Statistical Thermodynamics

Erwin Schrodinger

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318608
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

In this concise volume, one of the founder of quantum mechanics and one of the greatest theoretical physicists of the century (Nobel laureate, 1933) attempts to develop a simple, unified standard method of dealing with all cases of statistical thermodynamics (classical, quantum, Bose-Einstein, Fermi-Dirac, etc.)
The level of discussion is relatively advanced. As Professor Schrödinger remarks in the Introduction: `It is not a first introduction for newcomers to the subject, but rather a 'repetitorium.' The treatment of those topics which are to be found in every one of a hundred text-books is severely condensed; on the other hand, vital points which are usually passed over in all but the large monographs (such as Fowler's and Tolman's) are dealt with at greater level.`

CHAPTER VIII

EVALUATION OF THE FORMULAE. LIMITING CASES

TO determine the actual behaviour of such a degenerate gas requires the numerical evaluation of the two definite integrals for varying ζ. We indicate the general plan of this work.

First, from (7.24), viz.

we get the functional relation between

Then, from (7.26) (and (7.30)) we get

The latter gives us the departure from the ordinary gas laws, for it is just 1 for them. Indeed, if we divide (8.3) by (8.1), member by member,

Now for ζ very small we get for the ratio of the two integrals

So ζ very small gives (in both cases) the classical behaviour. (ζ is also called the parameter of degeneration.) Both integrals are then very small, and that means, from (8.1),

That is: high temperature, low density. This is at once (a) satisfactory, (b) disappointing, (c) astonishing.

(a) It is satisfactory because we have to find the classical behaviour for high temperature and low density (at least in the Bose case) in order not to contradict old, well-established experimental evidence.

(b) The densities are so high and the temperatures so low—those required to exhibit a noticeable departure—that the van der Waals corrections are bound to coalesce with the possible effects of degeneration, and there is little prospect of ever being able to separate the two kinds of effect.

(c) The astonishing thing is that the ‘new statistics’ which replaces just by 1 the factor

(very large in ‘the old one’, indeed its outstanding feature) should ever give the same behaviour as the old one (if at all, one might expect this rather at T → 0, where the factor would approach to 1 in the old theory!).

The solution of this paradox is, that this factor when worked out, applying classical statistics to the quantum levels of the single particles, is not just 1 but n!. And that ‘does no harm’, because it is constant (the harm it does work after all we shall see presently). In other words, the quantum cells are, at high temperatures and with low density, so numerous, that on the average, even in the ‘most populated region’, only every 10, 000th or 100, 000th is occupied at all. The n_s are either zero (most of them) or 1, hardly ever 2. And that is why it makes no difference whether the latter possibility is either excluded (Fermi-Dirac) or endowed with a greater statistical weight (Bose-Einstein)—it is negligible anyhow.

The above contention about the occupation numbers is made good by the following considerations. We recall the expression for the average of the occupation number n_s (7.16)

Now for

, since

, we can omit the

1 and have

showing at once that

when ζ is, and that proves the contention. Moreover, since in the ‘really most populated’ region, which is

, the exponential is still of the order of unity (not smaller), we can say that

gives the true order of magnitude also in the truly interesting region. It is worth while to inquire just how small it is! (I maintained above that it was about 1/10,000th or 1/100, 000th.)

That we easily get from (8.1)

or

This is expected to be a large number. Let us compute it for normal conditions (0° C. and 1 atm.) and for helium, the lightest monatomic gas, taking for convenience 1 mol.:

Hence, in these conditions

The occupation would remain extremely scarce even under strong compression and considerably lower temperature (see (8.7) and (8.8)).* But at the same time we can estimate that if a compression to about l/100th the volume and a refrigeration to about 1/100th the temperature (thus to 2–3° K.) could be performed without liquefaction, that would give a factor 1/100, 000th, and we would just reach the region where ζ ceases to be ‘very small’. So the region of noticeable gas degeneration is by no means outside the reach of experiment, only (as I said) its effects are inextricably mixed up with the ‘van der Waals corrections’.

The entropy constant. But eqn. (8.8) has also a direct and important application to experiment, viz. for computing the so-called entropy constant or chemical constant, or, to put it more concretely, the vapour-pressure formula of an ideal gas. And that it gives it correctly (while the classical theory gives pure nonsense) is the true justification of the new point of view.

Remember that we had found

from which the entropy

(Thus in (8.9) we have virtually computed the entropy; that is why I took the trouble to compute it exactly instead of merely estimating it.) But we are now interested in the general connexion, and using (8.8) we get

Please note in the first place that this expression is sound, as regards the dependence on V and n; if you increase n and V proportionally, S takes up the same factor. That may seem trivial, but that is just its first and supreme merit—it is just the point in which the classical point of view pitiably fails, as we shall see.

After having taken due notice of this soundness, we now refer to 1 gram-molecule, so that nk = R, the gas constant. In the argument of the first log we supply a 1/k (correcting for it in the constant) and then use

because it is more usual to speak of the pressure than of the volume in this connexion (viz. in the case of the saturated vapour, to which we sha...