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Lord Kelvin on the Age of the Earth

PROF. PERRY and I had not to wait long after the publication of his article “On the Age of the Earth” (Nature^ January 3, 1895, pp. 334 337) to learn that there was no ground for the assumption of greater conductivity of rock at higher temperatures, on which his effort to find that the consolidation of the earth took place far earlier than 400,000,000 years ago, is chiefly founded. In a letter of date January 13, most courteously written to me by Dr. Robert Weber in consequence of his having seen by my letter to Prof. Perry, of December 13, that we were anxious to find how far his experimental results regarding differences of thermal conductivity and specific heat at different temperatures could be accepted as trustworthy, he tells me that he had made farther experiments on an improved plan, and that on the whole his investigations do [not seem to prove augmentation of conductivity with temperature; and he kindly gives me, with permission to communicate to Nature, the following results, hitherto unpublished, of experiments which he made in the years 1885 and 1886 on the thermal conductivities (k) and specific heats (c) of five rocks. Density. Basalt 3'0144 c = 0'1763 + 0'000396t [between 0° and 60'J c = 0-1946 + 0'000575 (t — 60) [between 60° and 110°] k = 0'00317 1+0'0000lt - Marble 2'7036 c = 0'20279 + 0 000466. t k = O 005401—0'000005.t J- Rock salt 3-161 c = 0'2146 + 0 00017t k = 0'0137 1 — 0'0044t - Anhydrite of Jura 2'892 c = 0'1802 + 0 0003.t. k = 0 0133 1 — 0^0024.t - Quartz 3'638 c = 0'1754 + 0 0004. t k = 0'01576 1—0'0019.t - These results show practically no change of thermal conductivity with-temperature for basalt and marble. For rock salt, anhydrite of Jura, and quartz, they show diminutions of thermal conductivity amounting per 100° C. to 44 per cent., 34 per cent., and 19 per cent. respectively. They contrast curiously with the 75 per cent. augmentation of thermal conductivity per 100° C. (Nature, January 3, p. 336), used by Prof. Perry in his estimate of the age of the earth, and they form a practical comment on his statement (Nature, January 3, p. 336): “From the analogies with electric conduction, one would say, without any experimenting, that as a metal diminishes in conductivity with increase of temperature, so a salt, a mixture of salts, a rock, may be expected to increase in conductivity with increase of temperature.” Since the beginning of January I have myself been endeavoring to find by experiment the proportionate differences of thermal conductivity of rocks at different temperatures; and before the end of January I had made some preliminary experiments on slate and sandstone, from which I was able to tell Prof. Perry that the thermal conductivity of each of these two rocks is probably less at higher temperatures than at lower. Since that time I have been arranging for experiments on granite, in which as rapid progress as I would have liked has been impossible for many reasons, including the necessity of standardizing a Kew certificated thermometer of 1886, now for the first time being compared with an air thermometer in my laboratory. Unless its differences from the air thermometer are much larger than can be expected from what we know of the behavior of mercury in glass thermometers generally, it is already almost proved that the thermal conductivity of granite is less between 150° C. and 350' than between 50° and 150'. As to specific heats there can be little doubt but that they increase with temperature up to the melting point of rock, but the rate of augmentation assumed by Prof. Perry is about five times as much as that determined up to 1,300° by the experiments of Rucker and Roberts-Austen (Phil. Mag., 1891, second half year, p. 353) for basalt, and of Carl Barus (Phil. Mag., 1893, first half year, pp. 301-303) for diabase; these being apparently the only experiments hitherto made on specific heats of rock at temperatures beyond the range of the mercury-in-glass thermometer. Taking the primitive temperature as 4,000° C. and the thermal conductivity and the specific heat at this temperature respectively 30 times and 14^ times their values at the surface, and throwing in a factor 3 for threefold density at the greater depths (though the average density of the whole earth is scarcely double that of the upper crust), Perry takes the product of threa factors 30 X 14^ X 3 and so finds in round nutn- beis 1,300 times my estimate as his corrected estimate of the age of the earth !! (Nature, January 3, p. 337.) But even if the ratios of thermal conductivities and of specific heats at the higher and lower temperatures were as assumed, Prof. Perry's product of the two corresponding factors vastly overestimates the age. Of this I thought I had given a sufficient warning when I wrote to him (December 13), “But your solution on the supposition of an upper stratum of constant thickness, having smaller conductivity and smaller thermal capacity than the strata below it, is very far from being applicable to the true case in which the qualities depend on the temperature.” (Nature, January 3, p. 237.) It is obvious that the supposed higher thermal conductivity and the higher specific heat, if beginning suddenly at a short distance below the surface, and continuing constant to the great