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Unformatted text preview: T. Y. Tan 1 4. INTERPRETATIONS OF THERMODYNAMIC QUANTITIES USING STATISTICAL MECHANICS The laws of thermodynamics are summaries of experimental results of observing the macro- scopic physical systems. As such, the various thermodynamic quantities of the system are mani- festations of the time-averaged properties of the system constituents, which are nanoscopic in nature. With the recognition all matters are composed of the more fundamental particles atoms or molecules, it became possible to interpret the thermodynamic quantities, and hence also the thermodynamic laws, in terms of the properties of the atoms or molecules. Such interpretations constitute a justification of the thermodynamic laws. Because the number of particles constitut- ing the matter is extremely large, about 6x10 23 per mole, the methods used for such studies are those of statistical mechanics. In chapters 2 and 3, it has been mentioned that the thermodynamic laws do not yield expressions of the various state properties, i.e., E, S, G, etc., as functions of the state variables P, T, or V. Nor can the laws yield values of these properties for a given state, but only the difference in the state property values upon changing from one state to another. For ide- alized systems, statistical mechanics allows one to obtain such expressions as well as to calculate the values of the various thermodynamic properties, e.g., E, of a given state. Statistical mechanics does not treat a naturally occurring system. Instead, an idealized sys- tem, which has properties resembling as much as possible those of the naturally occurring one, is treated. The idealized system has certain exact and sharply defined characteristics that are nearly but frequently not exactly realized in the natural system. This is necessary, since mathematical abstractions must always be made when the calculation of the properties of any physical object is undertaken, and the more complicated the object, the more necessary this becomes. 4.1 Phase Space and the Equal Probability Theorem Statistical mechanics is aimed at studying the properties of an idealized system by applying the statistical methods to the system, based on laws of either classical (Newtonian) or quantum mechanics. For a system obeying the laws of classical mechanics, the most accurate description of the instantaneous state of the system consists in giving the values of all the coordinates of the system constituent particles and of all the momenta conjugate to the coordinates. If f, the number of de- grees of freedom of the system, is the total minimum number of coordinates, then the 2f dimen- sional-momentum space is called the phase space of the system. The point describing the system T. Y. Tan 2 moves through this phase space along a path and with a velocity determined by the laws of New- tonian mechanics. The calculation of this path and velocity is a very complicated problem, but it does not at all concern us. The only property of the motion of this point through the phase space does not at all concern us....
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- Fall '11