# Relationship between entropy and probability

### Boltzmann's entropy formula - Wikipedia

Entropy ~ a measure of the disorder of a system. A state of high order = low probability. A state of low . He was so pleased with this relation that he asked for it. Boltzmann's Relation Between Entropy and Probability Before deducing the relation, let us consider the actual situation. We know from kinetic theory that. 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 thermodynamic.

It must be stated once more that the microstate is determined by the totality of properties of all molecules of a system. It can be easily shown that for one and the same macro state of a system there may be a rather large number of microstates.

Consider a simple example. Let there be available a system which is a gas confined in a vessel of constant volume. Assume that, as was stated above, the macro state of the system is determined by its volume v and internal energy u. But the constancy of the internal energy of the system does not specify as yet the distribution of energy among individual molecules, i. Indeed, for the given macro state there may exist a microstate in which all molecules of the system have the same amount of energy equal to the internal energy of the system.

But for the same macro state there may also exist other micro-states.

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. 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.

### Thermodynamic Probability W and Entropy - Chemistry LibreTexts

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. 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. 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.

- 16.4: Thermodynamic Probability W and Entropy
- Boltzmann's entropy formula

Crystal A is at a higher temperature and contains 64 units of energy-enough to set 64 of the atoms vibrating. If the two crystals are brought together, the molecules of A lose energy while those of B gain energy until the 64 units of energy are evenly distributed between both crystals. In the initial state the 64 units of energy are distributed among atoms. Calculations show that there are 1. Thus W1, initial thermodynamic probability, is 1.

The atoms of crystal A continually exchange energy among themselves and transfer from one of these 1. At any instant there is an equal probability of finding the crystal in any of the 1.

## Heat and Thermodynamics by Anandamoy Manna

When the two crystals are brought into contact, the energy can distribute itself over twice as many atoms. The number of possible arrangements rises enormously, and W2, the thermodynamic probability for this new situation, is 3.

In the constant reshuffle of energy among the atoms, each of these 3. This example shows how we can use W as a general criterion for deciding whether a reaction is spontaneous or not. Movement from a less probable to a more probable molecular situation corresponds to movement from a state in which W is smaller to a state where W is larger.

In other words W increases for a spontaneous change.