Entropy Sites — A Guide

The content of this Web site has been selected for instructors in general and physical chemistry by Dr. Frank L. Lambert, Professor Emeritus (Chemistry) of Occidental College, Los Angeles (professional biography). It consists of copyrighted articles from the Journal of Chemical Education and the Chemical Educator that deal with a modern view of entropy change: a measure of how widely the energy in a process is dispersed or spread out in space or phase space (at T). Considerable non-published supplementary material concerning entropy and teaching it to beginners is also included.

Note: if you are using Mozilla/Firefox and having trouble viewing some figures on this site, please try another web browser.

 

January 2010

Entropy Re-mystified?

I have finally read a book that I was told had become popular, “Entropy Demystified”.  It is a 217-page disaster to anyone wanting to understand entropy and the second law.  Most of the lengthy evaluations that praise the book on Amazon.com seem to have been written by the author's best friends, several being mature physicists – rather than by persons trying to understand entropy for the first time.  The following is more objective than those "reviews":  giving it a rating of “no stars” out of a possible five.  (However, Amazon.com, for a reason you might guess J, increased that rating to “one star”.)

Fifty years ago, Arieh Ben-Naim, as every student in a physics or chemistry class of that era, was mystified by his introduction to entropy and the second law of thermodynamics.  Although he became a professor of chemistry at the University of Jerusalem before retiring 15 years ago, Ben-Naim has evidently not kept up with the teaching of those topics in current chemistry texts.  Thus, he seems unaware that most general chemistry texts currently published in the US (16) and three in physical chemistry now clearly and simply present entropy and the second law (Check “May 2009” in this website).

Therefore, his 217 pages of “Entropy Demystified” that are necessary to develop his personal viewpoint (an information theory variant, not present in any US undergraduate chemistry textbook) can be clarified by 3-4 pages in each of the chemistry texts listed in this site at “May 2009” with their ISBN numbers.

In fact, a conceptual summary of the second law and entropy for all chemistry students and many non-scientists can be abstracted from these texts in two sentences:  “Energy of all types in chemistry changes – if it is not hindered – from being localized in one volume to becoming more dispersed, spread out, distributed in space (and abstractly at one instant, in any one of many more energy quantum states, microstates, than were accessible before the change).”  Then, “entropy change is the quantitative measure of how much more widely distributed the initial energy becomes in a spontaneous process in chemistry.”  Thus, in real processes, energy literally spreads out in space, and abstractly, at each instant, is in one microstate of a maximally probable number of quantum states (microstates) that are consistent with a final macrostate at equilibrium.

Unfortunately, Professor Ben-Naim’s fundamental error, summarized on page 204 but vitiating all previous pages, is his misinterpretation of what happens in real systems of molecules, especially in the simple isothermal expansion of ideal gases or in their mixing or expansion.  These cases have misled him to focus on the lack of change in the total energy of the system, rather than on what is actually the fundamental cause of all thermodynamic entropy change in chemistry:  the increased spreading of the initial energy of actual molecules in space when constraints are removed – e.g., their spontaneously moving into a greater volume from a smaller volume (with unchanged energy) in a process such as expansion or mixing.  This is what traditional thermodynamic entropy readily measures and, as just stated, can be readily understood.

Ben Naim admits, in italics, the disconnect between information and the second law on page 203 of “Entropy Demystified” by writing “a measure of information cannot be used to explain the Second Law of Thermodynamics.”  This is true, indeed. The connection between the second law and information is tenuous. 

Contrast this with the modern view in beginning collegiate chemistry texts, e.g. “whenever a product-favored chemical or physical process occurs, energy becomes more dispersed...This is summarized in the second law of thermodynamics, which states that the total entropy of the universe ... is continually increasing.” (Moore, Stanitski, and Jurs; 3rd edition.)  A popular physical chemistry text that is used world-wide states “...the Second Law of thermodynamics, may also be expressed in terms of another state function, the entropy, S. ...entropy...is a measure of the energy dispersed in a process...” (Atkins and de Paula, 8th edition.)

The connection between the second law, spontaneous chemical reactions or physical processes, dispersal of energy, and entropy is integral, tight, and widely accepted in chemistry books.  It does not require 200 pages for its justification.

August 2009

In “what’s new” for August 2007 I described my article that showed how texts that introduced ‘positional’ (configurational) entropy to students would totally mislead them:  beginners are taught that “matter tends to become dispersed” and that there are two “types” of entropy rather than one.  Equally disastrous to students’ understanding is a focus on the ‘probability’ of molecules’ positions as the sole factor in entropy increase.

[Entropy increase is first enabled by molecular motional energy (rapidly moving or vibrating molecules); only then is entropy increase actualized by the probability of a maximal dispersal/distribution of that energy – in space, within each microstate of a greater number of accessible microstates.]

A far more fundamental article by Professor E. I. Kozliak has just been published in the September issue of the Journal of Chemical Education, “Overcoming Misconceptions  about Configurational Entropy in Condensed Phases”.   (He had previously resolved the old problem of incorrectly understanding “residual entropy” as simply due to molecules’ locations in space.)

May 2009

A minority of US general chemistry texts for majors still describe entropy in terms of “disorder” – an unfortunate subjective concept whose source appears to be a naïve statement by Boltzmann (http://entropysite.oxy.edu/boltzmann.html).  Now, however, most  ‘gen chem’ texts have discarded this non-scientific view and describe both entropy (e.g. standard molar entropy) and entropy change as measuring the result of energy becoming dispersed in physical or chemical processes – literally spreading more widely in space, while abstractly dispersing on additional energy levels in a conventional “particle in a box” diagram of one microstate.  (The latter, of course, then directly implies a greater number of microstates, W, in any final macrostate.)

It was eight years ago that the ms. outlining the above approach was accepted for publication (that now, revised and corrected, is available at this site: http://entropysite.oxy.edu/entropy_is_simple/index.html.  Accordingly, it is appropriate  that a list of ‘non-disorder’ texts, including physical chemistry as well as texts for non-majors, with their updated editions and ISBN numbers, be assembled from the scattered references in this website over the past years.

April 2009

There have been some noteworthy improvements in texts’ treatment of entropy in terms of energy dispersal.  A few will be mentioned here.  In May will be listed the 21 chemistry texts that no longer define entropy as “disorder” but rather emphasize molecular energy dispersal, concretely in space or abstractly on more energy levels in each microstate, as a useful approach to understanding standard entropy and entropy change.

Physical Chemistry

• Levine, in his new 6th edition of “Physical Chemistry”, well develops entropy, S, as a measure of the probability of a thermodynamic state.  On page 101, he states “…order and disorder are subjective concepts, whereas probability is a quantitative concept.  It is therefore preferable to relate S to probability rather than to disorder.”   Then, he summarizes,  “…it is the distribution of energy (which is related to entropy) that determines the direction of spontaneity.”

  The final sentence of the preceding section is ““The website entropysite.oxy.edu contains several articles criticizing the increasing-disorder interpretation of entropy increase and promoting the increasing-dispersal-of-energy interpretation.”  

