Pedagogical Approach to Microcanonical Statistical Mechanics via Consistency with the Combined First and Second Law of Thermodynamics for a Nonideal Fluid
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引用次数: 0
Abstract
Thermodynamics forms an important part of the science and engineering curriculum at the undergraduate and graduate levels. Over the years, the importance of statistical mechanics and molecular simulations in the curriculum has increased. In this work, we present a pedagogical approach to the microcanonical formulation of statistical mechanics via its consistency with the combined first and second law of thermodynamics. We start with Boltzmann’s entropy formula and use differential calculus to establish that dE = TdS – PdV for an isolated, nonideal fluid in an arbitrary number of dimensions, with a constant number of particles (N), volume (V), and energy (E) and with temperature T, pressure P, and entropy S. To this end, we write the partition function for an isolated monatomic fluid. Furthermore, we derive the average of the inverse kinetic energy, which appears in the microcanonical ensemble, and show that it is equal to the inverse of the average kinetic energy, thus introducing the system’s temperature. Subsequently, we obtain an expression for the pressure of a system involving many-body interactions and introduce it in the combined first and second law via Clausius’s virial theorem. Overall, we show that the statistical mechanics of an isolated (microcanonical) nonideal fluid is consistent with the fundamental thermodynamic relationship dE = TdS – PdV, thereby providing deeper insight into equilibrium statistical thermodynamics. We also demonstrate that this material resulted in favorable learning outcomes when taught as a 1.5 h lecture; therefore, it may be incorporated into graduate-level courses on statistical mechanics and/or molecular simulations.
期刊介绍:
The Journal of Chemical Education is the official journal of the Division of Chemical Education of the American Chemical Society, co-published with the American Chemical Society Publications Division. Launched in 1924, the Journal of Chemical Education is the world’s premier chemical education journal. The Journal publishes peer-reviewed articles and related information as a resource to those in the field of chemical education and to those institutions that serve them. JCE typically addresses chemical content, activities, laboratory experiments, instructional methods, and pedagogies. The Journal serves as a means of communication among people across the world who are interested in the teaching and learning of chemistry. This includes instructors of chemistry from middle school through graduate school, professional staff who support these teaching activities, as well as some scientists in commerce, industry, and government.