### Thermodynamic Probability W and Entropy - Chemistry LibreTexts

Ergodic Hamiltonian systems with an arbitrary number of degrees of freedom n are considered. A relation is derived connecting the distribution function of the. In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a W is sometimes called the "thermodynamic probability" since it is an integer greater than one, while. We have then studied the relationship between Keywords: Boltzmann entropy, thermodynamic probability, axiomatic derivation, information, statistical.

It can be assumed, for instance, that one half of the molecules possesses twice as much energy as the other half; but if all molecules are intermixed properly and their total energy is equal, as before, to the internal energy of the system, then this new microstate will correspond to the same macro state.

Thus, proceeding only from the distribution of energy between individual molecules, one and the same macro state can be shown to correspond to an enormous number of microstates, bearing in mind that the difference between microstates is not always due to the different distribution of energy among the molecules.

The difference between the microstates can also be traced to other factors, for instance, to the distribution of molecules in space and also to the difference in the velocities of molecules with respect to magnitude and direction.

It should also be noted that the invariability of a macro state determines in no way the invariability of a microstate.

### Boltzmann's entropy formula - Wikipedia

As a result of the chaotic motion of molecules and the continuous collisions between them, for each moment of time there is a definite distribution of energy among the molecules and, consequently, a definite microstate. And since not one of the microstates has any advantages over another microstate, a continuous change-of microstates takes place.

In principle, of course, it is possible that a microstate corresponding to a new macro state different from the preceding one may set in. For instance, a case is possible at least in principle when molecules of greater energies concentrate in one half of the vessel, and molecules with lower energies concentrate in the other half of the vessel. As a result we would have a new macro state in which a fraction of the gas would be at a higher temperature than the other. It should not be thought that as a result of the continuous change of microstates a system for instance, a gas in a vessel must necessarily undergo a change in microstates.

One of the microstates usually has a rather large number of microstates which realize exactly this macro state. It would, therefore, seem to an outside observer having an opportunity to determine the change of only thermodynamic properties that the state of the system does not change. Let us now turn to the concept of the thermodynamic probability of the state of a system. The term thermodynamic probability or statistical weight of a macro state, is the name given to the number of microstates corresponding to a given macro state.

As distinguished from mathematical probability, which is always expressed by a proper fraction, the thermodynamic probability is expressed by a whole, usually very large, number. Further in paper, we investigate equivalence of Sackur-Tetrode equation with classically obtained relation on thermodynamical probability.

However, the equation of state cannot be obtained from the laws of thermodynamics, experimental data play major role.

Such difficulties can easily be solved with statistical modeling, hence Statistical Mechanics is taken in use to successfully solve problems related to physical systems containing large number of particles.

Contrary to the existence of advanced statistics, the approximation and relevance at fundamental level still exist as improperly defined concepts. In this paper, we discuss the entropy for phase space positions for particle systems. Further, we reconsider the existing concept of Fig.

Paper is organized as follows. Section III equation can be written as, includes discussion on mathematical simulation followed by conclusion. While 1 represents the phase space shift from tothe returning route may not be 1. So, will also return to their initial phase spaces. This 1 1 where c is number of ensemble s clearly indicates a certain minute but not compartments and n indicates number of completely negligible change in microstate in compartmental divisions of the ensemble.

Further we consider the paths along which particle can move to attain different phase expressed with provided fluctuations. Contrarily, if particle Now with Clausius and Boltzmann-Maxwellian takes another route and in process spending more interpretations on entropy, we estimate the energy no matter it returns approximately to entropy difference for two random paths a and initial phase space finally.

Comparatively on large b between two specified phase spaces. Hence, consideration of ensemble s This yields, internal fluctuations is important.

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Entropy and thermodynamic probability distribution over phase spaces Nullifying factor, as defined, can be calculated with momentum, unlike the case in quantum physics fluctuation values which cannot be neglected. We where states are discrete and quantized.

Also, we will can easily correlated fluctuations with microstate- assume a three-dimensional space, small particle macrostate correspondence, so taking Boltzmann- density, and no intermolecular interactions. In the Maxwellian interpretations in consideration. Since we consider this situation in 3D container, we can claim that p is proportional to pas we 2.

All eight possibilities are shown in Fig. Also, we must realize that the crystal will not stay perpetually in any of these eight arrangements. Energy will constantly be transferred from one atom to the other, so that all the eight arrangements are equally probable. Let us now supply a second quantity of energy exactly equal to the first, so that there is just enough to start two molecules vibrating. There are 36 different ways in which this energy can be assigned to the eight atoms Fig.

Because energy continually exchanges from one atom to another, there is an equal probability of finding the crystal in any of the 36 possible arrangements.

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A third example of W is our eight-atom crystal at the absolute zero of temperature. This is true not only for this hypothetical crystal, but also presumably for a real crystal containing a large number of atoms, perfectly arranged, at absolute zero.

When two crystals, one containing 64 units of vibrational energy and the other at 0 K containing none are brought into contact, the 64 units of energy will distribute themselves over the two crystals since there are many more ways of distributing 64 units among atoms than there are of distributing 64 units over only atoms. The thermodynamic probability W enables us to decide how much more probable certain situations are than others.

Consider the flow of heat from crystal A to crystal B, as shown in Fig. We shall assume that each crystal contains atoms. Initially crystal B is at absolute zero. Crystal A is at a higher temperature and contains 64 units of energy-enough to set 64 of the atoms vibrating.