  To be cited by Levine is indeed an honor.

General Chemistry

• McMurry and Fay have completely discarded “disorder” from the 5th edition of their text.  However, they emphasize randomness as the key to understanding entropy rather than a start to that goal:  It is energy that is being carried by randomly moving molecules as far as they are allowed to move that results in a probable energy distribution, a final equilibrium state.

• Chang, in his 10th edition of “Chemistry” has also eliminated any mention of “disorder” in connection with entropy in this most recent of a distinguished series.

• Kotz, Treichel and Townsend in their 7th edition have markedly improved their previous introduction of spontaneity and entropy, with an even more clear and complete exposition.  They surpass several authors who still separate the dispersal of energy from the dispersal of molecules.  Matter, in chemistry, never spreads out without the intervention of energy (macro matter), unless it is a carrier of energy (molecular matter above 0K).

General Chemistry for non-majors

• Hill, Kolb, and McCreary in their 12th edition of “Chemistry for Changing Times” have now adopted their introduction of entropy as “a measure of the dispersal of energy in a system…” with no reference whatsoever to ‘disorder’.

  
  

March 2009

• Unfortunately, I had not seen a copy or excerpts from the 2008 2nd edition of Gilbert, Kirss, Foster and Davies “Chemistry: The Science in Context” until last week. The authors have remarkably improved their treatment of thermodynamics in this edition. Not only have they completely eliminated any mention of “disorder” in connection with entropy, but their presentation of spontaneous processes and entropy is unusually well done.
  
  Admittedly from anecdotal evidence, but not only from my geographical area, I am becoming convinced that the majority of students in general chemistry classes are ‘overtaught’ about microstates. In contrast to those texts emphasizing ‘positional entropy’ and spending an inordinate amount of time and space on probability to introduce microstates, Gilbert and his collaborators directly present energy levels and simple statements about arrangements on energy levels as their four-paragraph intro to the Boltzmann equation. This is characteristic of their entire thermo chapter — well-presented and sufficient — but not too much.

 

January 2009

• A standard molar entropy, S⁰, is the absolute (i.e., not ‘relative’) measure of the entropy of a substance at 298.15 K.  It is extremely important because these S⁰ values are essential for so many areas of chemistry and chemical engineering to predict the outcome of reactions via the Gibbs equation, its most simple form being  ∆G⁰ = ∆H⁰_formation - T∆S⁰.

  The January 2009 issue of the Journal of Chemical Education contains an unusually important article by physics professor Harvey S. Leff and me — important because it shows a surprisingly simple linear correlation.  The S⁰ of 77 solid substances, widely differing in type  —  e.g.,from Cu to NaCl to UF₄ to C₂₀H₄₂ —  and the quantity of enthalpy, ∆H⁰, added to them while they were heated from 0 K to 298.15 K are closely linked.
  
  The equation is:  S⁰ = 0.0066  ∆H⁰ /K    Notably, ∆H⁰  ≈ U⁰, the internal energy stored at 298.15 K.    (The deviations of “outliers” from the 0.0066 line are readily explained in the Supplement.)
  
  
  There is no “mystery” to entropy here, no place for old subjective terms like “disorder”, or such new terms as “missing information” ! Entropy in this most fundamental case is simply a measure of the spreading of energy within a cooler substance from its warmer surroundings when it is heated —  starting with the cooler substance near 0 K and summed up, degree by degree, to 298.15  K, the ‘standard’ temperature.
   
  The amount of spreading/energy dispersal depends upon the material in which the spreading occurs and the amount of energy added to it.

  This is indeed a nice corroboration of what I wrote in 2002, “The entropy of a substance at any temperature T is not complex or mysterious.  It is simply a measure of the total quantity of energy that had to be dispersed within the substance from 0 K to T so the substance could exist as a solid or liquid or gas at the designated temperature.”

   

November 2008

• A fundamental article about “residual entropy” by Professor E. I. Kozliak and me is now available here.  Residual entropy, the entropy remaining in crystals of some compounds even near absolute zero, e.g. CO, N₂O and H₂O, has traditionally been accounted for (and counted) just from the number of different molecular arrangements in the crystal. From this, chemists have felt that entropy can depend only on a “position in space”, regardless of a substance’s internal energy.  Unfortunately, residual entropy has been the ‘poster boy’ of an example of “positional entropy”.
  
  In a previous article, Kozliak showed that the results from counting procedures in residual entropy calculations are identical to those from considering the different forms on different energy levels — a novel focus on entropy values as related to energy distributions in substances like CO, rather than on the  number of arrangements of the molecules, their  “positions in space”.
  
  In this 2008 article, Kozliak shows how residual entropy is inherently coupled to the corresponding latent heat which would be released upon cooling if a reversible path were available.  That such a process is not fanciful for CO or N2O (although not yet achieved) is illustrated by the actual release of energy at very low temperatures by ice, when a small amount of KOH is added as a ‘catalyst’ and ice XI is formed that has no residual entropy.

August 2008

• In the January 2008 Journal of Chemical Education Dr. Frank B. Ellis and David C. Ellis published details of the construction and applications of a shaking apparatus with chambers in which beads can be used to analogize the behavior of molecules.  They described its versatility in demonstrating/analogizing 'molecular behavior' in equilibrium, bond energy, endothermic reactions and other areas, including entropy.  In describing entropy, they referred to an older text that stated "entropy is disorder".  However, when I wrote them about most modern texts having changed to say that entropy increase should be viewed as an increased spreading out of molecular energy, they agreed to a Letter to the Journal which corrected the obsolete statement about entropy. The Letter appeared in the just-published September issue of the Journal.

July 2008

• For the 20th Biennial Conference on Chemical Education held in July on the Indiana University campus, one of the symposiums was organized by Dr. Barbara Murray of the University of Redlands with the theme of "Engaging Students in Organic Chemistry".  There were a number of speakers and in her symposium Dr. Murray was kind to distribute copies of the following:
  
  The funniest lecture about chemical education (that could have been titled "Why Do Chemistry Profs Still Lecture, Since Gutenberg Made Printed Textbooks Possible?") was given by Professor Robert T. Morrison, co-author of the famous organic chem text, Morrison and Boyd.  We all know about the book that changed O-chem texts forever.  However, few instructors have ever heard of Morrison's 1986 lecture at the U of Chicago that applies to all beginning classes in chemistry, and perhaps even to physical chem.
  
  
• Details of my 'discovery' and the most effective implementation of the 'Gutenberg Method' are here.

June 2008

• I failed to announce a most important publication about entropy, “Entropy, Its Language, and Interpretation”.  It was written by Professor Emeritus of Physics Harvey S. Leff (of the California State Polytechnic University, Pomona) and appeared in Foundations of Physics, 2007, 37 (12), 1744-1766.  The abstract is available here and, of course, the full article is in the journal or online through one’s institutional library.

