What is entropy and probability

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Entropy and probability

The entropy is defined by the reversible amount of heat exchanged at a temperature:

The entropy can also be explained as the distribution of particles over a given number of places. The most likely arrangement is the one with the greatest number of implementation options (micro-states). In the following example the (mathematical) probability of gas particles is given to be only in a partial volume instead of in the total volume.

A gas molecule that is in a partial volume is separated from the rest of the total volume. You open the boundary so that the molecule can move in the entire volume. The probability of finding the molecule in the original volume is now

The (mathematical) probability is equal to the classical probability, which is a quotient of the number of favorable possibilities and the number of total possibilities.

If the process is repeated with two molecules, the probability of finding both molecules in the partial volume is given by the product of the individual probabilities:

With three molecules, the probability is that after the expansion all three are randomly located in the partial volume again at the same time:

In general, the probability of finding N molecules in a partial volume is:

With a large number of particles (e.g.) you will almost never find all particles again in the partial volume after an expansion. The expansion of gas into a vacuum is an irreversible change of state, because the probability that the gas particles are in the original volume is negligible. The entropy of the particles has increased during this process.