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character from H and OH, no such simple behavior is to be expected, nor is it found Rules for such systems have still to be discovered.


The reason for the above dependence of solubility upon compound formation becomes evident when we examine the freezing-point depression curve of a system such as Li2SO4-H2SO4. The absolute freezing-point of the salt is so high that, unless compound formation took place, solubility in the acid at ordinary temperatures would, according to the Schröder-Le Chatelier equation, necessarily be negligible. The removal of simple solute molecules by combination with solvent, however, results in an increased solubility, and the extent of such increase depends, naturally, upon the extent of compound formation. The further development of this topic and the presentation of quantitative results will be given in forthcoming articles in the Journal of the American Chemical Society.

* Contribution No. 350.

1 Kendall and Booge, J. Amer. Chem. Soc., 39, 1917 (2323).

2 Carpenter, Ph.D. Dissertation, Chicago, 1915; Gibbons, J. Amer. Chem. Soc., 37, 1915 (149); Booge, Ph.D. Dissertation, Columbia, 1916; Gross, Ph.D. Dissertation, Columbia, 1919; Landon, Ph.D. Dissertation, Columbia, 1920; Adler, Ph.D. Dissertation, Columbia, 1920; Davidson, Ph.D. Dissertation, Columbia, 1920.

Kendall and Booge, loc. cit.; Gross, loc. cit.

4 Roozeboom, Heterogene Gleichgewichte, 2, 1904 (270-84).




Read before the Academy, November 16, 1920


When a current of electricity goes from metal a, in which the freeelectron fraction of the conductivity is (kk) a, to the metal ẞ, in which the corresponding fraction is (kk), ionization or reassociation must take place at the junction, according as (k÷k), is greater or less than (kk) a. The resulting change of condition is the same whether we assume the ionization or reassociation to take place after or before the crossing of the interface between a and B. We shall use each of these hypotheses in turn, beginning with the first.

The gain of energy, and so the heat energy absorbed, by the unit quantity of electricity, 10 coulombs, in the form of electrons, in crossing the boundary from a to ẞ is

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· − (†). (P. - P.), + (†). (P. - P.) + (P, − P.)





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where (PB-Pa)f means the Pf of ẞ minus the P of a, etc., the nomenclature according in general with that of my preceding papers.

From my paper in these PROCEEDINGS for April, 1918, we get, by integration of equation (5) there given,

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Integrating equation (6) of the same paper, we get

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If we had used at the beginning the hypothesis that the ionization occurred in a, just before the crossing, we should have obtained the expression

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The total change of state being the same in the process which gives equation (5) as in that which gives equation (4), the total heat absorbed must be the same in both cases—that is, the II of (5) is the same as the II of (4). Subtracting (5) from (4) we get

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Equation (8) has a familiar form, and in fact it might have been written down at once without circumlocation as soon as the fact was recognized, as it was in my paper of April, 1918, that the free electrons in the one metal are in equilibrium with those in the other, there being no circulatory motion of the electrons at the junction if the metals are at one temperature. Making use of equation (7) we can change (4) to the form

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It is now in order to see how well values of II calculated by means of


this formula, with the tentative values of (kƒ ÷ k) and λ obtained for each metal in my paper on the Thomson Effect and Thermal Conduction,1 agree with those found by Bridgman from his experiments. Close agreement is, of course, not to be expected, even if my theory is everywhere correct, for the values of (ky÷k) and λ given in the paper just mentioned were obtained without any regard whatever for the Peltier effect, they being merely certain values, not the only values, that will, approximately at least, meet the requirements of the Thomson effect and thermal conductivity. Moreover, the experimental data used in arriving at these values are doubtless faulty in many particulars; for example, in a number of cases I had no observed value of the thermal conductivity at 100° C., and was obliged to make use of a value calculated, not very safely, by means of the Wiedemann-Franz law.

Departing from the common practice, I have substituted bismuth for lead as the reference metal, taking as my definition of the Peltier heat the amount of heat (ergs) absorbed in the passage of (1÷e) electrons from any given metal to bismuth. All the values in the table below are to be multiplied by 104. Con. is constantan and Man. is manzaniņ.

