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Basic Theoretical Physics: A Concise Overview P36

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Basic Theoretical Physics: A Concise Overview P36. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 48.2 The Entropy of an Ideal Gas from the Microcanonical Ensemble 365 The Stirling approximation was used again since Using one thus obtains s0 ln 4nm hr G 3.N 3N 2 N 3 2 5 _ _ 3 e 2 Eq UQ The entropy constant therefore only depends on the type of gas via the logarithm of the particle mass m. The above result is reasonable because with the factor N it explicitly expresses the additivity for the entropy of an ideal gas which as already mentioned depends crucially on the permutation factor for identical particles. Furthermore the law 1 _ dS T dU yields the relation between U and T 3 U - NkB T . Similarly p dS T dV yields pV NkB T. Only the expression for the chemical potential p is somewhat less apparent but we shall require it later From p dS T - dN it follows that p 5 kB T - T SS U PV - T S . p 2 B N N Thus p n i.e. the chemical potential p is identical to the free enthalpy G U pV - T S per particle. This identity is generally valid for a fluid system. 366 48 Canonical Ensembles in Phenomenological Thermodynamics 48.3 Systems in a Heat Bath Canonical and Grand Canonical Distributions In the previous section we treated closed systems. Now we shall concentrate on closed systems that consist of a large system a so-called heat bath to which a small partial system which is actually the system of interest is weakly coupled. For a canonical ensemble the coupling refers only to energy and only serves to fix the temperature. For a so-called grand canonical ensemble on the other hand not only exchange of energy takes place with the large system but also particles are exchanged. In this case not only the temperature T of the small system is regulated by the heat bath but also the chemical potential a. In the next section we shall consider how the transition from a micro-canonical ensemble to a canonical and grand canonical ensemble is achieved mathematically. For a canonical ensemble the appropriate variable of state is the Helmholtz free energy F T V N . For a magnetic .