Equilibrium and Stability
The first row of boxes shown in Fig. 3.1 depicts a number of identical systems differing only in their internal energies, \(E_\nu \) , volumes, \(V_\nu \) , and mass contents, \(n_\nu \) . The boundaries of the systems allow the exchange of these quantiti
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Equilibrium and Stability
3.1 Equilibrium and Stability via Maximum Entropy 3.1.1 Equilibrium The first row of boxes shown in Fig. 3.1 depicts a number of identical systems differing only in their internal energies, E ν , volumes, Vν , and mass contents, n ν . The boundaries of the systems allow the exchange of these quantities between the systems upon contact. The second row of boxes in Fig. 3.1 illustrates this situation. All (sub-)systems combined form an isolated system. We ask the following question: What can be said about the quantities xν , where x represents E, V or n, after we bring the boxes into contact and allow the exchanges to occur? According to our experience the exchange is an irreversible spontaneous process and therefore relation Eq. (1.50) applies to the entropy of the overall system. We can expand the entropy of the combined systems, S, in a Taylor series in the variables E ν , Vν , and n ν with respect to its maximum, i.e. ∂ Sν o ∂ Sν o ∂ Sν o E ν (3.1) S = So + + Vν + n ν ∂ E ν Vν ,n ν ∂ Vν E ν ,n ν ∂n ν E ν ,Vν ν ∂ o ∂ o ∂ o 1 E ν + + Vν + n ν 2 ∂ E ν Vν ,n ν ∂ Vν E ν ,n ν ∂n ν E ν ,Vν ν ,ν ∂ Sν o ∂ Sν o ∂ Sν o , × E ν + Vν + n ν ∂ E ν Vν ,n ν ∂ Vν E ν ,n ν ∂n ν E ν ,Vν o where S = μ Sμ (E μ , Vμ , n μ ). The quantity S is the maximum value of the entropy. Notice that this quantity is somewhat hypothetical. The usefulness of this approach relies on the differences between time scales on which certain processes take place.
R. Hentschke, Thermodynamics, Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-36711-3_3, © Springer-Verlag Berlin Heidelberg 2014
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3 Equilibrium and Stability
Leaving a cup of hot coffee on the table, we expect to find coffee at room temperature upon our return several hours later. If we come back after some weeks of vacation the coffee has vanished, i.e. the water has evaporated. Only the dried remnants of the coffee remain inside the cup. After waiting for a much longer time, how long depends on numerous things including the material of the coffee cup, the cup itself has crumbled into dust. However, if we are interested merely in the initial cooling of the coffee to room temperature, we may neglect evaporation and we may certainly neglect the deterioration of the cup itself. In this sense we shall use the expression of equilibrium. For all practical purposes equilibrium is understood in a “local” sense, i.e. the time scale underlying the process of interest is much shorter than the time scale underlying other processes influencing the former. In the case at hand equilibrium means that all variables, E ν , Vν , and n ν , have assumed the values E νo , Vνo , and n oν corresponding to maximum entropy. However, we may impose deviations from these values in each subsystem, xν , as illustrated in the bottom part of Fig. 3.1. The long dashed line indicates the equilibrium value(s), which is the same in all (identical) systems. The short dashed lines indicate the
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