depth, would greatly prolong the time of cooling to the same surface gradient, beyond what it would be with these qualities increasing continuously with temperature. For the' simple case of conductivity assumed to increase in the same proportion as specific heat, Prof. Perry has himself since given in a later communication (Nature, February 7, pp. 341-342) the necessary correction of his previous mathematics; and in an example of his own choosing (50 per cent. augmentation of each quality per 100' elevation of temperature), he now finds 131 times my estimate for the age of the earth, instead of 441 times as by the formula which he used in hi.. first article. When the ratio of thermal conductivity to specific heat per unit bulk varies with tbe temperature, the problem of secular cooling presents mathematical diffi- Bulties, which, so far as I know, have not been hitherto attacked; but I find it quite amenable to analytical treatment, and I hope before long to be able to offer a paper to the Royal Society of Edinburgh on the subject, as an appendix to my original paper “On the Secular Cooling of the Earth,” published in its Transactions (1863). I have already worked out numerically two cases, in one of which both conductivity and specific heat increase with temperature, and in the other the specific heat increases with the temperature but the conductivity is constant. The first of these is at present only interesting as a mathematical exercise because, according to present knowledge, it is more probable that the thermal conductivity decreases than increases with increasing temperature. To the results of the second I shall refer later as substantially helping us toward a revised estimate of the time which has elapsed since the consolidation of the earth. Twelve veal's ago, in a laboratory established by Mr. Clarence King in connection with the United States Geological Survey, a very important series of experimental researches on the physical properties of rocks at high temperatures was commenced by Dr. Carl Barus for the purpose of supplying trustworthy data for geological theory. Mr. Clarence King, in an article “On the Age of the Earth,” published in the American Journal of Science (vol. xlv, January, 1893), used data thus supplied, to estimate the age of the earth more definitely than was possible for me in 1863 with the very meager information then available as to specific heats, thermal conductivities, and temperatures of fusion. I had taken 7,000' F. (3,871° C.) as a high estimate of the temperature of melting rock. Even then I might have taken something between 1,000° C. and 3,000° C. as more probable, but I was most anxious not to underestimate the age of the earth, and so I founded my primary calculation on the 7,000° F. for the temperature of melting rock. Now we know from the work of Carl Barus (Phil. Mag., 1893, first half year, pp. 186, 187, 301-305) that diabase, a typical basalt of very primitive character, melts between 1,100° C. and 1,170° and is thoroughly liquid at 1,300°. The correction from 3,871° C. to 1.300° or 1/3 33 of that value, for the temperature of solidification, would, with no other change of assumptions, reduce my estimate of 100,000,000 to 1/(3'33)2 of its amount or a little less than 10,000,000 years; but the effect of pressure on the temperature of solidification must also be taken into account, and Mr. Clarence King, after a careful scrutiny of all the data given to him for this purpose by Dr. Barus, concludes that without farther experimental data “4 we have no warrant for extending the earth's age beyond 34,000,000 of years.” By the solution of the conductivity problem to which I have referred above, with specific heat increasing up to the melting point, as found by Rucker and Roberts-Austen and by Barus, but with the conductivity assumed constant, and by taking into account the augmentation of melting temperature with pressure in a somewhat more complete manner than that adopted by Mr. Clarence King, I am not led to differ much from his estimate of 34,000,000 years. But, until we know something more than we know at present as to the probable diminution, or still conceivably possible augmentation, of thermal conductivity with increasing temperature, it would be quite uninteresting to publish any closer estimate. In the latter part of Mr. Clarence King's paper on the “Age of the Earrh,” the estimates of the age of the sun's heat by Helmholtz, Newcomb, and myself, are carefully considered, and the following sentences with which the paper is brought to a conclusion will, I am sure, be interesting to readers of Nature: “From this point of view the conclusions of the earlier part of this paper become of interest. The earth's age, about twenty-four millions of years, accords with the fifteen or twenty millions found for the sun. “In so far as future investigation shall prove a secular augmentation of the sun's emission from early to present time in conformity with Lane's law, his age may be lengthened, and further study of terrestrial conductivity will probably extend that of the earth. “Yet the concordance of results between the ages of sun and earth certainly strengthens the physical case and throws the burden of proof upon those who hold to the vaguely vast age, derived from sedimentary geology.” Kelvin.

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