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that they employed a method differing from that employed by the latter physicist, and also that the substances examined could not be considered perfectly identical. Had De la Rive and Marcet examined more thoroughly into these differences, they must easily have perceived the incorrectness of their mode of accounting for them (the method of cooling must give larger, not smaller numbers than Regnault's method of mixtures; small impurities, trivial differences in the physical state of an element, may well alter the specific heat of that element 1 per cent. or so, but surely not 30 to 60 per cent.), and that remarkable property of carbon the announcement and examination of which forms a part of the following communication would thirty years ago have most probably been discovered.

In his comprehensive research on the specific heat of solid bodies*, Kopp estimated anew the specific heats of carbon, boron, and silicon, using a modification of the method of mixtures; his results were as follows:

Gas-coke .... 0.185 | Amorphous boron 0-254 | Amorphous silicon 0.214
Furnace-coke.. 0.166 Crystallized boron 0.230 Fused silicon .... 0138
Native graphite 0.174
Crystallized silicon 0.165

These numbers are smaller than those of Regnault. Kopp explained the different numbers obtained by using different modifications of carbon by supposing that carbon has in reality but one specific heat (that of diamond, 0.1469), and that the other varieties give higher numbers inasmuch as, being porous substances, they absorb gases, and on coming into contact with the water of the calorimeter evolve a small quantity of heat. Kopp believed that all the allotropic modifications of each element possess the same specific heat, and that variations in the number actually obtained are due to the errors of experiment, or to the use of impure materials.

Several years later (1868) Wüllner and Bettendorf attempted to show that Kopp's hypothesis was untenable, that Regnault's numbers were perfectly reliable, and that the smaller numbers obtained by Kopp did not justify the conclusion which he had drawn. The following are the numbers obtained by these authorst :

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These numbers agree very closely with Regnault's. Willner

*Liebig's Annalen, Ser. 3. Sup. vol. pp. i & 289.
† Pogg. Ann. vol. cxxxiii. p. 293.

and Bettendorf concluded that the different modifications of carbon have really different specific heats. In obtaining these numbers a small error was introduced by the following circumstance. The substance to be examined (1 to 5 grms.) was warmed in a glass with water to about 70°, and then allowed to cool to about 20° in the calorimeter. In calculating the specific heat of the substance, it was assumed that the specific heat of the water remained equal to unity throughout the experiment. Now the average specific heat of water between 20° and 70°, according to Bosscha*, is 1.0099. Taking this number into account, Wüllner and Bettendorf's numbers become:

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If we use the number for the specific heat of water between 20° and 70°, obtained by Jamin and Amaury†, these numbers become still smaller. We may say, then, that the numbers obtained by Wüllner and Bettendorf stand midway between those of Kopp and those of Regnault.

From a consideration of the results of all the researches upon the specific heats of carbon, boron, and silicon, it may safely be averred that the different allotropic modifications of these elements possess different specific heats, and that no one of these elements in any of its modifications obeys the law of Dulong and Petit. These three solid elementary bodies differ in this respect, therefore, from all the other elements. It is also, however, evident that the numbers obtained by the different experimenters diverge considerably from one another. The four series of determinations have not presented us with any two exactly similar numbers. The differences in the individual results are so great and so general as to preclude us from believing that they are due to errors in the methods of experiment, or to impurities in the substances themselves. Some circumstance really conditioning the specific heat of these elements, and the value of which differs in the four series of experiments, must be present.

With this idea I undertook, in the winter of 1871-72, an analysis of the estimations hitherto made of the specific heats of carbon, boron, and silicon, and arrived at the following conclusion:-The different observers have determined the specific heats of these elements for entirely different intervals of temperature; and the greater the interval of temperature for which the specific heat is determined, the greater is the number representing that specific heat. The following Table shows that this is the case. Column C * Pogg. Ann. Jubeibd. p. 545. † Comptes Rendus, vol. lxx. p. 661.

contains the observed specific heats; the column AT gives the degrees of temperature between which the experiment has been

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From these facts I concluded that the specific heats of the different allotropic modifications of these three elements increase in an altogether surprising manner. Inasmuch as the experi

ments upon carbon are the most numerous and the most trust. worthy, the foregoing conclusion can be most confidently applied to this element. To boron and silicon the inference cannot be applied with so much certainty. Kopp's numbers for a temperature ranging from about 20° to 50° are certainly smaller than Regnault's for the interval of temperature 20° to 100°; yet inasmuch as Kopp worked with very small quantities and with a not very exact method, it is possible that the differences arose from circumstances other than the mere difference of temperature. The analogous behaviour of carbon and boron towards Dulong and Petit's law permits us to believe that the great variability in the specific heat of the former element will find its counterpart in a like variation in the specific heat of the latter.

