Chapter 1 - The Statistical Basis of Thermodynamics

Abstract

According to quantum mechanics, in a physical system composed of N identical particles, the single-particle energies ɛ i , are discrete and their values depend crucially on the volume V to which the particles are confined. Accordingly, the possible values of the total energy E are also discrete. However, when the value of V is high, the spacing of the different energy values is so small in comparison with the total energy of the system that the parameter E might well be regarded as a continuous variable. The specification of the actual values of the parameters N , V, and E then defines a macrostate of the given system. At the molecular level, however, a large number of possibilities still exist because, in general, there will be numerous different ways in which the macrostate ( N , V , E ) of the given system can be realized at that level. In other words, there will be numerous ways in which the total energy E of the system can be distributed among the N particles constituting it. Each of these (different) ways specifies a microstate—or complexion—of the given system. This chapter gives a brief account of the significance of parameters N , V , and E and the uses of these parameters in statistical approaches to define various topics such as ideal gas, entropy of mixing, and the Gibbs paradox. The chapter also uses this approach to give an enumeration of the microstates of a system.