Entropy as Disorder: History of a Misconception

By “uniform” I mean “uniform on a macroscopic scale.” All phases of all materials are of course granular on the atomic scale.

Todd P.

 

Silverstein

, “,”  , – ().

Thomas A.

 

Moore

and

Daniel V.

 

Schroeder

, “,”  , – ( ).

D. F.

 

Styer

, “,”  , – ( ).

“Wheeler’s first moral principle. … Guess the answer to every puzzle. Courage: no one else needs to know what the guess is. Therefore make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise.”

Edwin F.

 

Taylor

and

John Archibald

 

Wheeler

, , 1st ed. (, , ) p. ; 2nd ed. (1992), p. 20.

This conclusion cannot be avoided by invoking “the law of large numbers” or “the thermodynamic limit.” For if we had a million times more squares scattered on a million times more sites, then the configurations in Fig. 2 would be absolutely typical for a window containing one-millionth of the total sites.

A more extreme example comes from family size. In the 2010 census, the average American family had 3.14 members—π to three significant digits—despite the fact that no family whatsoever had 3.14 members. See Table II in

Daphne

 

Lofquist

,

Terry

 

Lugaila

,

Martin

 

O’Connell

, and

Sarah

 

Feliz

, “,” , issued April , at http://www.census.gov/prod/cen2010/briefs/c2010br-14.pdf.

See, for example,

Ihsan

 

Barin

, (, , ).

Richard P.

 

Taylor

,

Adam P.

 

Micolich

, and

David

 

Jonas

, “,”  , ( 3, ).

Parker

 

Tyler

, “,”   (), – ( ).

Ray Nelson

 

Faulkner

, (, , ), p. .

Stephen G.

 

Brush

, (, , ).

Given the vagueness of the word “disorder,” we must ask the exact meaning of the phrase “a gas at equilibrium is disordered.” First consider an ideal gas subject to no external forces: An atom is equally likely to occupy any position within its container, and pairs of atoms are uncorrelated. The velocity of the atom is equally likely to point in any direction. But the speed of the atom is very much not “equally likely to have any value”: indeed, the probability of the atom having speed between v and v + dv is given by the Maxwell-Boltzmann distribution $2π(mkBT)3v2e−mv2/2kBTdv.$ In contrast, for a real gas subject to no external forces, an atom is equally likely to occupy any position within its container, but pairs of atoms are correlated. (See for example David Chandler, Introduction to Modern Statistical Mechanics [Oxford University Press, New York, 1987], Chap. 7.) The velocity remains equally likely to point in any direction, but the speed probabilities do not follow the Maxwell-Boltzmann distribution (which is derived assuming that the only energy is kinetic).

Sadi

 

Carnot

, (, , ).

Rudolf

 

Clausius

, “Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,” read at the Philosophical Society of Zürich on April 24, 1865. (Translated by John Tyndall as “On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat” in [, , ], pp. –, at p. 357.)

Ludwig

Boltzmann

, “,” , , – ().

(Translated by

Stephen G. Brush

 

as

“Further Studies on the Thermal Equilibrium of Gas Molecules” in [, , ], pp. –; excerpts quoted here appear on pp. 262, 263, and 291.)

Perhaps because he was at the time engaged in a priority quarrel with Clausius. See

Giovanni

 

Gallavotti

, “,”  , – ().

Hermann

von Helmholtz

, “,” , , ( 2, ).

(Translated as “The Thermodynamics of Chemical Processes.” See

Ralph

Baierlein

and

Clayton

Gearhart

, “,”  , [ ].

See also the translation by

E. F. J.

 

Love

in [, , ], pp. –, at p. 56.)

Satyendra

 

Ray

, “,”  , – ().

Karl K.

Darrow

, “,”  , – ( ).

Jeffrey A.

Phillips

, “,”  , – ( ).

See the appendix at TPT Online, http://dx.doi.org/10.1119/1.5126822 under the Supplemental tab for 16 related critiques with publication dates between 1944 and 2016.

R. H.

 

Fowler

and

E. A.

 

Guggenheim

, (, , ) does not mention disorder at all, but remarks that the related concept of absolute entropy “has caused much confusion and been of very little assistance” (p. ).

Henry

 

Adams

, (, , ).

John

 

Adams

, “” ( 3-4, 1770), see http://founders.archives.gov/documents/Adams/05-03-02-0001-0004-0016.

Reference 22 at p. 451.

Henry

 

Adams

, “The Rule of Phase Applied to History,” written Jan. 1, 1909, published on pp. 264–311 of (, , ), excerpts quoted here appear on pp. , 305, and .

Henry

 

Adams

, “A Letter to American Teachers of History,” written Feb. 16, 1910, published on pp. 134–263 of (, , ), excerpts quoted here appear on pp. and .

Gilbert L.

 

Wedekind

, (, , ), p. .

Using the rusting of iron to illustrate entropy arguments is particularly common and particularly specious. The rust reaction is 4 Fe + 3 O₂ → 2 Fe₂O₃. Anyone can look up the entropy values of iron, oxygen, and iron oxide at room temperature and atmospheric pressure. (See Ref. 8, pp. 675, 1239, and 702.) The entropy of the reactants is 724.6 J/K, the entropy of the products is 174.8 J/K. The rust reaction results in an entropy decrease! Of course, this doesn’t mean that when a car rusts, the entropy of the universe decreases: although the entropy of the iron plus oxygen decreases, the entropy of the surroundings increases by even more. But the same observation holds for the reverse reaction: in a blast furnace, oxygen is stripped from iron oxide to produce elemental iron. In this process, too, the entropy of the universe increases.

Abigail

 

Van Buren

, “,’” ( 31, ), p. .

This statement was particularly damaging to me, because I read it the summer after I was introduced to the concept of entropy in my high school chemistry class. My understanding of entropy was weak at the time, and after reading this statement by S.A.S. I felt that I would never grasp it. How could morality be related to the entries in a table of entropy values? How could morality be measured in units of joules/kelvin? Little did I realize, at age 15 years, that the weakness lay in S.A.S., not in me.

Alessandra

 

Stanley

, “,” ( 18, ).

, “,” ( 6, ).