Energy and Entropy: An Intimate Relationship — © 2020 Harvey S. Leff

Selected Published Articles On Energy, Entropy and Thermodynamics

The following articles are more technical than the essays. Perhaps they will be most helpful to teachers seeking explanations of energy and entropy. Hopefully, they offer an opportunity to gain an appreciation of some of the fundamental issues related to energy and entropy. In each item on the list below, I show the article’s abstract and/or other descriptive paragraph.

Note: Each link to an article will open in a separate browser tab or window. To return to this page, simply close the article’s tab or window.

• “The Mayer–Joule Principle: The Foundation of the First Law of Thermodynamics” (2011) Ronald Newburgh and Harvey S. Leff

To most students today the mechanical equivalent of heat, called the Mayer–Joule principle, is simply a way to convert from calories to joules and vice versa. However, in linking work and heat—once thought to be disjointed concepts—it goes far beyond unit conversion. Heat had eluded understanding for two centuries after Galileo Galilei constructed an early thermometer. Independently, Julius Robert Mayer and James Prescott Joule found the connection between heat and work, the Mayer–Joule principle. Read more…

• “Removing the Mystery of Entropy and Thermodynamics – Part I” (2012)
Harvey S. Leff

Energy and entropy are centerpieces of physics. Energy is typically introduced in the study of classical mechanics. Although energy in this context can be challenging, its use in thermodynamics and its connection with entropy seem to take on a special air of mystery. In this five–part series, I pinpoint ways around key areas of difficulty to reduce that mystery. In Part I, the focus is on building an understanding of fundamental ideas of thermodynamics, including its connection with mechanics, and how entropy is defined and calculated. A central thread is that energy tends to spread within and between macroscopic objects, and this spreading is a surrogate for entropy increase. Specific questions are posed and answered, building on foundations laid in prior articles. A question–answer format is used throughout, with major results enumerated in numbered Key Points 1.1–1.5. Read more…

• “Removing the Mystery of Entropy and Thermodynamics – Part II” (2012)
Harvey S. Leff

Part II of this five–part series is focused on further clarification of entropy and thermodynamics. We emphasize that entropy is a state function with a numerical value for any substance in thermodynamic equilibrium with its surroundings. The interpretation of entropy as a “spreading function” is suggested by the Clausius algorithm. The Mayer–Joule principle is shown to be helpful in understanding entropy changes for pure work processes. Furthermore, the entropy change when a gas expands or is compressed, and when two gases are mixed, can be understood qualitatively in terms of spatial energy spreading. The question–answer format of Part I1 is continued, enumerating main results in Key Points 2.1–2.6. Read more…

• “Removing the Mystery of Entropy and Thermodynamics – Part III” (2012)
Harvey S. Leff

In Part III of this five–part series of articles, simple graphic properties of entropy are illustrated, offering a novel way to understand the principle of entropy increase. The Boltzmann entropy is introduced and shows that in thermal equilibrium, entropy can be related to the spreading of a system over accessible microstates. Finally, constant–temperature reservoirs are shown to be idealizations that are nevertheless useful. A question–answer format is continued here and Key Points 3.1–3.4 are enumerated. Read more…

• “Removing the Mystery of Entropy and Thermodynamics – Part IV” (2012)
Harvey S. Leff

In Part IV of this five–part series, reversibility and irreversibility are discussed. The question–answer format of Part I is continued and Key Points 4.1–4.3 are enumerated. Read more…

• “Removing the Mystery of Entropy and Thermodynamics – Part V” (2012)
Harvey S. Leff

Part V ends this five–part paper series. We discuss the interpretation of entropy as uncertainty and connections between spreading and uncertainty. The too commonly used disorder metaphor for entropy is roundly rejected. Finally, a generalization of the equity concept that was introduced in Part III is presented. The question–answer format is continued and Key Points 5.1–5.3 are enumerated. Read more…

• “The Correlation of Standard Entropy with Enthalpy Supplied from 0 to 298.15 K” (2009)
Frank L. Lambert and Harvey S. Leff

It takes energy to heat a substance from near absolute zero to standard room temperature, 298.15 K. As heating progresses, the substance’s entropy increases from zero (as dictated by the third law of thermodynamics) to a standard entropy S° for the substance. For each substance, the standard entropy is the sum of reversible small increments dH divided by temperature T for the heating process. That is, the standard entropy S° is a function of the added energy, namely, the total enthalpy ∆H° delivered during heating. This energy spreads spatially throughout the solid and is stored within it. The entropy function can be usefully interpreted as a spreading function, with the symbol S connoting spreading, as clarified below. Along similar lines, the term energy dispersal, rather than spreading, has been used. Read more… In addition, a supplement on the best fit line is here.