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Column I gives the observed, or experimentally found, values of II, with respect to bismuth, at 0° C. Column II gives the values calculated,

by means of equation (10), from the values of (ky÷k) and λ, for 0° C., given in my already printed paper on the Thomson Effect and Thermal Conduction. Comparison of these two columns shows that the calculated values are three or four times as great as the observed values, but this disparity, before any adjustments have been attempted, is not discouraging. Column V gives the observed values for 100° C., while VI gives those found by means of equation (10) from the values of (ky÷k) and λ, for 100° C., given in the paper just mentioned. Here again the calculated values are some-fold larger than the observed values.

Columns III and VII, which are to be compared with I and V, respectively, are obtained from columns II and VI, respectively, merely by using new values of (kƒ÷k) and λ, for bismuth, these new values serving quite as well as the former ones, on the whole, for the Thomson effect and the thermal conductivity of bismuth.

Columns IV and VIII, which are to be compared with I and V, respectively, are obtained from III and VII, respectively, by using new values of (ky÷k) and λ for each of the other metals than bismuth, due regard being paid to the Thomson effect and the thermal conductivity in the selection of these new values.

The agreement between columns I and IV is as good as need be. The agreement between columns V and VIII is not so good, the values in V being in every case the larger, as column IX shows. A new readjustment in the case of bismuth, making [(k, ÷ k)]100 for this metal about 14% greater than the last re-adjustment left it, would reduce most of the differences shown in column IX to a negligible size. But such a re-adjustment cannot be made without introducing disagreements at other points. The disparities between columns V and VIII, together with equally serious disparities, at 100° C., between thermal conductivities given by my formulae and those found by other means, must stand for the present.

A plausible explanation of these discrepancies, where they are not to be acounted for by mere imperfection of experimental data, may be found in the crudity of my assumption that the number, n, of free electrons per cu. cm. of a metal, can be expressed by the formula

n = zT

where z and q are constants. In making this assumption I did not expect it to hold for so great a range of temperature as that between 0° and 100° C. In fact, I was greatly surprised to find, a year or so ago, that I could make the jump from 0° to 100° with any measure of success. It is interesting to inquire how the theory of the Peltier effect set forth in this paper can deal with Bridgman's observations on the magnitude and sign of this effect between compressed and uncompressed pieces of the same metal. Testing twenty metals, including two alloys, he measured the heat absorbed by unit quantity of "positive electricity in passing from uncompressed metal to [the same] metal compressed" or, as I prefer to

put it, by (1e) electrons in passing from the compressed to the uncompressed metal. Working from 1 atm. to 2000 kgm./cm. and from 0° C. to 100° C., he found this absorption to be positive throughout in fourteen of the metals; negative throughout in three, cobalt, magnesium, and manganin; mixed, sometimes positive and sometimes negative, in three, aluminium, iron, and tin.


In terms of my theory a positive effect here can be accounted for by a decrease of (kk)λ under pressure, and a negative effect by an increase. It seems probable that (kƒ ÷ k) is generally decreased by pressure, and that the increase of (k÷k)λ indicated for certain cases by Bridgman's experiments is to be attributed to an increase of A sufficient to overbalance the decrease of (ky÷k). A priori one might expect A to decrease with increase of pressure, causing reduction of volume, since, according to the formula ' ' + sRT, it increases with rise of temperature, causing expansion. But it is unsafe to assume that a contraction caused by pressure will have the same effect on the properties of a substance as a contraction caused by fall of temperature. Thus Bridgman says: "The volume of many metals at 0° C. and 12,000 kg. [per cm.2] is less than the volume at atmospheric pressure at 0° Abs. The resistance of most metals tends towards zero at 0° Abs., but at 0° C. at the same volume the resistance is only a few per cent less than under normal conditions.”

A change of about 8% in the value of (ky÷k) λ would account for the maximum Peltier effect between compressed and uncompressed bismuth, as observed by Bridgman, and a still smaller per cent change, in most cases much less than one per cent, would serve for the other metals dealt with in this paper.

1 These PROCEEDINGS, October, 1920, p. 613.

2 Proc. Amer. Acad. Arts & Sci., 52, No. 9 (638).



DEPARTMENT of Mathematics, University of CaliFORNIA

Communicated by E. H. Moore, December 13, 1920

The roots of the equation J(2) = 0 are known to be all real if n < -1. The methods of Sturm when applied to the function J, (2) show that the roots are increasing functions of n if n>0, that is to say, denoting by (n) the kth positive root (n')>(n) if n'>n>0. In the following it will be shown that (n')>¥Â(n) if n'>n>-1 so that the inequality holds as well when -1<n<0.

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so that J,(z)

They satisfy the recurrence

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