A system of physical chemistry - Index of

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A system of physical chemistry - Index of

ENTROPY AND THERMODYNAMIC FROBABILITY \i

can interchange squares I and II in the other two arrangements, so

that in all, there are six arrangements possible in this system, and in

all, thirty-two different ivays in which these six arrangements or distributions

may be carried out. Each of these ivays is analogous to

a " complexion " on Boltzmann's nomenclature. Every way or complexion

is to be regarded as equally probable. It is very necessary to

distinguish between arrangement or distribution and complexion. Thus

in the first case, the arrangement is five letters in square I, none in

square II. There is only one way or complexion of doing this. In

the second arrangement, which consists of lour letters in square I and

one letter in square II, there are five ways or complexions of doing

this. It is simply a question of combinations. Thus there are five

letters to be divided in such a way that four are in one square, one in

another square. The number of possible ways of doing this particular

distribution is ,-^=7- = 5. In the third arrangement or distribution

we have to distribute the letters so that there are always three in

square I and two in square II. There are ten ways or complexions

corresponding to this single distribution. This number is given by


, =10.

Note that the problem is not which letters are in the

squares, but how many different ways can they bed ivided to correspond

with any particular arrangement, such as three letters in one

square, two in the other.

If instead of five letters we had N letters and divided them between

two squares, in such a way that n letters are in square I and N-« letters

in square II, the possible ways or complexions possible to this particular

distribution are—

jN

|N - « \ri

If, instead of two squares, we had m squares, the total number of

complexions in a particular distribution would be given by —

|N

1^1 1^2 1% '• • • |^»t

where n^ \- n-i -\- n-^ \- , . . «„, = N.

To return to the simple case of five letters and two spaces. We

have seen that there are six possible arrangement or distributions, viz.

(5) (o); (4) (i); (3) (2); (o) (5); (i) (4); (2) (3). of these

.Each

arrangements has its own number of ways or complexions. Thus for

arrangement (5) (o), the number of complexions is one. For arrange-

ment (4) (i), the number of complexions is five for ; arrangement

is ten. Similar numbers of

(3) (2), the number of complexions

complexions are found in the remaining arrangements or distributions.

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