• “Melding Two Approaches to Entropy” (2010)
Harvey S. Leff and Frank L. Lambert

Fortuitously, an interesting article by Ben–Naim follows an article by us in the January 2009 issue of the Journal of Chemical Education. Ben–Naim discussed the second law of thermodynamics and entropy using a probabilistic–information–theoretic approach. We illustrated a strong connection between entropy and energy for solids. Here, we suggest how the two modes of thought can be usefully melded. Read more…

• “Entropy, Its Language, and Interpretation” (2007)
Harvey S. Leff

The language of entropy is examined for consistency with its mathematics and physics, and for its efficacy as a guide to what entropy means. Do common descriptors such as disorder, missing information, and multiplicity help or hinder understanding? Can the language of entropy be helpful in cases where entropy is not well defined? We argue in favor of the descriptor spreading, which entails space, time, and energy in a fundamental way. This includes spreading of energy spatially during processes and temporal spreading over accessible microstates states in thermodynamic equilibrium. Various examples illustrate the value of the spreading metaphor. To provide further support for this metaphor’s utility, it is shown how a set of reasonable spreading properties can be used to derive the entropy function. A main conclusion is that it is appropriate to view entropy’s symbol S as shorthand for spreading. Read more…

• “Thermodynamic entropy: The spreading and sharing of energy” (1996)
Harvey S. Leff

A new approach to thermodynamic entropy is proposed to supplement traditional coverage at the junior–senior level. It entails a model for which: (i) energy spreads throughout macroscopic matter and is shared among microscopic storage modes; (ii) the amount and/or nature of energy spreading and sharing changes in a thermodynamic process; and (iii) the degree of energy spreading and sharing is maximal at thermodynamic equilibrium. A function S that represents the degree of energy spreading and sharing is defined through a set of reasonable properties. These imply that S is identical with Clausius' thermodynamic entropy, and the principle of entropy increase is interpreted as nature's tendency toward maximal spreading and sharing of energy. Microscopic considerations help clarify these ideas and, reciprocally, these ideas shed light on statistical entropy. Read more…

• All about work" (1992)
A. John Mallinckrodt & Harvey S. Leff

A comprehensive "taxonomy of work" is developed to clarify the confusing potpourri of work-like quantities that exists in the literature. Seven types of work that can be done on a system of particles interacting internally and/or with its environment are identified and reviewed. Each work is defined in terms of relevant forces and displacements; mathematical connections between the works are delineated; work-energy relationships are derived; and the Galilean transformation properties of the works and corresponding energy changes are obtained. The results are applied to several examples, illustrating subtle distinctions between the various works and showing how they can be used to bridge the conceptual gap between the "pure" mechanics of point particles and the thermodynamics of macroscopic matter. Read more…

• Reversible and irreversible heat engine and refrigerator cycles (2018)
Harvey S. Leff

Although no reversible thermodynamic cycles exist in nature, nearly all cycles covered in textbooks are reversible. This is a review, clarification, and extension of results and concepts for quasistatic, reversible and irreversible processes and cycles, intended primarily for teachers and students. Distinctions between the latter process types are explained, with emphasis on clockwise (CW) and counterclockwise (CCW) cycles. Specific examples of each are examined, including Carnot, Kelvin and Stirling cycles. For the Stirling cycle, potentially useful task-specific efficiency measures are proposed and illustrated. Whether a cycle behaves as a traditional refrigerator or heat engine can depend on whether it is reversible or irreversible. Reversible and irreversible-quasistatic CW cycles both satisfy Carnot’s inequality for thermal efficiency, η ≤ η(Carnot). Irreversible CCW cycles with two reservoirs satisfy the coefficient of performance inequality K ≤ K(Carnot). However, an arbitrary reversible cycle satisfies K ≥ K(Carnot) when compared with a reversible Carnot cycle operating between its maximum and minimum temperatures, a potentially counterintuitive result. Read more…

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