  As indicated by its length and the eminence of the journal, this article is both thorough and theoretically sound in presenting the increase in entropy as fundamentally a spreading of energy spatially during processes and its greater temporal spreading over accessible microstates after such an entropy increase.

  “Temporal spreading” means that a system’s total energy (e.g., that of an ideal gas) in a single microstate at one instant in time — a particular arrangement of its molecular energies on quantized energy levels —  will change, in the next instant of time, to a different arrangement/microstate.  Then, if after a process there is a greater number of accessible microstates than before, a few or many different microstates are more likely to be reached from an initial microstate in a second of time or so over which a measurement is made.  It can be seen that this is a “greater temporal spreading of the system's energy” in the sense of a “temporal dance” over the same number but probably a few or many more different microstates than prior to the process. 
            The notion of temporal speading that increases with the number of accessible microstates (W) is consistent with, and provides a new interpretation for the Boltzmann entropy, S = k_B ln W, where k_B is Boltzmann’s constant.
            It is appropriate to see entropy’s symbol S as shorthand for spreading, a coincidental memory aid for beginning students.

  Professor Leff published the first article that quantitatively presented the idea of entropy involving the dispersal or spreading of a system’s energy, with entropy as a measure of such spreading, in 1996 (Am. J. Phys. 1996, 64, 1261-1271).  I inserted a Note to it in my “entropy is a cracked crutch” ms. in 2001 while giving more proper credit as the lead reference in my next article, “Entropy is Simple, Qualitatively”.   When the focus is on energy spreading out — or dispersing as I happened to choose before reading his article, entropy can indeed be seen correctly as being relatively simple:  It is a passive measure/’meter stick’ both of how much energy has spread (and how it has spread, i.e., at what heat capacity and at what temperature(s)) and how widely in space it has spread out (in isothermal processes such as gas expansion or in mixing).

  I am indebted to Professor Leff for aid over the past six years.  We have just collaborated on a very important article (to be published within the next five months) that should convince the last doubters of the value of our viewpoint.

April 2008

• Joshua Floyd, with a background in engineering and now teaching at the Australian Graduate School of Entrepreneurship in the Swinburne University of Technology, Victoria, Australia, has published a brilliant and scholarly article, "Thermodynamics, entropy and disorder in futures studies". Its abstract is below. (The article is accessible via college or university library e-journal services.) He discusses the manifold misuse of the second law of thermodynamics over many years and by many economists, IT specialists, serious professionals involved in futures studies, and not-as-serious popularizers. The major error in most of their writing is an assumption that this law, derived from studies of the energetic behavior of atoms and molecules, applies to “order or disorder” in the material objects with which we all must deal

  Futures  39 (2007) 1029-1044. Abstract

  The conceptual bases of futures studies are constrained by physical reality only to the extent that we construct these according to our best understanding of physical principles. This places a burden on futures practitioners to ensure that engagement and use of these principles is sufficiently robust to protect the plausibility of their work. The second law of thermodynamics is widely recognized as having fundamental implications for the nature of our physical reality. It is also widely misinterpreted, leading to distorted understanding of this reality. Thermodynamic principles are frequently referred to in the futures literature, and are sometimes fundamental to the futures thinking underlying the work. Reflecting the widespread misunderstanding of the second law, usage in the futures literature is usually problematic. This has implications for the value of the work, and also for the credibility of the field. In this article, the problem is demonstrated, and an updated interpretation of the second law is introduced. The origin of the problem is examined from historical and scientific perspectives within the thermodynamics field. The updated interpretation's implications are examined in the context of futures and other transdisciplinary perspectives.

  Article Outline

  1. Introduction: thermodynamics in futures studies
  2. An introduction to the problem, and to a response
  3. Understanding the problem: some history and science of thermodynamics
  4. Reinterpreting the second law: the futures studies context
  5. Conclusion
  References

   

October 2007

• In his new book “Notes From the Holocene (A Brief History of the Future), Dorion Sagan, science author and eldest son of the late Carl Sagan, writes:

  “Publications in the Journal of Chemical Education by Occidental College professor emeritus Frank L. Lambert have drawn attention to the century-old non-scientific mantra, entropy-as-disorder, altering the modern textbook landscape.  [By April 2007, authors of eighteen chemistry textbooks] had deleted their previous identification of entropy as “disorder”.  The new texts describe the second law as being fundamentally a matter of energy dispersal.  If energy is not hindered, it spreads out.”

  Invited to speak as a registrant at the International Thermodynamics Symposium at MIT on October 4-5,  Sagan opened his remarks in the Session, “Foundations of the Second Law”, with the above quotation from his book.

August 2007

• The September issue of the Journal of Chemical Education has my article which shows why the statement in some texts (e.g., those numbered 11-13 in December 2005) that "Matter tends to become dispersed" is seriously misleading, It is a misinterpretation of a result from configurational (positional) entropy — itself a viewpoint that should not be introduced to general chemistry students because they then believe that there are two "types" of entropy rather than one. The article concludes with the important statement, "Two factors are necessary for entropy change in chemistry. An increase in thermodynamic entropy is enabled in a process by the motional energy of molecules (that, in chemical reactions, can arise from the energy released from a bond energy change). However, entropy increase is only actualized if the process results in a larger number of arrangements for the system's energy, that is, a final state that involves the most probable distribution for that energy under the new constraints"

June 2007

• In the July issue of the Journal of Chemical Education, Dr. M Sözbilir reports the results of an unusually detailed survey of individual Turkish undergraduates’ understanding of entropy. Not only had all the students taken a first course in chemistry, but all of them in these physical chemistry classes were being trained to be chemistry teachers. However, the basic conceptual model in both of those courses was that an increase in thermodynamic entropy is related to an increase in disorder.  Thus, it is hardly surprising that  “[s]tudents’ understanding of the basic aspects of [entropy] are—in many cases–limited, distorted or wrong.”  Dr. Sözbilir’s findings are supported by his references to similar conclusions from Scotland, South Africa, and Portugal.  Clearly, students everywhere need modern texts that present the concept of entropy as a measure of the dispersal of energy within a system or between the system and its surroundings!  (See the 1898 origin of the confusing idea of ‘disorder’ being associated with entropy here.)                       

April 2007

• In addition to the 16 chemistry textbooks described in March 2006 and December 2005, two more texts now introduce entropy in terms of energy dispersal within a system or in the universe of a system plus its surroundings.

  17. A thorough new general chemistry text by Tro (Prentice-Hall, 2008) introduces the concept of entropy early in his presentation of solutions.  He describes the mixing of two gases, initially separated, as due to the kinetic energy of each kind of molecule becoming more spread out in the final larger volume. Immediately, he mentions the parallel of the thermal energy in one end of an iron bar inevitably becoming dispersed throughout the bar. In his later thermodynamic chapter, he mentions the possibility of considering the criterion for spontaneous chemical change as involving “disorder”.  However, in the next sentence he discards that possibility by defining and consistently using “Entropy (S) is a thermodynamic function that increases with the number of energetically equivalent ways to arrange the components of a system to achieve a particular state.”
  18. In the 7th edition of “General Chemistry” by Whitten, Davis, Peck, and Stanley (8th edition, Brooks/Cole, 2007) entropy change was described only in terms of “disorder”. I have not seen the new edition, but the publisher states, “The discussion of entropy in Chapter 15 has been expanded to include not only molecular disorder but also the concept of energy dispersal.”  In my admittedly biased opinion, that is a bit like saying “the reactions with oxygen and a metal have been expanded to include not only phlogiston but the formation of metal oxides with the evolution of heat.”  Nevertheless, it is indeed a step forward in describing entropy scientifically.

