B) CALCULO DE CARGA POR MOTORES ELECTRICOS
4.2 PROGRAMA DE MANTENIMIENTO A LAS UNIDADES DEL SISTEMA
4.2.3 POSIBLES FALLAS DEL EVAPORADOR Y SU SOLUCIÓN
The arguments of this section do not allow us to determine the absolute entropy of any macrostate. An expression for absolute entropy is one that, given com-plete information about a particular macrostate, provides a determinate value of the entropy of that macrostate. Equations ( 2.15 ) and ( 2.16 ) obviously pro-duce only relative entropies. And only relative entropies emerge from the i rst and second laws of thermodynamics.
This limitation to relative entropies originates with our inability, given clas-sical presuppositions, to uniquely identify and count microstates. Furthermore, reference microstates in classical statistical mechanics are necessarily arbitrary and so classical macrostates are also necessarily arbitrary. Only the introduction of quantum physics allows us to uniquely identify microstates. But, of course, Boltzmann and the other classical statistical mechanicians were unaware of the possibilities inherent in quantum physics.
What Boltzmann did know was how to arbitrarily discretize space in such a way as to construct the arbitrary microstates we call classical microstates . Boltzmann then applied the fundamental postulate to these arbitrarily con-structed classical microstates by asserting that S∝ ln Ω . Boltzmann’s method of discretizing phase space allowed him and will allow us to determine clas-sical versions of statistical entropies from which it is possible to derive equa-tions of state. Of course we will use S
( )
Ω ccc kk Ω , in place of Boltzmann’s S∝ ln Ω in order to ensure that the entropy is always an additive function of state variables.Example 2.2 Joule expansion
Problem: Consider, as in Example 2.1, a system with left and right, equal-volume chambers containing a total of N identical but classical, and so dis-tinguishable, particles. Suppose this system is initialized in a macrostate in which all its particles are contained within its left chamber. Then the par-ticles are allowed to move freely between both chambers and achieve a i nal equilibrium macrostate in which the particles are distributed throughout both chambers as illustrated in Figure 2.5 . How much does the entropy of the sys-tem increase?
Solution : The number of spatial microstates available to a single parti-cle increases by a factor of 2 during this process. And the number of spa-tial microstates available to two distinguishable particles increases by 2 2 . Therefore, the multiplicity of the N -particle system increases by 2N during this Joule expansion. Consequently, the ratio of the i nal to the initial macrostate multiplicity is
and the entropy is incremented by
Apparently each particle contributes k ln 2 to the entropy increment.
Example 2.3 A paradox
Problem: Consider a system of 2 N identical, classical particles initialized with N particles in the left of two equal-volume chambers and N in the right. The particles are allowed to move freely and achieve a i nal, equilibrium state.
What is the entropy increment?
Solution: One response is to say that the entropy of the N particles initially in the left chamber increases, as in Example 2.2, by Nk ln 2 as those particles spread throughout both chambers and likewise for the particles initially in the right chamber. In this way the total entropy would increase by 2Nk l 2 .
Time
Figure 2.5 Joule expansion. After the barrier that keeps the particles in the left hand side of two equal-volume chambers is removed, the particles occupy both chambers.
2.5 Maxwell’s demon 47
But this is absurd. For, if we were then to reinsert a barrier between the two chambers, remove that barrier, and allow the particles on each side to again mix with each other, the system entropy would again increase by 2Nk l 2 and so on.
The key to resolving this paradox is to recall that a system’s entropy only increases when the multiplicity of its i nal macrostate is larger than the multi-plicity of its initial macrostate. In this problem the initial and i nal macrostates are the same because they have the same description: N classically identical molecules occupy each chamber. Therefore the initial and i nal macrostate multiplicities and entropies are the same, and there is no entropy increment.
True, the i nal microstate of the system may differ from its initial microstate, but this is of no consequence for entropy. The entropy increment is a function only of macrostate multiplicity and not of the particular microstates that com-pose the macrostate.
Example 2.4 Entropy of mixing
Problem: A system consisting of 2N classical particles is initialized in a mac-rostate in which N particles of one kind, for example, nitrogen molecules, are all in the left of two equal-volume chambers and N particles of another kind, for example, oxygen molecules, are all in the right chamber. The particles are allowed to mix. What is the entropy increment?
Solution: The microstates available to each kind of particle are doubled dur-ing this process and this doubldur-ing does increase the number of microstates available to the system in its i nal macrostate. Therefore, the initial and i nal multiplicities of the 2 N -particle system are in the ratio
Ω
Ω
f
i
= 22N.
Accordingly the entropy increment ΔS k
(
Ω Ωf Ω of the system is i)
ΔS 2NkNNNkNN l 2.
2.5 Maxwell’s demon
While Boltzmann was the i rst to suggest that a relation exists between the entropy S of an isolated system and its macrostate multiplicity Ω , it was James Clerk Maxwell (1831–1879) who i rst suggested that entropy is a probabilistic concept – i rst privately in a letter to his friend Peter Guthrie Tait in 1867 and
later in his 1871 text Theory of Heat . Maxwell died before he could develop this idea, but his way of arguing for it had important consequences for the way in which the concept of statistical entropy has developed. (See the entry Maxwell’s Demon in the Annotated Further Reading.)
Maxwell’s conceit was to imagine a small, “neat i ngered” being, later called a demon and much later called Maxwell’s demon , who attended a door between two chambers of air. (See Figure 2.6 .) The air in the two chambers initially has the same temperature and density, but even so its individual molecules move at various speeds – a phenomenon i rst discovered by Maxwell. The demon’s job is to open the door in order, say, to allow the fastest moving particles in the right chamber to move to the left chamber and to allow the slowest mov-ing particles in the left chamber to move to the right chamber. In this way, the demon could cause the air in the left chamber to become hotter than the air in the right chamber while keeping their densities the same. If we consider the two chambers, the air they contain, and the demon as an isolated system, this process directly violates Clausius’s version of the second law assuming the entropy of the demon does not change. Admittedly, the latter is a signii cant assumption.
Maxwell claimed that his inhabited thought experiment demon-strated that the second law of thermodynamics “has only statistical certainty,”
that the second law can at times be violated, and that the statistical entropy of an isolated system can momentarily decrease. For if a demon can cause the entropy of an isolated system to decrease, possibly an inanimate automa-ton like a spring-loaded, one-way door can do the same. Indeed, subsequent analyses and numerical simulations of many cleverly constructed, inanimate
“demons” have shown that the statistical entropy of an isolated system may l uctuate – but only l uctuate. And the larger the system, the smaller is the
Figure 2.6 Maxwell’s demon. By opportunely opening and closing a door con-necting two chambers of gas, the demon can increase the temperature of the gas in one chamber over that of the other.
2.6 Relative versus absolute entropy 49