Physical Units
"... nearly a third of what you have to learn ..."

In Fond Memory of
Richard P. Feynman (Nobel 1965)
May 11, 1918 - February 15,1988

The Feynman's Lectures on Physics are based on a famous course of undergraduate lectures given at Caltech by Professor Richard Phillips Feynman in the early 1960's.  What Dick Feynman had to say to undergraduates about various physical units was considered too trivial by the editors and was not included in the published version of these lectures.  We resurrect it here, from the audio record, as a tribute to Richard P. Feynman.

NOTICE: The text quoted below was transcribed by Dr. Gerard P. Michon from a copyrighted 6-hour audio record (© 1997 by the California Institute of Technology), published by Helix Books (Addison-Wesley, Reading, MA) with a printed collection of edited lectures by Richard P. Feynman, entitled "Six not-so-easy pieces".  Applicable copyright laws allow this short excerpt to appear here, but a formal permission from the California Institute of Technology and/or other parties may be required to reproduce any part of this text in a broader context.

Before I begin the lecture [on spacetime], I wish to apologize for something that is not my responsibility: Physicists and scientists all over the world have been measuring things in different units, and causing an enormous amount of complexity.  As a matter of fact, nearly a third of what you have to learn ¹ consists of different ways of measuring the same thing, and I apologize for it.  It's like having money in francs, and pounds, and dollars...  with the advantage over money that the ratios don't change, as time goes on.

For example, in the measurement of energy, the unit we use here is the joule (J), and a watt (W) is a joule per second.  But there are a lot of other systems to measure energy.  There are at least three different ones for engineers, which I have listed here.²

The physicists do something else when they want to talk about the energy of a single atom, instead of the energy of a gross amount of material.  The reason is, of course, that a single atom is such a small thing that to talk about its energy in joules would be inconvenient.  But instead of taking a definite unit in the same system (like 10^-20 J), they have unfortunately chosen, arbitrarily, a funny unit called an electronvolt (eV), which is the energy needed to move an electron through a potential difference of one volt, and that turns out to be about 1.6 10^-19 J.  I am sorry that we do that, but that's the way it is for the physicists.

The chemists also talk about the energy per atom.  Since they don't use the atoms individually but large blobs of them, in cans and barrels, they've chosen a certain number of atoms as a unit.  This number of things is called a mole (mol), and it is 6.023 10²³  objects.  The more precise definition, which is now correct or soon ³ will be, is that one mole of carbon-12 atoms has a mass of exactly 12 grams.  A mole is just a certain number of things.  So, instead of giving the energy per atom, the chemists give the energy per mole.  It's good, therefore, to know how much energy is a mole of electronvolts.  In other words, if each atom had one electronvolt of energy, a large number of atoms would have a reasonable amount of joules, namely 96500 joules per mole.  Incidentally, a mole of electrons has a total charge of 96500 coulombs (C); these numbers are equal for a reason you have to figure out.⁴

Now, there is an additional unit that the physical chemists use, the kilocalorie per mole (kcal/mol), and 23 of those is an electronvolt per atom.

Finally, unfortunately, you have another system for measuring masses.  The mass of an atom, from a chemist's point of view, is given by the mass of a mole of these atoms.  For example, the mass of carbon-12 is called 12 "atomic mass units" (u), because a mole of carbon-12 "weighs" 12 grams (or rather "has 12 grams of mass").  One atomic mass unit represents one gram for every mole of objects, one gram per mole.  We can measure that in electronvolts also.  "You can't measure mass in electron volts!" ⁵  Sure you can, because of the relation E = mc² ...  It is useful to know how much energy corresponds to the consumption of one atomic mass unit of material: That turns out to be about 931 million electronvolts (MeV).  Incidentally, the rest mass of a proton is 938 MeV, while the rest mass of an electron corresponds to 0.511 MeV.  The number 938 differs from 931, because a proton has a mass of about 1.008 amu.

I am sorry about the confusion produced by all these systems of units.  I left out, obviously, a large number of different things.  For example, when measuring luminous ⁶ energy, the lumen (lm) is used, which corresponds to about 1.5 mW of power in the "most visible" light, around 5500 Å (ångströms).  It's all very annoying, but don't worry about it now.  When you need to measure light, just look up in a book what a lumen is.

That's an unfortunate fact that we measure things in a whole series of different kinds of units.  This causes a lot of confusion.

It's too bad, but I have already apologized, and there is nothing else I can do...

1. In 1935, the thermochemical calorie was defined as exactly equal to 4.184 J. Historically, the calorie used to be defined as the energy needed to raise a gram of water by 1°C, just like the Btu (British thermal unit) was the energy to raise a pound of water by 1°F.  Such definitions depend on the initial temperature and thus gave rise to a multiplicity of units with the same name.  Also, using a finite increase in temperature (like 1°C or 1°F) in either definition made the factor of 1.8 reflect only approximately the effect of switching from the Fahrenheit to the Celsius scale (the factor is exact for the slopes of the relevant curves, or for equal temperature increments, not for different finite increments, no matter how small).  This is why the calorie is now defined precisely with the above mechanical equivalent in joules.  Unfortunately, this definition competes with the 1956 IST definition discussed below.  The trend seems to be that, unless otherwise specified, the calorie is to be understood as having the above thermochemical definition, whereas the Btu has the IST definition given in the following note.
    