• The new 9th edition of “General Chemistry” (Prentice-Hall, 2007) by Petrucci, Harwood, and Herring continues the superb compact presentation of the previous edition,“…the way in which the energy of a system is distributed among the available microscopic energy levels is called entropy,” is immediately followed by the Boltzmann entropy equation, its relation to numbers of energy levels, and unusually clear depictions of levels in different ‘boxes’ and in a heated system.

March 2007

• " 'Disorder' in Thermodynamic Entropy", the historical origin of the introduction of 'disorder' by Boltzsmann, is now available here in response to many questioners.  The brief article also is an introduction for instructors who are not familiar with my approach to understanding entropy change. It closes with a description of the clear distinction between thermodynamic entropy and Shannon information "entropy".
  
  
• Professor E. I. Kozliak has published a noteworthy article in the March issue of the Journal of Chemical Education in which he resolved the old problem of understanding "residual entropy" , the entropy remaining in crystals of compounds such as CO, N₂O, FClO₃ and H₂O even as they approach absolute zero.  The entropy present in two or more arrangements of molecules in such crystals had only been considered in terms of "configurational" or "positional" entropy. Kozliak shows that the counting procedures in these entropy calculations are identical to what would result from considering the different forms on different energy levels -- a considerably more fundamental focus on entropy values as related to energy distributions.

November 2006

• A rather remarkable article by Arnd Jungermann, a secondary school educator in Germany, is in the November issue of the Journal of Chemical Education. For a number of years, he has treated entropy as a physical property that measures the amount of energy stored in a substance. His use of simple shelves and calculation of the occupancy of energy levels by a few particles, leads students to the conclusion that entropy is a measure of the occupied (accessible) energy levels. Then, development of atomic entropy values from S/R, shows relations of entropy to trends in the periodic table. "The way in which a material stores supplied heat or work is characteristic of its internal structure and is reflected in its molar entropy value."

June 2006

• My two sections, "The Second Law" and "Entropy", for the thermodynamics chapter of the Wikibook "General Chemistry" have just been completed. Click on the titles to reach copies that are here in the Supplements.  Although both contain material that is scattered on this entropysite, it is presented in somewhat different format, designed to be readily readable by beginners in chemistry and accessible by students not majoring in science.
  
  
• The new "Physical Chemistry" (Benjamin Cummings 2006) by Engel and Reid, both of the University of Washington, wisely omits any mention of the word 'disorder' in its exemplary definition of entropy on a phys chem level: "Entropy serves as a measure of the number of quantum states accessible to a macroscopyic system at a given energy."

March 2006

• Atkins' "Physical Chemistry" has been the best selling text worldwide in this subject for many years.  The new 8th edition was published in the US March 16. However, in previous editions Atkins described systems and entropy change in terms of order and disorder or chaos.  Even though he long has used the phrase, "the dispersal of energy", it was confined to an order-disorder view of thermodynamics — for example, to spontaneous changes being "always accompanied by a dispersal of energy into a more disordered state".   (Or "to chaos" in his book "The Second Law".)

  In contrast to the Second Law chapter of the 7th edition, which had some 27 instances of using "order to disorder" as a rationale for change, "disorder" and "disorderly" are mentioned only 3 times in the new 8th edition.  Atkins, with co-author dePaula, now state that their view of entropy "summarized by the Boltzmann formula is consistent with our previous statement [earlier in the chapter, re the dispersal of energy in classical thermodynamics] that the entropy is related to the dispersal of energy.

January 2006

• In 2005, the five sites about the second law and entropy change (see Links section) had a total of over 510,000 visitors and more than two million "hits".


• http://secondlaw.oxy.edu is now ranked #1 in the Google list of over 10 million sources under the search term thermodynamics, as well as being ranked #1 out of some 1,500,000 results in a Google search of "second law". (That it is also #1 out of an astonishing 201,000,000 results from a search for second law without quotes is less noteworthy for http://secondlaw.oxy.edu than for the apparent speed of Google in finding so many citations with both "second" and "law" in them!)


• http://2ndlaw.oxy.edu is ranked #1 in some 9,000,000 Google results for entropy.

December 2005

First-year college chemistry textbooks since about 1960 have used the 1898 description of thermodynamic entropy as “disorder”. In the February 2002 issue of the Journal of Chemical Education I showed that treating entropy change as “disorder” was not based on modern science and could mislead students. In the October 2002 Journal I urged that entropy be presented as the quantity of dispersal of energy/T or by the change in the number of microstates.

Textbooks do not alter their presentation of basic concepts readily nor rapidly. Thus, for the following 15 texts to delete “entropy is disorder” from their new editions within three years of my calling for such a drastic change is perhaps without precedent. Further, for all of them now to describe the meaning of entropy in various terms of the spreading or dispersing of energy (in some, quantified by Boltzmann's number of microstates) shows the utility of this concept in good teaching.

Textbooks for science majors

1. Moore, Stanitski, and Jurs' "Chemistry: The Molecular Science", whose first edition was the best-selling new text in a decade, has a 2005 2^nd edition (Thomson). The authors state that the new edition is improved because, among other features, the "...treatment of entropy in Chapters 14 and 18 has been rewritten to make it clear that entropy measures dispersal of energy" rather than "disorder". This text most thoroughly and most extensively applies my concept of "follow the energy flow" in aiding students to understand the concept of entropy.
   
2. Silberberg, in the 2006 4^th edition (McGraw-Hill) of his #1 or #2 best-selling "Chemistry" writes, "[The thermodynamics chapter] has been completely rewritten to reflect a new approach to the coverage of entropy. The vague notion of "disorder". has been replaced with the idea that entropy is related to the dispersal of a system's energy." and acknowledges my advice.
   
3. The 3^rd edition (Wiley, 2000) of Brady & Senese's "Chemistry" for science majors used "disorder"/order 65 times to describe entropy. However, in the 2005 4^th edition Senese told me that "disorder" is entirely omitted. In featuring their improvements for this edition, the authors state "We have changed our approach to presenting Thermodynamics... [by explaining] entropy as a measure of the number of equivalent ways to spread energy through a system."
   
4. Oxtoby, Gillis and Nachtrieb's 5^th edition of "Principles of Modern Chemistry" (Brooks/Cole) has removed any references to entropy as a measure of "disorder" that appeared in the 4 th edition. This text's relating of entropy increase to greater numbers of microstates as shown by the Boltzmann entropy equation is perhaps the most thorough in any general chemistry text.
   
