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 is kept fixed, the thermal efficiency η of each cycle considered here can be increased by enlarging the heat engine and operating it at less-than-maximum work output per cycle. Formally, for such a fixed work output, η can approach the Carnot efficiency ηc in the limit of infinite engine size.
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.
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.
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.
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