SECTION C

DETERMINATION OF A PRECISE, NUMERICAL, EXPRESSION FOR THE ENTROPY OF ANY PHYSICAL CONFIGURATION

PLANCK modestly says that everything essential in the determination of this expression has been done by L. BOLTZMANN, in his wide range of physical investigation. PLANCK'S discussion, however, is so compact and lucid that it is best suited for our purpose. Keeping this purpose in mind we will here also abbreviate by dispensing with parts of PLANCK'S fuller proof and content ourselves with the main steps which lead to the desired expression. These main steps are as follows:

(a) Determination of the general expression for the of a given physical configuration of a known macroscopic state;

(b) Determination of the general expression for the Entropy of a given physical configuration of a known macroscopic state;

(c) Special case of (b) namely, expression for the Entropy of the state of thermal equilibrium of a monatomic gas;

(d) Confirmation, by equating this value of with that found thermodynamically and then deriving well-known results.

(e) PLANCK'S conversion of the expressions of (b) and (c) into more precise ones by finding numerical values of in C. G. S. units; in F. P. S. units;

(f) Determination of the dimensions of the universal constant and therefore also of entropy in general.

Step a
Determination of the Number of Complexions of a given Physical Configuration of a Known Macrostate

We will, for simplicity's sake, consider here an ideal gas in a given macro-state and consisting of N-like, monatomic, molecules. By generalizing the meaning of our co-ordinates, the following presentation could be made equally applicable to the more general case of Physics contemplated under this heading.

Of course we must here have clearly in mind what is meant by the state of a gas. For this we may refer to p. 10 (lines 13 to 24) and to p. 19 (lines 8 to 24). The conditions there imposed are all fulfilled if we suppose the state given in such a way that we know: (1) The number of molecules in any macroscopically small space (volume element); and (2) the number of molecules which lie in a certain macroscopically small velocity region (soon to be more fully described). To have the Calculus of Probabilities applicable, each of the tiny regions contemplated under (1) and (2) must still contain a large number of molecules and their motions must besides have all the features of haphazard detailed on pp. 25, 26; all this is necessary in order that the contemplated motions may possess all the characteristics of "elementary chaos."

Before proceeding further on our main line, we will define more fully what is meant by the two elementary regions in which lie respectively the molecules and their velocity ends. After this has been done we will, for convenience, combine these two regions into a fictitious elementary region, say, a six-dimensional one.

First there is the volume element , in which any molecule having co-ordinates lying between is located; this element can be conceived as a parallelopipedon whose edges are parallel to the co-ordinate axes; this is the simplest of the elementary regions here to be considered. To conceive of the elementary region containing the velocity ends of the molecules, let us suppose any origin for velocities in a unit volume and from this as a pole lay off as vectors the molecular velocities lying between the limits, where are the components of said velocities parallel to the respective co-ordinate axes. Then, under the velocity limitations imposed, the end of the velocity of each such molecule will lie in the elementary parallelopiped , one vertex of this parallelopiped having of course the co-ordinates . This parallelopiped can be regarded as a constructed volume within which the velocity end must lie. We have therefore here two elementary volumes and , which are independent of each other. Now remembering that the probability of any properly endowed molecule being found in one of these volumes is in each case equal to the number of molecules belonging or corresponding to the volume considered. Assuming, for the moment, an equal distribution of molecules and velocities throughout the whole volume, we may say that the number of molecules occurring, in each of the said elementary volumes, is proportional to their respective sizes; this is here equivalent to saying that the probability of any molecule thus occurring in said elementary volumes is proportional to their respective sizes. Having stated the probability of each contemplated occurrence, we can now say the probability of these two events concurring is equal to the product of the probabilities of said two separate occurrences. Moreover, as the probability of each occurrence is proportional to the size of its own elementary volume, the product of said probabilities will likewise be proportional to the product of the two elementary volumes. Here can be regarded as a sort of fictitious volume or region, constructed, say, in a six-dimensional space.[20]

The extent of such an elementary region is very minute in comparison with the total space under consideration, but still it must be conceived as sufficiently large to embrace many molecules, otherwise its state would not be one of "elementary chaos." On account of the equivalence here of probability and number of concurring molecules, we may for the present say that the number of the latter is proportional to the magnitude of this elementary region . But before we proceed further this last statement must be subjected to a correction, for we temporarily assumed above that there was an equal distribution of molecules and velocities throughout the whole volume. Now at the start, in defining the contemplated state, it was distinctly announced that there was an unequal distribution of such elementary conditions, the law of their distribution being given by the known number of molecules in each elementary volume and in each constructed elementary velocity volume . This correction is effected by the introduction of a finite proportionality factor, which can be any given function of the location and velocity co-ordinates, so long as it fulfills the one condition (put in abbreviated form), where .