5. Petrucci, Harwood and Herring in the 8^th edition of "General Chemistry: Principles and Modern Applications" (Prentice-Hall) have an unusually readable development of entropy as increasing when there are more microstates among which the energy of a system can be distributed. This is accompanied by a simple introduction to increased density of energy levels (and therefrom, more microstates) when the volume of a gas spontaneously increases.
   
6. The 2005 4^th edition of Hill, Petrucci, McCreary and Perry's "General Chemistry" (Prentice-Hall) still employs the word "disorder" in referring to entropy change in several places, but it is primarily as a bridge for those students who have heard the expression. Overall, the authors use my approach to entropy change as a dispersal of energy.
   
7. The 2005 8^th edition of Ebbing and Gammon's "General Chemistry" (Houghton Mifflin) includes some references to "disorder" in their treatment of entropy, but they emphasize that, fundamentally and scientifically, entropy involves energy dispersal as a function of temperature.
   
8. Ebbing, Gammon, and Ragsdale's 2006 (Houghton-Mifflin) "Essentials of General Chemistry" (785 pages rather than the 1200 in Ebbing and Gammon) has a similar treatment of entropy to the larger text, an emphasis on energy dispersal as essential to understanding entropy change.
9. Moog, Spencer and Farrell (Houghton Mifflin) have developed three paperbacks as a novel “Guided Inquiry” technique in areas of physical chemistry. Their 2004 “Thermodynamics” completely omits the references to “disorder” of “messy desks” in a previous trial edition and replaces them with viewing entropy as related to how energy can be spread out in a system.
10. A new text, “Physical Chemistry for the Life Sciences” by Atkins and de Paula (Freeman, 2006) omits the definition of entropy as disorder that was present in Atkins’ previous general chemistry and physical chemistry textbooks. Repeatedly, the emphasis in describing entropy change is on the dispersal of energy in the process. However**
11. The novel approach by Bell and his ten collaborators uses simple experiments or thought-experiments of “Investigate This” in developing concepts in “Chemistry: A General Chemistry Project of the American Chemical Society” (Freeman, 2005). Disorder is ignored as a definition or code word for entropy. Rather the student is led to consider arrangements of molecular energy in developing the Boltzmann relation. However**
12. The new 6th edition of “Chemistry and Chemical Reactivity” by Kotz, Treichel and Weaver (Brooks/Cole, 2006) have deleted their description of entropy increase as disorder that was in previous editions. They state that “spontaneous change results in dispersal of energy”. However**
13. Although previous editions of Olmsted and Williams “Chemistry” had 89 uses of “disorder” vs. “order”, including the definition of entropy, the 2006 4th edition (Wiley) defines entropy only in terms of energy dispersal. The word “disorder” is rigorously avoided in any context. However**
    
    **However, the preceding four texts each have the unfortunate concept of “the dispersal of matter” as though there were no motional energy considerations associated with such dispersal (as in gas expansion, or any type of mixing wherein the initial motional energy of the molecules becomes more widely dispersed in space). One even states that “Things tend to become dispersed.” The source of this error is dealt with here.
    
14. Brown, LeMay and Bursten's 2003 9^th edition (Prentice Hall) defined entropy only as "disorder". In a preliminary ms. of the thermodynamics chapter in their 2006 10^th edition all references to "disorder" were eliminated by one of the authors and the concept of energy dispersing or 'spreading out' more for increased entropy was used throughout. Although the published 10 th edition presents energy dispersal as a view of entropy, it includes the "extent of randomness" as equal, later stating “Each of these descriptions [of entropy] (randomness, disorder, and energy dispersal) is conceptually helpful if applied correctly.” This ‘trifecta' is an insurmountable challenge to beginning students who are readily confused even by a singular presentation of the concept.
    

Textbooks for non-science majors

15. The first edition of Suchocki's "Conceptual Chemistry" (Benjamin Cummings) introduced the second law as "Order Tends to Disorder". His 2^nd edition (2004) does so as "Entropy Is a Measure of Dispersed Energy"..."This fits with our everyday experience...." Then, with ΔS^o_React, Suchocki can lead even this group of students to understand the direction of chemical reactions.

 

... more in the news section

 

To aid students and others who have reached this Web site and who do not teach chemistry, here are links to reliable sites that introduce entropy in an easily understandable manner:

• Entropy for students in general chemistry is developed at http://secondlaw.oxy.edu. (In Q and A form, and built on a simple approach to the second law, http://secondlaw.oxy.edu/six.html is the specific page that introduces entropy — a measure of the dispersal of energy as a function of temperature.)
  
  
• The molecular approach to entropy for more advanced general chemistry students is at http://2ndlaw.oxy.edu/entropy.html.
  
  
• The shortest intro to the second law and entropy for students is in the supplemental material of this site, here.
  
  
• Adult liberal arts students who are somewhat curious about entropy should visit http://entropysimple.oxy.edu, "Entropy Is Simple, If We Avoid the Briar Patches!". Qualitative, emphasizing the generality of application of the second law to environmental phenomena (but including why cream mixes in coffee), this site is designed for general readers rather than the technical requirements of chemistry students.
  
  
• Adults who are in business, law, or other non-scientific fields can readily see the importance of the second law of thermodynamics in everyday life — that C. P. Snow never mentioned when he talked about "the two cultures" — via http://shakespeare2ndlaw.oxy.edu. The concept of entropy is not introduced.
  

The "news" section is for chemistry instructors. It reports recent developments related to the theme of this site: the spontaneous spreading-out of energy (if it is not hindered) as a key to understanding molecular behavior, entropy change, and the second law.

November 2005

• The Chemistry seminar on November 8 at the California State Polytechnic University, Pomona was  “A Conversation About Entropy”. It dealt with  the general topic of considering entropy as a measure of the dispersal of energy at T. A new subtopic was “Clarification of the meaning of `configurational' entropy: a measure of energy dispersal in statistical mechanics calculations”. A summary of that clarification is below.

August 2005

• The 2002 article in the Journal of Chemical Education, "Entropy Is Simple, Qualitatively" has been completely rewritten and expanded to amplify points that were novel at that time and to correct errors. The section describing microstates has new Figures and a far more detailed explanation.

July 2005

• The just-published book from the University of Chicago Press, "Into The Cool" by Dr. Eric Schneider and Dorion Sagan, has a subtitle that describes its theme: "Energy Flow, Thermodynamics, and Life". Emphasizing what lies behind the second law, energy's inevitable tendency to change from being concentrated in one place to becoming spread out [not just in space], Schneider and Sagan show that energy-gradients are vital to life. "[L]ife's increase in evolutionary complexity ...makes sense when we consider life along with other natural systems of energy flow." In describing how activation energies in chemical kinetics delay and obstruct energy flow and act as protective barriers to immediate second law action, the authors use my "Chemical kinetics firmly restrains time's arrow in the taut bow of thermodynamics for milliseconds to millennia" that corrects Eddington's overly simplified 1925 aphorism "The second law of thermodynamics is time's arrow ". (See article below.)

