By Bruno Linder
During this transparent and concise advent to thermodynamics and statistical mechanics the reader, who may have a few prior publicity to thermodynamics, can be guided via all the disciplines individually first and foremost to supply an in-depth figuring out of the realm and thereafter the relationship among the 2 is gifted and discussed.
moreover, mathematical suggestions are brought at acceptable instances, highlighting such use as: distinctive and inexact differentials, partial derivatives, Caratheodory's theorem, Legendre transformation, and combinatory analysis.
* Emphasis is positioned both on basics and applications
* numerous difficulties are incorporated
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Extra resources for Thermodynamics and Introductory Statistical Mechanics
Example text
3 GENERALIZATION: ARBITRARY CYCLE The above result obtained for a reversible Carnot cycle can be generalized to an arbitrary cycle. To accomplish this, we suppose that, when the system traverses the cyclic path, it exchanges heat with a series of reservoirs at temperatures T1, T2, etc. 4), each of which operates quasistatically. It is seen that, when all the small Carnot cycles are completed, each adiabat has been traversed twice, once in the forward direction and once in the reverse direction, effectively cancelling each other.
1 MATHEMATICAL INTERLUDE IV: LEGENDRE TRANSFORMATIONS Equation 5-2 is the well-known representation of a straight line in the x; y frame. x and f are, respectively, the slope and the intercept of the line. y ¼ xx þ f ð5-2Þ If the line is curved, the equation can be represented as y ¼ xðxÞx þ fðxÞ ð5-3Þ Here, x ¼ dy=dx. The curve (Eq. 5-3) can be completely described by specifying x and the corresponding y or by specifying the slope x and the corresponding intercept f; that is (as is obvious) fðxÞ ¼ y À xx or fðxÞ ¼ yðxÞ À xðdy=dxÞ ð5-4Þ The variable x is considered to be the independent variable, and the function f(x) is the Legendre transformation of y.
2a Schematic representation of a Carnot cycle running in a clockwise direction and transferring heat from the high temperature reservoir to the lower one. respectively at y2 and y1, and q may have either a positive or a negative value, depending on whether heat is absorbed or emitted. 1. The efficiency of an engine is defined as the work done by the engine in a complete cycle divided by the heat absorbed at the higher temperature reservoir; that is x ¼ Àw=q2 ð4-1Þ The work done by the system on the surroundings must be represented as Àw, because, by our convention, w represents the work done on the system.