A system of physical chemistry - Index of

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A system of physical chemistry - Index of

ENTROPY AND THERMODYNAMIC PROBABILITY ii

demonstrated by Boltzmann. Hitherto we have regarded the second

law as a law of experience, its validity depending upon the fact that no

contradiction to it has been met with in nature. It is important to see

that this law possesses at the same time a mechanical basis. The

demonstration consists in showing the connection between the entropy

of a system — the concept of entropy involving necessarily the concept

of the second law—and a statistical quantity known as the thermo-

dynamic probability of the system.

It is necessary to recall first of all what is meant by thermodynamical

equilibrium, as stated in terms of entropy, that is as stated in terms of

the second law. Planck's definition of such equilibrium is as follows

{cf. Planck, Theory of Heat Radiation, English ed., p. 22) : "A system

of bodies of arbitrary nature, shape, and position, which is at rest and

is surrounded by a rigid cover impermeable to heat, will, no matter what

its initial state may be, pass in the course of time into a permanent

state in which the temperature of all bodies in the system is the same.

This is the state of thermodynamic equilibrium, in which the entropy

of the system has the maximum value, compatible with the total energy

of the system as fixed by the initial conditions. This state being reached,

no further increase in entropy is possible."

We know that heat, from the kinetic molecular point of view, is

represented by the kinetic energy of the molecules of a system, the

molecules moving about in a completely chaotic manner as a result of

collisions. Owing to collisions any ordered arrangement which the molecules

might be conceived of as possessing initially would be quickly

annulled, and completely disordered distribution, both as to position

and to molecular velocities, would ensue. This represents the direction

of change in any spontaneous or naturally occurring process. That is,

from the molecular standpoint a system always changes from an ordered

to a chaotic state, and the change will go on until the molecular motion

has become as disordered as possible. When this stage is reached,

there is no longer any reason for further change. When equilibrium is

reached the system has at the same time reached a maximum disorder

or " mixed-up-ness ". This involves the idea that a system in equili-

brium possesses a maximum value of the probability of the state, the

probability here referred to dealing with possible modes of molecular

arrangement and velocity. We may call this the thermodynamic prob-

ability.

According to Boltzmann the thermodynamic probability of an ideal

monatomic gas is a number which denotes by how many times or by

how much the actual state of a gas system is more probable than a

state of the same gas system {i.e. possessing the same total energy and

volume) in which the molecules are equally spaced and all possess the

same velocity. This " standard " state represents perfect order or

arrangement of the molecules. It is of course never realised in practice

owing to the disorder brought about as a result of collisions. The

standard state represents the stage farthest away from the equilibrium

state finally attained by the gas,

in which final state the system is

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