November 2004

• The November issue of the Journal of Chemical Education has two important articles using the “energy dispersal” approach to understanding entropy and one that is pertinent to the RT ln Q entropy function in the Gibbs free energy equation. (Also, in this issue is the formal letter to the editor that dealt with the article discussed in the “news” for April 2003). The article that upgrades the qualitative approach to entropy of this site to its quantitative application in physical chemistry by Professor E. Kozliak (University of North Dakota) is especially noteworthy.

  Their abstracts are presented as Numbers 6 – 8 of the articles section with links directly to the print version, courtesy of the Journal of Chemical Education.

October 2004

• For instructors in general chemistry, a detailed description of the dispersal of energy as fundamental in understanding the cause of entropy change — from gas expansion to chemical reactions — has just been completed. See "supplemental material" below.

May 2004

• A letter from Professor William Jensen, a chemistry professor and expert in chemical history, that supplemented and supported my approach to teaching entropy change appeared in the May issue of the Journal of Chemical Education. It is reproduced below under "articles".

March 2004

• At the National Meeting of the American Chemical Society, publishers exhibited new editions of their first-year college textbooks. The majority of them had discarded the definition "entropy is disorder" that was in previous editions and replaced it with a statement involving entropy as a measure of the dispersal of energy/T.
  

2003

Articles

• In the November JChemEduc, pp. 1271-74, Professor Robert Hanson (St. Olaf, MN) reports on “modeling” the equilibrium constant in a (catalyst-aided) isotope exchange equilibrium of H₂ + D₂ <-> 2HD with playing cards. He well develops concepts and details of the mathematical relation of probability to an equilibrium constant. However, as shown in “Shuffled Cards…” (Article 1 on this Web site), using playing cards to analogize chemical behavior can be misleading. Students and professors in the past often have not realized that cards cannot analogize the energetic movement of molecules (that leads to chemical equilibrium, or other results measured by entropy change) unless the cards also are constantly moving. In other words, only when cards are being vigorously shuffled do they become analogs of energetically colliding molecules. In a personal email, Professor Hanson expressed it concisely : “the shuffling [illustrates] the equilibrium, and counting [the probabilities from the card arrangements] is only taking snapshots.” I believed that his otherwise excellent article would be misunderstood by many readers because it may seem to them that chemical equilibrium is based on probability. However, probability only measures the results (“is a snapshot”) of the maximal spreading out of the energy of ceaselessly moving molecules. Card shuffling is enabling in   the analogy involving the probability of molecular equilibrium described in the article.

 

1. "Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms — Examples of Entropy Increase? Nonsense!"
   from the Journal of Chemical Education, Vol. 76, pp. 1385-1387, October 1999.

   Changes in the arrangement of ordinary objects do not change their entropy. Entropy depends on the dispersal of energy at a specific temperature, not on a  pattern. (Information "entropy" with no inherent or integral energy factor therefore is only related in form, and not in function, to  thermodynamic entropy that must have an enabling factor of energy.

   
   

2. "Disorder — A Cracked Crutch for Supporting Entropy Discussions" from the Journal of Chemical Education, Vol. 79, pp. 187-192, February 2002.

   "Entropy is disorder" is an archaic, misleading definition of entropy dating from the late 19th century before knowledge of molecular behavior, of quantum mechanics and molecular energy levels, or of the Third Law of thermodynamics. It seriously misleads beginning students, partly because "disorder" is a common word, partly because it has no scientific meaning in terms of energy or energy dispersal. Ten examples conclusively demonstrate the inadequacy of "disorder" in general chemistry.

   
   

3. "Entropy Is Simple, Qualitatively" originally published in the Journal of Chemical Education, Vol. 79, pp. 1241-1246, October 2002.

   Note: the article as presented here has been extensively revised and expanded, most recently in August 2005.

   Energy disperses from being localized to becoming spread out if it is not hindered. This is the enabling factor  for all spontaneous physical and chemical events. Entropy change measures the dispersal of energy in a process: how much is spread out or how widely spread out that energy becomes. This is discussed in terms of macro thermodynamics, q(rev)/T, and molecular thermodynamics, k_B ln [microstates_final / microstates_initial ].

   
   

4. " "Disorder" in Unstretched Rubber Bands?" from the Journal of Chemical Education, Vol. 80, p. 145, February 2003.

   The well known experiment of stretching a rubber band has often been used as an example of entropy increase toward greater "disorder" in the unstretched band as a cause  for a stretched band to contract. Instead, from a scientific point of view, the unstretched rubber has greater entropy than the stretched form because of the increased possibilities for energy dispersal among the more freely-moving portions of rubber molecules in unstretched rubber compared to an extended rubber band. Thus, spontaneously a stretched band will change to unstretched.




5. "Entropy and Constraint of Motion" from the Journal of Chemical Education, Vol. 81, pp. 639-640, May 2004.

   Professor William B. Jensen, a chemistry professor and historian at the University of Cincinnati, has independently developed an approach to teaching entropy that involves interpreting entropy change as a change in the dispersion of energy. His additional contributions are that only kinetic energy can become dispersed and that examination of the constraints to dispersion clarify how/what mode energy dispersion takes. In my response here, I call attention to the dispersal of kinetic energy to potential energy (due to bond breaking) at fusion and vaporization temperatures.

   
   

6. "Teaching Entropy Analysis in the First-Year High School Course and Beyond", Thomas H. Bindel, from the Journal of Chemical Education, Vol. 81, pp. 1585-1594, November 2004.

   A novel and creative 16-day teaching unit is presented that develops chemical thermodynamics at the introductory high school level and beyond — exclusively from  an entropy viewpoint referred to as entropy analysis. Many concepts are presented, such as: entropy, spontaneity, the second law of thermodynamics, qualitative and quantitative entropy analysis, extent of reaction, thermodynamic equilibrium, coupled equilibria, and Gibbs free energy. Entropy is presented in a nontraditional way, using energy dispersal.




7. "Introduction of Entropy via the Boltzmann Distribution in Undergraduate Physical Chemistry: A Molecular Approach", Evguenii I. Kozliak, from the Journal of Chemical Education, Vol. 81, pp. 1595-1598, November 2004.

   Several problems that hinder optimal communication with students in the conventional introduction to thermodynamics are identified. Even though students from their first course focus on chemistry as a molecular science, most texts in physical chemistry begin with the phenomenological Clausius formulation, thereby emphasizing its macroscopic aspect; the others concentrate on so-called "positional" entropy thus decoupling it from the entropy of heat exchange. The suggested approach uses simple examples based on the Boltzmann distribution to introduce the concept of entropy consistently on a molecular basis by emphasizing energy distribution due to the number of accessible microstates but bypassing the complexities of statistics. Thereby, a connection between the increase of entropy on expansion as well as on heating can be shown. A clear illustration is provided for the basic tenet of the second law, the spontaneous transfer of thermal energy from hot to cold bodies.




