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