly diffuse liquid water.

That is what happens in terms of classical physics; but what makes Feynman’s lecture so entertaining, is that he describes things (including pressure) in terms of what the individual atoms are actually doing. So perhaps later in the lecture he includes the necessity of pressure for the existence of water. Let us see:

“The atomic hypothesis also describes processes, and so we shall now look at a number of processes from an atomic standpoint. The first process that we shall look at is associated with the surface of the water. What happens at the surface of the water? We shall now make the picture more complicated—and more realistic—by imagining that the surface is in air. Figure 1–5 shows the surface of water in air. We see the water molecules as before, forming a body of liquid water, but now we also see the surface of the water. Above the surface we find a number of things: First of all there are water molecules, as in steam. This is water vapor, which is always found above liquid water. (There is an equilibrium between the steam vapor and the water which will be described later.) In addition we find some other molecules—here two oxygen atoms stuck together by themselves, forming an oxygen molecule, there two nitrogen atoms also stuck together to make a nitrogen molecule. Air consists almost entirely of nitrogen, oxygen, some water vapor, and lesser amounts of carbon dioxide, argon, and other things. So above the water surface is the air, a gas, containing some water vapor. Now what is happening in this picture? The molecules in the water are always jiggling around. From time to time, one on the surface happens to be hit a little harder than usual, and gets knocked away. It is hard to see that happening in the picture because it is a still picture. But we can imagine that one molecule near the surface has just been hit and is flying out, or perhaps another one has been hit and is flying out. Thus, molecule by molecule, the water disappears—it evaporates.”

As you can see; even when describing water at a molecular level, Feynman ignores the effect of pressure. The fact is, that every molecule on the surface of the water is being hit around half a billion times a second by air molecules which would be travelling at an average speed of 500 m/s at room temperature. The air molecules are moving in all different directions; but as they are coming from above the surface of the water, they will all have a component trying to knock any water molecules that try to escape back into the liquid.

Each water molecule on the surface of the liquid, gets pounded by air molecules that have nearly twice their mass; so they in turn bump into the water molecules underneath, and as such the pounding of the air molecules keeps the body of water together, presumably with the assistance of the molecular attraction between the water molecules.

In a sense the pounding of the water molecules by the air molecules actually makes them jiggle. One could perhaps say that if both the water and the air are at the same temperature, then on average the collisions between the air molecules and the water molecules would transfer no energy; but if the air is hotter than the water then on average the collisions would add energy to the water and heat it or put another way make the water molecules jiggle faster, whilst if the air was colder the collisions would on average transfer energy to it and heat it.

Of course it was not possible for Feynman to cover everything in the lecture, and it is understandable that he would make errors when covering physics in an original manner. However his characterisation of the existence of water as merely a battle between jiggling and molecule attraction, is far from the truth; and his later description of what happens at the surface of the water on a molecular level, does not begin to cover what really happens.

The idea that the surface of water requires a continual bombardment of air molecules to keep it in place, is fairly basic physics. Nonetheless it is not something I had ever thought about before. Normally one thinks of whether a substance is found as a gas, liquid or solid, in terms of temperature and the characteristics of the substance, as Feynman did. From now on when I look at the surface of a lake, I will think about the air molecules battering it into existence. If all the air were to disappear from the atmosphere, all the water on earth would not simply boil away; rather some would boil away to produce a new atmosphere of water-vapour, such that the bombardment of water-vapour molecules would be sufficient to keep the surface of the water in place.

My estimate that each molecule on the surface of the water is hit a billion times is based on the following calculation:

There are about 3 x 1028 molecules of water in 1 cubic metre, so a 1 square metre area of water-surface would contain about 1019 molecules.

Atmospheric pressure is about 105 kgm/s/s per square metre at sea-level. That means that every second each square metre will receive 105 kgm/s of momentum from the air molecules. At room temperature air molecules are flying around at about 500 m/s, and their mass is about 5 x 10-26 kg. So the number of collisions (n) on a 1 square metre area is given by the equation: 105 = 5 x 10-26 x 500 x n, so n = 4 x 1027. So the number of collisions per molecule = 4 x 1027 divided by 1019 = 4 x 108, which is 400 million.  (That calculation assumed that all the air molecules were heading directly perpendicular to the surface of the water, which obviously is not the case.)  So even at the triple point of water, the lowest pressure at which liquid water can exist, each surface water molecule would be hit about 400 million x 0.006 = 2.4 million times a second, or slightly more often as the air molecules would be flying around a bit slower than 500 m/s at the lower temperature.

The odd thing is, that I have explained how air pressure allows liquid water to exist, and even calculated the number of collisions, without mentioning the fact that atmospheric air pressure relies on gravity. Air pressure is the product of the mass of the column of air above, times the acceleration due to gravity. The mass of the column of air is about 10,000 kg per square metre at sea-level, and the acceleration due to gravity near the surface of the earth is roughly 10 m/s/s, hence the atmospheric pressure at sea-level is about 100,000 kg m/s/s per square metre.