12 A SYSTEM OF PHYSICAL CHEMISTRY completely chaotic. The probability of disorder is very much greater than the probability of complete order. The thermodynamic probability of this "standard" state (that is the probability of complete order) is taken to be unity. It follows therefore that the thermodynamic probability of a real equilibrium state is an integral number usually much greater than unity. Thermodynamic probability differs therefore from mathematical probability in that the latter is always a fractional quantity, i.e. it denotes the ratio of the number of favourable cases to the total number of possible cases. Thermodynamic probability is proportional mathematical probability. to but is not identical with It has just been stated that a system in equilibrium possesses a dis- tribution or arrangement which is characterised by a maximum value for the thermodynamical probability of the state. It is necessary before going further to give a somewhat more concrete idea of what we mean by states or arrangements and the probability of arrangements. Let us leave the problem of molecules and turn to a very simple kind ofsystem which can undergo various arrangements. Let us suppose that we have two squares or areas denoted by the symbols I and II, and further let us suppose that we have five letters, a, b, c, d, e, and we wish to distribute or arrange these letters between the squares in every possible way. It is evident that all possible ways are included in the following :—
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.