Strictly speaking, the expression for the fictitious elementary region , formed by the product of and the constructed-volume element , should be replaced by the expression , where is the mass of a molecule. The reason for this substitution is found in the fact that the magnitude of the constructed-volume element varies with time due to the variation of velocities effected by molecular collisions. Now this variation of magnitude is not permissible with the probability considerations which here obtain. For the probability of a state which necessarily follows from another state must be like that of the latter. As the momenta after collision are the same as before collision, we have now in the momenta, co-ordinates which do not vary with time like their constituent velocities. Therefore if we substitute in (12) for the velocities their corresponding momenta, the variation with time of the constructed-volume will cease and the objection cited will no longer be a valid one.

Now let us take up the determination of the number of complexions in the given state. For this purpose think of this whole state as represented by the sum total of all these equal elementary regions ; for convenience of reference let us call this whole state the "state-region." The probability that a particular molecule will belong to a particular elementary region is equally great for all the elementary regions. Let represent the number of these equal elementary regions. Now we will proceed with the help of a parallel case. Let us think of as many dice as there are molecules and let each die be provided with faces. On each of these faces we will write in their order the digits 1, 2, 3, ... , so that each of the faces will designate a particular elementary region. Then each throw of the dice will result in representing a particular state of the gas, because the number of dice which show uppermost a particular digit will give the number of molecules belonging to the elementary region represented by said digit. In this parallel case each die is equally likely to show up any one of the digits 1 to , corresponding to the circumstance that each individual molecule is equally likely to belong to any one of the elementary regions. The desired probability of the given state of the molecules corresponds therefore to the number of different kinds of throws (complexions), by which the given distribution can be realized. For example, if we take molecules (dice) and elementary regions (dice faces), and assume that the state is so given that it is represented by: Then this state can be realized by one throw, in which the 10 dice show the following digits:

Under each of the 10 dice stands the digit shown uppermost in the throw. In fact, we see

In like manner the same state can be realized by many other such complexions. The desired number of all possible complexions can be found by considering the digit row designated above by (15). For, since the number of molecules (dice) is given, the digit row will have a particular number of places (). Besides, since the number of molecules belonging to each elementary space is given, each digit will occur equally often in the row in all permissible complexions. Moreover every change in digit arrangement effects a new complexion. The number of the possible complexions or the probability of the given state is therefore equal, under the conditions specified, to the possible "permutations with repetition."[21] In the simple example chosen we have for such permutation, according to a known formula, Consequently in the general case, we have where the sign signifies the product extended over all the elementary regions.

Result contained in (16) is equally true for any other physical system, say, one involving radiant energy. For the conditions and the variables are similar to those of the molecular system just employed. The chosen model, the dice system, which served as an easily conceived parallel case, would be equally serviceable in dealing with the elements of radiation.

Step b
Determination of a General Expression for the Entropy S of any given Natural State

This step is an easy one. We have in Eq. (10) the relation expressing the universal dependence of entropy on probability . Substituting and writing out the logarithm of the quotient given in (16), we have

The summation must be extended over all the elementary regions . With the help of STIRLING'S formula, and remembering that both and are constant for all changes of state, the above expression (17) is reduced to the form This magnitude is numerically the same as for which BOLTZMANN proved that it changed in a one-sided way in all changes of state. We must bear in mind too that function represents, for every state of the gas, the given space and velocity distribution of the gas molecules. The permanent, stationary, state of the gas known as thermal equilibrium is only a special case of the general case (18), this special case being widely known as MAXWELL'S Law of Distribution of Velocities.