2. The 5th International Conference on the Properties of Steam (London, July 1956) adopted a set of mechanical equivalents to thermal units, now identified by the abbreviation IT or IST (International [Steam] Tables), which made 1 Btu/lb exactly equal to 2326 J/kg.  One kilocalorie per kilogram is 1.8 times this ratio, because there are 1.8°F in 1°C (this factor applies to ideal units although the actual energy needed to raise a given mass of water, say, 10°C may be slightly different from 1.8 times what's needed to raise it 10°F).  An IT calorie is thus exactly 4.1868 J.  When the avoirdupois pound was finally defined in metric terms as exactly 0.45359237 kg  (effective January 1, 1959), the IT Btu became equal to exactly 1055.05585262 J (namely, 2326 J times the ratio of the pound to the kilogram).  The rarely used "centigrade heat unit" (chu) is 1.8 Btu.
    
3. The definition of the candela (adopted by the 16th CGPM in 1979) makes the "Mechanical Equivalent of Light" exactly one watt per 683 lumens (at 540 THz).
    
4. The speed of light in a vacuum (c) is Einstein's constant (the name was advocated by Kenneth Brecher of Boston University, in April 2000).  Because of the (fifth and final) definition of the meter adopted by the 17th CGPM in 1983, Einstein's constant is now exactly equal to 299792458 m/s.
         In 1948, the definition of the ampere by the 9th CGPM gave the magnetic constant mo (the permeability of the vacuum) an exact value of 4p 10^-7 H/m. (One henry per meter is the same as one newton per square ampere.)
         The second has been defined in absolute terms since the 13th CGPM, in 1967.  (The previous astronomical definitions are obsolete.)
         The final fundamental step in the construction of the SI system would be to redefine the kilogram by giving yet another fundamental constant an exact value.  For this transition to happen seamlessly, current precision on the known value of such a constant should be about as good as the precision of comparisons with the international prototype of the kilogram.  We are almost there:  A large device (2 stories high) called a watt-balance compares a mechanical watt (proportional to a calibrated mass) to an electric watt, known in terms of the Planck constant.  The best watt-balances have already determined Planck's constant [namely 6.626 068 76(52) 10^-34 J/Hz] to an accuracy (1 s) of about 0.078 ppm, which is within a factor of 2 or 3 from the best mass comparisons.  The end is in sight.

The Feynman Webring Next Site Previous Site List Sites Join

We are proud to have this page belong to the Feynman Webring, which connects a number of fine pages which perpetuate the legacy of Richard Feynman in various ways.  Some are more controversial than others:

At this writing, the next site in this ring happens to feature an essay where James G. Gilson presents his own formula (involving two integer parameters) for the value of the so-called Fine-Structure Constant  a = 0.007297352533(27) [whose reciprocal is 1/a = 137.035 999 76(50)].  This dimensionless fundamental constant of Nature was apparently first introduced in 1915 or 1916, by Arnold Sommerfeld (1868-1951).

Sommerfeld's Fine-Structure Constant  may be viewed as the only free parameter in QED, the relativistic quantum theory of the interactions between electrons and photons (for which Feynman, Schwinger and Tomonaga shared the 1965 Nobel Prize).  In QED, the coupling constant's effective limit is simply the square root of a, and Feynman was understandably annoyed that QED was thus based on an unexplained numerical value.  He expressed the wish that a deeper understanding of Nature would eventually explain that value and/or allow it to be expressed in terms of known constants, like p or e.

Before and after Feynman, others have tried to guess such a relation, possibly hoping that it could be a clue to whatever more general theory may underly our current understanding of the laws of Nature.  Around 1923, Sir Arthur Eddington (1882-1944) "proved" a to be 1/136, in agreement with early rough estimates.  When subsequently faced with incompatible experimental data, he amended the "proof" to show that the value had to be 1/137, so that Punch magazine dubbed him Sir Arthur Adding-One.  See 137 by Charles C. Mann, or look up the description by Robert Munafo of the so-called Eddington number [the outrageously asserted total number of electrons in the Universe, as the inverse of the fine structure constant multiplied by a power of two].

The two integers in Dr. Gilson's formula may be tuned to accurately reflect modern experimental data: The pair (137,25) was the best match for the midrange of the previous (1986) CODATA value of a [namely 0.00729735308(33)] and (137,29) is a good match for the current one (CODATA 1998, as of 2002).  Interestingly, Gilson quotes Michael Wales, who had argued that the cube of a should be the reciprocal of some integer (namely 2573380).  The 1986 CODATA value of a placed Wales' number at 2573379.99(35), which encouraged the conjecture.  However, the more precise 1998 CODATA update gives 2573380.571(29), which does not stand any reasonable chance of being an integer!

Gilson's formula (and its justifications) may be intriguing at first, but Gilson is [at best] guilty of wishful thinking when he presents his formula as a generally accepted fait accompli.  This is simply not so.  You've been warned; you may proceed.

2005

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