8. "The Concentration Dependence of the ΔS Term in the Gibbs Free Energy Function: Application to Reversible Reactions in Biochemistry", Ronald K. Gary, from the Journal of Chemical Education, Vol. 81, pp. 1599-1604, November 2004.

   Biochemistry students must use the concept of free energy change to understand reaction reversibility and the energetics of metabolism. The theory is founded on the Gibbs free energy function: ΔG = ΔH - TΔS.
   
   Reactant and product concentrations affect the ΔS term and therefore determine whether ΔG is positive or negative at a standard temperature. However, most biochemistry texts do little to connect the sign of ΔG in this function to the concentration variables that determine it, and instead rely exclusively on the equation to relate these parameters. This can have the undesirable effect of rendering the Gibbs equation irrelevant for these students. For the biochemistry instructor, the challenge is to clarify the role of entropy in determining reaction directionality without digressing into aspects of thermodynamic theory that would be more appropriately covered in other courses. A model to explain the concentration dependence of the ΔS term is presented in a format that is appropriate for an audience of biochemistry students, and the concepts are illustrated using an aqueous phase reaction, the anomeric conversion of glucose.




9. "Playing-Card Equilibrium", Frank L. Lambert, from the Journal of Chemical Education, Vol. 81, p. 1569, November 2004. Complete letter to the Editor reproduced below, with permission from the JCE.

   From experience, I am hypersensitive to the misconceptions of students and instructors that can be caused when playing cards are used in teaching chemistry (1). The root of such errors lies in overlooking the non-mobile, non-energetically-interacting nature of pieces of cardboard. Only while they are being shuffled can cards serve as some sort of analogy to molecular behavior in chemistry.
   
   Thus, I found Hanson's "Playing-Card Equilibrium" of special interest (2). To me, his otherwise excellent treatment of probability in relation to chemical equilibrium lacked emphasis on shuffling as a vital element in the analogy. However, in a personal email, Professor Hanson said that his experience with teaching teachers did not show that they overlooked the importance of constant shuffling to simulate the interacting state of molecular movement. His summary is my view also: "The shuffling illustrates the equilibration, and counting the probabilities from the card arrangements at any moment is like taking snapshots of that dynamic process."
   
   Literature Cited:
        1. Lambert, F. L. J. Chem. Educ. 1999, 76, 1385-1387.
        2. Hanson, R. M. J. Chem. Educ. 2003, 80, 1271-1274.




10. "``Order-to-Disorder'' for Entropy Change? Consider the Numbers!", Evguenii I. Kozliak and Frank L. Lambert, from The Chemical Educator (an Online Journal) 10 (2005) 1, pp. 24-25 © The Chemical Educator 2005.

    Click title above to download the article in Acrobat (pdf) format. Abstract is below. The text in brackets is an addendum to the original abstract.

    Defining entropy increase as a change from order to disorder is misleading at best and incorrect at worst. Although Boltzmann described it this way in 1898, he did so innocently in the sense that he had never calculated the numerical values of W using ΔS = k_B ln (W/W₀) (because this equation was not stated, k_B was not known, and W₀ was indeterminable before 1900–1912). Prior publications have demonstrated that the word “disorder” is misleading in describing entropy change. In this paper, convincing evidence is provided that no starting system above ca. 1 K can be said to be orderly so far as the distribution of its energy (the fundamental determinant of entropy) is concerned. This is supported by a simple calculation showing that any system with “a practical state of zero entropy” has an incomprehensibly large number of microstates.
    [The calculation is from K. L. Pitzer “Thermodynamics” (3rd ed.; McGraw-Hill, 1995), p.67, (5-3) and shows that any molar system even at temperatures as cold as 1 K has about 10^{26,000,000,000,000,000,000} different microstates. This is not “order” or “orderly”!]

    
    



11. "Chemical Kinetics: As Important As The Second Law Of Thermodynamics?" Frank L. Lambert, from the Chemical Educator (an Online Journal) 3 (1998) 2, 6 pages © The Chemical Educator, 1998.

    Note: In the article above, the intended marginal summary on the first page was "Chemical kinetics firmly restrains "time's arrow" in the taut bow of thermodynamics for milliseconds or for millennia."

    The second law may be “time’s arrow” but activation energies (chemical kinetics) prevent second law predictions from occurring for femtoseconds to eons. This is humanly important: Activation energies not only protect all the organic chemicals in our bodies and our oxidizable possessions from instant combustion in air, but also our breakable skis and surfboards (and legs) from disastrous fracture. Murphy’s Law is often applied to chemical and physical mishaps — things going wrong. But things do not always follow the second law and burst into flame or break! Chemical kinetics is the reason Murphy’s Law usually fails.




12. "Entropy and the Shelf Model: A Quantum Physical Approach to a Physical Property", Arnd H. Jungermann, from the Journal of Chemical Education, Vol. 83, pp. 1686-1694. November 2006

    For a number of years Jungermann has presented standard molar entropy to his students as energy that is stored in substances — using shelves as energy levels and S/k_B = ln W as an introduction to the number of particles and their 'energy' distributions on various levels. With S⁰/R as a dimensionless but mass and attractive-force related property, Jungermann shows how these 'atomic entropy' values are related to trends in elements and compounds in the periodic table. His procedures and concepts well fit our "The standard molar entropy of a substance at temperature T is a measure of the quantity of energy that must be dispersed in that substance for it to exist at T, that is, it is ΔS from 0 K to T."

13. "Consistent Application of the Boltzmann Distribution to Residual Entropy in Crystals", Evguenii I. Kozliak, from the Journal of Chemical Education, Vol. 84, pp. 493-498, March 2007.

    Resolution of the old problem of understanding "residual entropy" , the entropy remaining in crystals of compounds such as CO, N₂O, FClO₃ and H₂O even as they approach absolute zero.  The entropy present in two or more arrangements of molecules in such crystals had only been considered in terms of "configurational" or "positional" entropy. Kozliak shows that the counting procedures in these entropy calculations are identical to what would result from considering the different forms on different energy levels — a considerably more fundamental focus on entropy values as related to energy distributions.

14. “A Study of Turkish Chemistry Undergraduates' Understanding of Entropy”, Mustafa Sözbilir and Judith M. Bennett, from the Journal of Chemical Education, Vol. 84, pp. 1204-1208, July 2007.

    “This study explores Turkish chemistry undergraduates' understanding of entropy and identifies and classifies their misunderstandings. For this purpose, a diagnostic questionnaire and semi-structured interviews were used—before and after teaching [about entropy in the physical chemistry course – to students who had also been taught entropy in their first-year course]….[Students were] from two different chemistry education departments in two different universities in Turkey…The misunderstandings identified were categorized into these five broad headings: (i) Defining entropy as "disorder" and considering visual disorder and entropy as synonymous; (ii) Inaccurate connection of entropy to the number of inter-molecular interactions; (iii) Inaccurate connection of entropy of a system and the accompanying entropy changes in its surroundings; (iv) Entropy of the whole system decreases or does not change when a spontaneous change occurs in an isolated system; and (v) Entropy of carbon dioxide is bigger than that of propane or the same at the same temperature. The findings have implications for tertiary-level teaching, suggesting that a substantial review of teaching strategies is needed.”