Step c
Special Case of (b), Namely, Determination of Entropy S for the Thermal Equilibrium of a Monatomic Gas

This case PLANCK derives very easily from the general case represented by (18). As the desired result has already been found in another way in pp. 48-53 when dealing with MAXWELL'S Law (of distribution of molecular velocities), we will not repeat PLANCK'S derivation of the law from (18). It will suffice here to give the results: The law of distribution is given by function where and are constants and the total energy. As this expression for function is free from all location co-ordinates , we see that this state of thermal equilibrium is independent of these space co-ordinates and conclude that in this state the molecules are uniformly distributed in space; only the velocities are variously distributed, all of which accords with the earlier presentation. Substituting the results of (19) in the general equation (18) there results for the entropy S of the state of equilibrium of a monatomic gas, To make Eq. (20) practically serviceable we need to know the constants and and they will be found later on.

Step d
Confirmation, by Equating this Probability Value of S with that found Thermodynamically and Securing well-known Results

We know from Thermodynamics that the change of entropy is defined in a perfectly general way (for physical changes)[22] by

Deriving the partial differential coefficients and making use of (20), there follows: where = number of gram-molecules (referred to ) and = absolute gas constant [1545 in F.P.S. system]. Here the first of Eqs. (22), represents the combined laws of BOYLE, GAY-LUSSAC, and AVOGADRO. We get besides from the equating of (20) and (21), the additional relations, where mechanical equivalent C.G.S. system.

From this follows as is known for monatomic gases.

Furthermore, we find for the mean kinetic energy of a molecule

We also have

With the help of the specific heats and the characteristic equation of the gas, the whole thermodynamic behavior of the gas is disclosed. All this has resulted from the identification of the mechanical and thermodynamic expressions for entropy and is an indication of the fruitfulness of the method pursued. PLANCK also shows that this method leads to the finding of results heretofore unknown.

Step e
Conversion of the General Expressions in (b) and (c) into more Precise ones by Finding and Inserting the Numerical Value of the Universal Constant k; Some of the Results

From the consideration of certain phenomena of radiation PLANCK found where and are constants found by experiment while and are exactly known values, mathematically derived. The present accuracy of (27) therefore rests on the accuracy of the experiments from which and were found. In discussing Eq. (10) it was pointed out that was a universal constant, applicable to all physical systems and consequently may be used for the molecular configurations mainly considered in this presentation. But before introducing numerical value of in the general expressions contained under headings and , we will add other numerical values of interest.

PLANCK gives = number of molecules in 1 ccm. of an ideal gas at freezing-point ( C.) and atmospheric pressure; he also gives for the ratio = number of molecules per grams; the corresponding numbers in English units are, approximately, = number of molecules in one cubic foot of an ideal gas and = number of molecules in one pound of an ideal gas. Assuming air to be an ideal gas and its "apparent" molecular weight about 28.88, the number of molecules in one pound of air would be .

Substituting the numerical value of universal constant in Eq. (10) we get Eq. (28).

For C. G. S. system, Entropy of any natural state, Eq. (28) is

For F. P. S. system, Entropy of any natural state, Eq. (28) is

To each of these may be added an arbitrary constant. In Eq. (20) we may substitute directly the equivalent of the product found from Eq. (22), and then get for the entropy of a monatomic gas in the state of thermal equilibrium,

When the volume is known we can now readily find and then numerically, and place this number as a coefficient in Eq. (20).

Step f
Determination of the Dimensions of k or of the Entropy S

It is at once evident from an inspection of the perfectly general Eq. (10) that the dimensions of Entropy depend solely on those of the universal constant . The relation given in Eq. (21) shows at once that dimensions of depend upon the quotient found by dividing energy by temperature and the relations given in Eqs. (22) and (25) that the dimensions of constant also depend on this same quotient. The dimensions of Entropy and of constant are therefore identical and this might suffice to show that here neither nor is to be regarded as a mere ratio or abstract number. A word further in this connection may, however, be helpful. In reversible processes we have the well-known relation . To simplify matters, let us suppose heat supplied while volume is kept constant, then or Here Entropy has the same dimensions as ; now in the relation if we regard as an abstract number then, in order that the equation shall be homogeneous the factor () must represent heat energy like , and this is sometimes done; in such case (if retains its ordinary meaning) the quotient in Eq. (30) is no longer a mere ratio or abstract number, but a quotient of the dimensions of energy divided by temperature. On the other hand, if be regarded as of the dimensions of the quotient of energy divided by temperature, then we may consider in (30) as an abstract number or ratio and of the same dimensions as . When an absolute system of units is employed, which possesses as one of its features the expression of temperature in units of energy, then , and will all be mere ratios or abstract numbers.[23]