Entropy Boltzmanns View

We will soon see that the criterion for spontaneous change can be expressed in terms of a thermodynamic quantity called entropy. Let s first focus our attention on developing a conceptual model for understanding entropy. Then, we will be able to use entropy, more specifically entropy changes, to explain why certain processes are spontaneous and others are not. 

Let s explore the concept of a microstate by considering a system of five particles confined to a one-dimensional box of length L. (We discussed the model of a particle in a box on page 326.) To start, we use the energy level 

Boltzmann s famous equation is inscribed on the tomb. At the time of Boltzmann s death, the term log was used for both natural logarithms and logarithms to the base ten the symbol In had not yet been adopted. 

Notice that, for the situation just discussed, the state of the system can be described in two ways. At the macroscopic level, the state of the system is described by specifying the total energy, U, and the length, L, of the box. At the molecular level, the state of the system is described in terms of a microstate having all particles in the n = 1 level. If we use the symbol W to represent the number of microstates, we have for this case W = 1. Notice that for this total energy, one energy level is accessible to the particles, namely, the n = 1 level. 

The point of this discussion was to illustrate not only the enumeration of microstates through the distribution of particles among the available energy levels but also that W, the number of microstates, increases with both the total energy and total space available to the particles of the system. The number of accessible energy levels also increases with the total energy and total space available. Now we make the connection between the number of microstates, W, and entropy. 

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Entropy Boltzmann s View 13-2 Entropy Change Clausius s View 13-3 Combining Boltzmann s and... 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

Boltzmann, following Clausius, considered entropy to be defined only to an arbitrary constant, and related the difference in entropy between two states of a system to their relative probability. An enormous advance was made by Planck who proposed to determine the absolute entropy as a quantity, which, for every realizable system, must always be positive (third law of thermodynamics). He related this absolute entropy, not to the probability of a system, but to the total number of its possibilities. This view of Planck has been the basis of all recent efforts to find the statistical basis of thermodynamics, and while these have led to many differences of opinion, and of interpretation, we believe it is now possible to derive the second law of thermodynamics in an exact form and to obtain... 

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

One can discard the claim that the relatively primitive assumptions about the structure of the gas model also give a correct picture of the phenomena even over very long time intervals. This point of view was, of course, also considered by Boltzmann. He emphasized very early (1871)156 that in the further development of the kinetic theory one has to consider the interaction of the molecules and the ether (i.e., the influence of radiation on the thermal equilibrium). However, in the discussions about the F-theorem, he was right to insist on the first point of view to its final consequences. In this case a reference to, for instance, thermal radiation would easily lead to a premature condemnation of Boltzmann s ideas, as if the increase in entropy for processes during an observable time interval could not be interpreted without invoking radiation. 

Molecular mechanics still gives an inherently enthalpic view of the world. Molecular dynamics (MD) calculations should give more reliable and rigorous predictions of thermodynamic properties, as they deal with entropy by averaging over time and temperature. Ideally, a full atomistic model, including solvent, should be used. This can be prohibitively expensive, so solvent-continuum models such as Poisson-Boltzmann" or are... 

Applying Boltzmann s formula (Equation 1.4-1) for the entropy of a molecular coil, and in view of Equations 104 and 106, we write... 

Interlude 3.2 Poincare Recurrence Times We have seen that Boltzmann s entropy theorem leads not only to an expression for the equilibrium distribution function, but also to a specific direction of change with time or irreversibility for a system of particles or molecules. The entropy theorem states that the entropy of a closed system can never decrease so, whatever entropy state the system is in, it will always change to a higher entropy state. At that time, Boltzmann s entropy theorem was viewed to be contradictory to a well-known theorem in dynamics due to Poincare. This theorem states that... 

The concept of entropy was developed so chemists could understand the concept of spontaneity in a chemical system. Entropy is a thermodynamic property that is often associated with the extent of randomness or disorder in a chemical system. In general if a system becomes more spread out, or more random, the system s entropy increases. This is a simplistic view, and a deeper understanding of entropy is derived from Ludwig Boltzmann s molecular interpretation of entropy. He used statistical thermodynamics (which uses statistics and probability) to link the microscopic world (individual particles) and the macroscopic world (bulk samples of particles). The connection between the number of microstates (arrangements) and its entropy is expressed in the Boltzmann equation, S = kin W, where W is the number of microstates and k is the Boltzmann constant, 1.38 x 10 JK L... 

Since Ludwig Boltzmann (1844-1906) introduced a statistical definition of entropy in 1872, entropy is associated with disorder. The increase of entropy is then described as an increase of disorder, as the destruction of any coherence which may be present in the initial state. This has unfortunately led to the view that the consequences of the Second Law are self-evident or trivial. This is, however, not true even for equilibrium thermodynamics, which leads to highly nontrivial predictions. Anyway, equilibrium thermodynamics covers only a small fraction of our everyday experience. We now understand that we cannot describe Nature around us without an appeal to nonequilibrium situations. The biosphere is maintained in nonequihbrium through the flow of energy coming from the sun, and this flow is itself the result of the nonequilibrium situation of our present state in the universe. 

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