    Dr. Sozbilir has told me that in his future writing about entropy he is adopting our approach to entropy and eliminating all reference to macro or molecular "disorder".   (If you do not know how the idea of "disorder" came to be associated with entropy, see the link to Boltzmann's first erroneous deduction about "order" in nature here.)

15. "Configurational Entropy Revisited", Frank L. Lambert, from the Journal of Chemical Education, Vol. 84, pp. 1548-1550, September 2007

    Entropy change is categorized in some prominent general chemistry textbooks as being either positional (configurational) or thermal. In those texts, the accompanying emphasis on the dispersal of matter — independent of energy considerations and thus in discord with kinetic molecular theory — is most troubling. This article shows that the variants of entropy can be treated from a unified viewpoint and argues that to decrease students' confusion about the nature of entropy change these variants of entropy should be merged. Molecular energy dispersal in space is implicit but unfortunately tacit iin the cell models of statistical mechanics that develop the concept of configurational entropy change. Two factors are necessary for entropy change in chemistry. An increase in thermodynamic entropy is enabled in a process by the motional energy of molecules (that, in chemical reactions, can arise from the energy released from a bond energy change). However, entropy increase is only actualized if the process results in a larger number of arrangements for the system's energy, that is, a final state that involves the most probable distribution for that energy under the new constraints. Positional entropy should be eliminated from general chemistry instruction and, especially benefiting "concrete minded" students, it should be replaced by emphasis on the motional energy of molecules as enabling entropy change.

16. “Residual Entropy, the Third Law and Latent Heat”, Evguenii I. Kozliak and Frank L. Lambert, from Entropy (an Online open access Journal), Vol. 10 (3), pp. 274-284, 2008

    A thermodynamic treatment of residual entropy in crystals, involving the configurational partition function, is suggested, which is consistent with both classical and statistical thermodynamics. It relates residual entropy to the inherent latent heat which would be released upon cooling if the reversible path were available. The nature of this heat is that if the crystal possessing residual entropy freezes above its Boltzmann’s characteristic temperature of molecular alignment, the difference in energy between different molecular arrangements is overcome by the kT heat bath to form a nearly-ideal solution. However, upon cooling below this characteristic temperature, they would separate with a concomitant release of the corresponding energy, provided the reversible path were available.

17. "The Correlation of Standard Entropy with Enthalpy Supplied from 0 to 298.15 K”, Frank L. Lambert and Harvey S. Leff, from the Journal of Chemical Education, Vol. 86, pp. 94- 98, January 2009.)

    As a substance is heated at constant pressure from near 0 K to 298 K, each incremental enthalpy increase, ΔH, alters entropy by ΔH/T, bringing it from approximately zero to its standard molar entropy, S^o. Using heat capacity data for 32 solids and CODATA results for another 45, we found a roughly linear relationship between S^o and ΔH^o. The plot showing the relationship S^o ≈ (constant) ΔH^o, with constant = 0.0066 K^–1, for 77 solids can serve as an enlightening visualization of this relationship for students in general chemistry. The near-linearity can be understood qualitatively in terms of lattice vibrations and internal vibrations within polyatomic units, which are reflected by molar heat capacities and Debye temperatures. This study supports the thesis that thermodynamic entropy and stored internal energy in a solid are intimately related and that entropy can be usefully interpreted as a spreading function, as described in the text.

18. “Overcoming Misconceptions about Configurational Entropy in Condensed Phases”,  Evgenuii I. Kozliak, from the Journal of Chemical Education, Vol. 86, pp. 1063-1068, September 2009.)

    Configurational and thermal entropy yield identical numerical values for ΔS only when the system's "dimensionless" energy gaps (Δε /kT ) between the accessible quantized energy levels are minimized by temperature to nearly infinitesimal values so that the spreading of energy among the system's microstates becomes effectively classical and equiprobable. The molecular partition function provides the numerical value for the effective number of both accessible states and spatial configurations per molecule, for which this condition is valid at a given temperature. Considering the phenomenon of mixing and standard molar entropy values leads to the conclusion that configurational entropy calculations are significant and thermodynamically valid because of their fundamental connection to the process of random energy re-distributions in a system, via the available modes of molecular motion.
                          .          .          .          .          .          .          .          .          .
    Kozliak’s thorough connection of classical and quantum mechanics is not easy reading but his conclusions about the profound differences between the “entropy” of playing cards and thermodynamic entropy, as well as other variants of over-hyped “probability”, are more explicitly and strongly supported than any previously in the literature.  FLL

 

1. "Entropy Is Not "Disorder"; It Is a Measure of the Dispersal of Energy"

   A complete summary of the concept and its application to macro and molecular thermodynamics for chemistry instructors.

   
   

2. "What is a microstate?"

   Two approaches to understanding a microstate, a description of one arrangement of a system’s energy.

   
   

3. "A Student's Approach to the Second Law and Entropy"

   A short introduction to the second law and entropy for students. Written with the hurried student in mind.

   
   

4. "Teaching Entropy Is Simple — If You Discard "Disorder" "

   An introduction for AP teachers to the concept of entropy as measuring the dispersal of energy at a specific temperature.

   
   

5. " 'Configurational' Entropy: A Measure of Energy Dispersal in Statistical Mechanical Calculations"

   A subtopic in the chemistry seminar at California State Polytechnic University, Pomona on November 8, 2005.
   
   'Positional' or 'configurational' entropy change in general chemistry texts misplaces the nature of the change due to a probable increase in molecular positions. Actually, those positions represent the increased numbers of microstates, the spreading out of the initially more localized energy of the components

   
   

6. "The Second Law of Thermodynamics" and "Entropy in General Chemistry"

   Written by Dr. Lambert for Wikibooks, these two articles contain material that is scattered on this site but is presented in somewhat different format, designed to be quite readily readable by beginners in chemistry and accessible to students not majoring in science. An article based on the concepts developed in this seminar was published in the Journal of Chemical Education in September 2007,available here in pdf format.

7. "'Disorder' in Thermodynamic Entropy"

   The historical origin of the introduction of 'disorder' by Boltzsmann, reproduced in response to many questioners.  The brief article also is an introduction for instructors who are not familiar with my approach to understanding entropy change. It closes with a description of the clear distinction between thermodynamic entropy and Shannon information "entropy".

 

I thank the Journal of Chemical Education for permission to reprint these articles on this Web site. The Journal serves chemistry instructors worldwide and across the span of education from K-12 teachers to professors in graduate school. Permission by the JCE to display the logo below does not constitute any sort of endorsement of this site by the Journal or the American Chemical Society. The logo link is reproduced here only to aid the reader in learning more about the JCE and its remarkable print and software contributions to chemical education.