8. ENTROPY

A new physical quantity related with the direction in which physical processes take place. 


 


It has been seen in Section 2 Pressure that a gas occupying partially a container evolves freely until its particles are uniformly distributed over the entire container, without changing its internal energy. We have also seen that two systems at different temperatures when held in thermal contact evolve until both reach the same temperature. In both cases the First Principle does not predict the direction in which the systems will evolve. The first Principle is compatible with the fact that the gas particles occupy all the space available in the container or just a half of it, as far as there is no work involved. It does not take care wether the system at higher temperature releases or gains energy as far as the change in the internal energy equals that energy transferred. A new concept is needed to describe all this phenomenology.


 


These questions are essentially related with the fact that macroscopic systems are made of a huge number of particles.

Let us consider the statement:

A system in equilibrium has the most probable configuration.

Consider a system like that shown in the simulation, but much simpler: 5 particles in a container that is divided into 5 cells. Assume that the probability for a particle to occupy a given cell is the same for all cells. We could find the five particles in one cell, or three in one cell and two in another one, etc. Which is the most probable distribution?
 



 


Some of the possible distributions are shown in the figure. Click here to see the answer.

The differences in the probabilities of the different configurations are not extremely large with five particles, though they are significant. But real systems are made of a number of particles of the order of Avogadros's number, a number with 24 significant digits!

A new concept is defined, related to the number of different ways in which the particles of the system can be distributed

The entropy of the system is the natural logarithm of the number of different possible distributions.
and the following Principle is stated
A system in equilibrium has the maximum entropy.
which is equivalent to say
The microscopic state of a system in equilibrium is that one of maximum probability.
This Principle allows to cope with phenomena like those discussed at the beginning of this Section. A gas in equilibrium occupies uniformly the container because this is the most probable distribution, i.e. that with a larger number of associated configurations of the particles.

The new concept is related with probability. This would seem to be something not very deterministic in principle, but considering the large number of particles in real systems it may be understood that some deterministic statements could be made with very high accuracy. It would not be impossible that all the molecules of the air in this room be concentrated in a half of the room, but nobody has died until now by that reason and we certainly should not worry about that!
 
08.Q1. QUESTION:
The entropy of a system
Depends only on the number of particles of the system.
Is directly related with the number of ways of distributing the particles according to the specifications on the system.
It is independent of the energy of the system.
Another problem that can be approached in a similar way is the distribution of a given amount of energy among a number of particles that can suffer collisions and then exchange energy. This will be the subject of next Section.

Velocity Distribution
 
Index 1.Introduction 2.Pressure 3.Temperature 4.internal Energy
5.Heat 6.Work 7.First Law 9.Velocity Distribution 10.Specific Heat