In an experimental inquiry into the specific heat of the diamond*, I showed that the specific heat of this body increases, with increase of temperature, more quickly than that of any other substance; the values at 0°, 100°, and 200° were almost in the ratios 1:23. The number expressing the relation of the specific heat of the diamond y, to the temperature t was calculated from the following equation:


* Ber. der deut. chem. Ges. 1872, p. 305; and Pogg. Ann. vol. cxlvii. p. 317.

Perfectly analogous results were obtained for graphite in two series of researches. From experiments made in the autumn of 1872, I concluded that the specific heat of crystalline boron increases with the temperature exactly as that of diamond and graphite increases, and that, at any rate for low temperatures, the specific heat of silicon varies with the temperature. Following up these preliminary researches, I have now for two years busied myself with an inquiry into the relation subsisting between the specific heats of the various modifications of carbon, silicon, and boron, and the temperature at which these are determined. As the investigation advanced I found that comprehensive determinations of the specific heat of these elements, as a function of the temperature, might lead to a removal of the idea that these bodies form an exception to the law of Dulong and Petit, and also to a new point of view from which to regard this law itself—that these numbers might also be useful in further determinations of the specific heat of the various modifications of any element, or of the specific heats of the elements when in chemical combination. I therefore spared no trouble in obtaining exact and trustworthy determinations of the specific heats of these elements at as many and as various temperatures as possible. Over a hundred careful measurements have been made. As regards-carbon in the free state, all the problems which presented themselves have been solved; the specific heat of the different carbon-modifications has been determined for all temperatures between -80° and +1000°. Only in connexion with the elements boron and silicon a few questions yet remain unanswered. I prefer, however, to publish the results which I have obtained; and in the next Part I hope to correct any errors and to extend the investigation.

The greater part of this research was carried out in the Physical Institute of the University of Berlin, between December 1872 and July 1873, the remainder being conducted in the present year at Hohenheim and Stuttgart. (The author expresses his thanks to Professors Helmholtz, Rammelsberg, and G. Rose of Berlin, Tschermack of Vienna, and Marx of Stuttgart.)


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Let W,, W. W... W be the amounts of heat which G weightunits of a body give up when cooled from the initial temperature T1, T... T to the final temperature To in a calorimeter; n different values for the specific heat of the body yr in relation to T+To T2+T T+Tthe temperatures





may be deduced

(and hence what function of the temperature the specific heat represents), provided that the temperature-differences T1-To,

T2-T1, ... T2-T2-1 fulfil certain conditions. Between yr and the


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To T, T2, T... the relations hold good:








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If the differences T1-To, T2-T1,... Tn-T2-1 of the limits of integration be taken so small that the parts of the area between them are represented as trapezes, we may set down the following equations:

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How great the differences of the values To, T1, T2,... T, may be taken without introducing material differences between the true values of yr and the values as calculated depends upon the nature of the functiou yr. Preliminary experiments proved that the average W-W, n-1 specific heat of the elements carbon and boron G(T-T-1) within the temperature-intervals 0° to 100°, 100° to 200°, and 200° to 300°, altered almost with the temperature. For these two elements, therefore, the temperature-differences T1-To, T2-T1,... may rise to 50°. At high temperatures, between 500° and 1000°, experiment showed that the specific heat of carbon changed but very slightly as the temperature rose; within these limits, therefore, the temperature-differences may amount to 200°. For silicon, as the result of experiment, from 100° to 200° a temperature-difference of 50° was allowed; from 0° to 100° this difference must not exceed 30° to 40°.

The methods and the apparatus by means of which the values of

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