Harvey S. Leff

Thermal efficiency at maximum work output: New results for old heat engines

What is the thermal efficiency of a heat engine producing the maximum possible work per cycle consistent with its operating‐temperature range? This question is answered here for four model reversible heat engine cycles. In each case, the work is maximized with respect to two characteristic temperatures that are intermediate between the maximum and minimum cycle temperatures T+ and T−. The maximum‐work efficiencies are found to equal or be well approximated by η*=1−(T−/T+)1/2. Because this efficiency is a function solely of the extreme cycle temperatures, it can be compared easily with the corresponding reversible Carnot cycle efficiency ηc =1−T−/T+. Here, η*, which is a much better guide to the performance of actual heat engines than ηc, is the same efficiency found by Curzon and Ahlborn [Am. J. Phys. 43, 22 (1975)] for a model irreversible heat engine operating at maximum power output. The present results show that η* is more ‘‘universal’’ than had been realized previously. If the work output per cycle i...

All about work

A comprehensive ‘‘taxonomy of work’’ is developed to clarify the confusing potpourri of worklike 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.

EER, COP, and the second law efficiency for air conditioners

It is pointed out that there is a close relationship between the energy efficiency ratio (EER) of an air conditioner unit and the coefficient of performance (COP) of its refrigeration cycle. This connection helps to bridge the gap between pure thermodynamics and practical energy‐related problems. In this spirit, two other efficiency parameters, the total COP and total EER, measured relative to the energy extracted by a primary energy source (e.g., a fossil fuel), are defined. A comparison of the actual total COP (or total EER) relative to its maximum allowed value, consistent with the second law of thermodynamics, leads to an estimate for air conditioners of the recently proposed second law efficiency.

Thermodynamic entropy: The spreading and sharing of energy

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.

Efficiency and efficacy of incandescent lamps

Planck’s radiation formula is used to estimate the dimensionless efficiency of incandescent lamps as a function of filament temperature, with typical values of 2%–13%. Similarly, using the known spectral luminous efficiency of the eye, the efficacy of incandescent light bulbs is estimated as a function of temperature, showing values of 8–24 L W−1 for bulbs of 10–1000 W. The efficiency and efficacy results compare favorably with published data and enable estimation of the filament temperature for any lamp of known efficacy.

Resource Letter MD‐1: Maxwell’s demon

This Resource Letter provides a comprehensive guide to the voluminous literature that has developed around Maxwell’s demon, and offers a perspective on issues for which the hypothetical character Maxwell introduced over 120 years ago has inspired continuing research and debate. The code (E) indicates elementary level or general interest material useful to persons just learning the field; (I) indicates intermediate level or somewhat specialized material; and (A) indicates advanced or highly specialized material. No accompanying AAPT reprint book will be available, because an extensive reprint collection (Ref. 29) edited by the authors will be published separately.

Class of Ensembles in the Statistical Theory of Energy‐Level Spectra

The investigation of a large class of ensembles in the statistical theory of energy‐level spectra is initiated. Each member of this class is characterized by a joint probability density for N consecutive eigenvalues of the form PNβ(λ1,…λN)=ΩNβ−1{ ∏ i=1N f(λi)}  ∏ k<l |λk−λl|β, where a ≤ λi ≤ b, and β may be 1, 2, or 4. Formal calculations of the nearest‐neighbor spacing distribution and the level density are made for β = 2. Results are in terms of asymptotic properties of orthogonal polynomials. It is conjectured that spacing distributions are relatively insensitive to the function f(λ) and the interval [a, b]. When f(λ) = 1 and b = −a = 1, the resulting (Legendre) ensemble has the same spacing distribution as the Gaussian and Dyson ensembles. The level density is concave upward and rapidly increasing for λ ≥ 0, qualitatively resembling actual nuclear and atomic densities. This feature is not present in previously investigated ensembles. Certain invariant matrix ensembles introduced by Dyson, which are of...

Stopping objects with zero external work: Mechanics meets thermodynamics

Although the work‐energy theorem of pure, nondissipative mechanics states that the work done stopping a body equals its kinetic energy change, the work done stopping a body via an inelastic, dissipative collision can be zero. This counter‐intuitive result is used to motivate the development of thermodynamic ideas as a direct extension of classical mechanics. The approach leads to a natural introduction of internal energy, the path dependence of work, and dissipation. It also offers an opportunity for early exposure to powerful symmetry and frame‐invariance arguments. The main presentation addresses one‐dimensional highly symmetric collisions, with a generalization in the Appendix.

What if entropy were dimensionless

One of entropy’s puzzling aspects is its dimensions of energy/temperature. A review of thermodynamics and statistical mechanics leads to six conclusions: (1) Entropy’s dimensions are linked to the definition of the Kelvin temperature scale. (2) Entropy can be defined to be dimensionless when temperature T is defined as an energy (dubbed tempergy). (3) Dimensionless entropy per particle typically is between 0 and ∼80. Its value facilitates comparisons among materials and estimates of the number of accessible states. (4) Using dimensionless entropy and tempergy, Boltzmann’s constant k is unnecessary. (5) Tempergy, kT, does not generally represent a stored system energy. (6) When the (extensive) heat capacity C≫k, tempergy is the energy transfer required to increase the dimensionless entropy by unity.

Translational Invariance Properties of a Finite One‐Dimensional Hard‐Core Fluid

A formalism is developed for expressing the n‐particle distribution functions Dn(x1 ≤ x2 ≤ … ≤ xn) explicitly in terms of the configurational partition function for one‐dimensional fluids with hard‐core repulsive and nearest‐neighbor attractive forces. The translational invariance properties of the Dn functions are investigated for the case of no attractive forces when the system is finite. When the number density is less than half the close packing density, there exists a central region in which D1(x) is constant and all the Dn functions, n ≥ 2, are functions of the (n−1) nearest‐neighbor separation distances. Several relevant theorems are proved and limiting cases are investigated.