A Novel Approach to the Teaching of Entropy Based on a Recent Single Particle Heat Engine Model

In a recently developed simple particle mechanics model in which a single particle represents the working fluid (gas) in a heat engine (exemplified by a piston engine) a new approach was outlined for the teaching of concepts to thermodynamic students. By mechanics reasoning a model was developed that demonstrates the connection between the Carnot efficiency limitation of heat engines and the Kelvin-Planck statement of Second Law requiring only the truth of the Clausius statement. In this paper the model is extended to introduce entropy. Here the particle's entropy is defined as a function of its kinetic energy and the space that it occupies that is analogous to that normally found in classical macroscopic analyses. INTRODUCTION In a previous paper [5], simple, one dimensional theoretical models of heat engines were presented that bridge the conceptual gap between the elegantly abstract Second Law approaches and physical/mechanical principles governing the limiting efficiency of heat engines. The models demonstrated clearly how heat engines harness the kinetic energy of the particle and convert it to useful work. They showed that the engines must operate between two kinetic energy levels (the source level and the sink level) in order to produce net useful work. The maximum efficiency achievable with the engines was shown to be in agreement with the Carnot efficiency. Also demonstrated was that Carnot’s analogy to the water wheel of his day was an appropriate approach, even though he falsely reasoned that the caloric flowing from the source (to drive the engine) had to be rejected at the sink. The models demonstrated that if the “flow analogy” is to be used then it is kinetic energy that flows, and that it flows from a higher kinetic energy level to the working fluid and part of it must be rejected at a lower kinetic energy level. Fortunately a part of this (limited by the Carnot formula) may be converted to useful work. Simple mechanical concepts allow students to grasp complex concepts in terms of readily understandable particle mechanics, without introducing the full intricacies of the kinetic theory of gases or statistical thermodynamics. The thermodynamic claims of the paper were limited to situations involving ideal gases. The discussion centered on a single particle contained in one dimensional oscillation. Traditional macroscopic thermodynamic symbols such as P, T, V, H, S and Q were not used. Yet many of the basic relationships and demonstrations of the Kinetic Theory of Gases were derived. Whilst some engineering thermodynamics textbooks introduce a discussion of kinetics theory of gases and some statistical thermodynamics [12], the majority of authors concentrate on developing macroscopic concepts [4,6,9,11,13]. However, in our experience the majority of students find thermodynamic concepts difficult to grasp whether they are introduced through the classical approach, or with the aid of kinetic theory of gases and statistical thermodynamics. The significance of the approach developed in [5] is that that the cause of the Carnot limitation clearly shown in an ideal heat engine operating in a cycle, the Carnot limitation is purely due to mechanical and spatial constraints without the need for complex new concepts. Because of environmental and other concerns, there are intense efforts to develop fuel cells, thermoelectric and photovoltaic systems [3, 10] as well as many other processes for energy generation. Mechanical engineering students need to be well acquainted with such devices so that a grounding in traditional heat engines [6, 13] may not be sufficient for an adequate understanding of the phenomena involved in these new fields which all require an understanding of the principles of the kinetic theory of gases, statistical and a modicum of quantum mechanics. We believe that many mechanical engineering students have difficulties understanding the abstract concepts involved, so that a visual, more mechanistic approach is now essential in teaching the fundamentals of thermodynamics. As an example consider the Clausius statement of the Second Law of Thermodynamics which proposes that heat can only be transferred unaided from a higher to a lower