Elements of Physical Biology

Ay = a;,je™* ss a; ,2e “Pl ao. =F a; se >;* (33) The series contains only linear terms, and breaks off at the term in Ajj. If \;, is (numerically) the least of the \’s, then, evidently, after a sufficient lapse of time the term in \, outweighs all other terms, which thus become negligible, so that The coefficients a are easily determined? as functions of the d’s._ It is thus found that * For a somewhat remarkable method of integration (by a multiple integral), see A. Debierne, Les idées modernes de la Matiére, 1913, p. 328.

Thus, after a sufficient lapse of time, the substances S1, So, S;, are always present together in constant proportion, so that we have a moving equilibrium of a very simple type, illustrated graphically in figure 56, in accordance with some of the constants given in tables 28 and 29. The most slowly decaying substance here acts as Radioactive equilibrium of radium in contact with its disintegration products on the basis of data in Jour, Am. Chem. Soc., 1923, vol. 45, pp. 872-873

ae ee 1/\ = MEAN LIFE FERS GANAT TY eras whcaen koa Soca Ge woe eS eee 1.00 ton 2,440 years* RUC OT I sR PG eo iene APs. ts« no i01e Fe las = Cotes 6.23 grams| 5.55 days FEO NEA Reese oe some ea en ae Nee pete ee ee 3.37 mgm. | 4.32 minutes PMAGHUNICIE EShg BA oie 3 Lie ake coos Cees F 30.18 mgm. | 38.7 minutes VENTS RUIW gl © 5 tiie a ig eee Cee 21.91 mgm. | 28.1 minutes ea AR TUNIC ee seh ee aves BIS, ce cin REE 9.75 kgm. | 23.8 years Peep Onya Dieser 2 atu tal see GUase a 8.08 grams| 7.2 days Testig Gea ee Aes, ee ea en eee ee 220.01 grams|196 days

the controlling, slowly variable parameter, and sets the pace with which all the subsequent members in the transformation chain keep step, so that the polygons representing the system in its successive stages are geometrically similar. (Compare fig. 58 on p. 277.) The second of the two expressions for the ratio a;,/a,, calls for brief comment. Ifi < k, that is to say if the substance S; precedes, in the transformation chain, the substance S;, which has its lowest disintegration rate, then S; does not appear at all in the equilibrium. Hence when an aggregation of substances in radioactive equilibrium is found in nature, the substance at the head of the chain (the ‘“parent substance”’) is always the one of slowest disintegration rate. But, obviously we cannot from this draw any conclusion as to whether or not it is itself a product of disintegration of a pre-parent of more rapid

decay rate. Any such pre-parent would, as it were, have been weeded out in the evolution of the system, as being ‘unadapted”’ to present conditions, like the extinct species of biology.* Fia. 56. Equrnrprtum PoLyGoN FoR RADON IN Contact wiTH Its DIsIN- TEGRATION PRopUCcTS The successive amounts, at intervals of 3.85 days (half-decay period), of the four substances radon (radium emanation, a gas), radium A, B and C are laid off along the four radial axes. Owing to the wide disparity in the relative amounts of these substances in equilibrium, it is necessary to employ widely different scales to represent these amounts. This must be duly taken into account in construing the diagram. The geometric similarity of successive polygons is the graphic expression of the constancy of the ratio of the several substances. The linear dimensions of successive polygons form a geometric series.

The differential equations representing the course of radioactive transformation are readily solved by direct integration, and there is therefore no necessity to employ the method of successive approximations to determine the moving equilibrium. However, the first approximation is so simple, that it is commonly applied. Equating the right hand member of (32) to zero we find immediately ie ee ADET ees ¥ (37) and by an obvious extension xy rn Xi ( )

It is easily shown that L;, the reciprocal of },, is the ‘‘mean length of life” of an atom of the substance S;. We may write (38) which expresses the fact that, in first approximation, the amounts of several substances present together in radioactive equilibrium are in the ratio of the respective mean lengths of life.6 This result can also be read out of (35) if |X,|, the least of the |\|’s, is negligible in comparison with all the other |A\’s, so that the denominator reduces to the product

It is thus seen that the closeness of the first® approximation depends on the relative magnitude of \, and the remaining )’s. In many chains of radioactive transformation the parent substance is very slow in its disintegration, and the first approximation (giving what Rutherford has termed the secular equilibrium) is exact within the limits of experimental error. But if one of the more rapidly decaying members is isolated and is then allowed to come into equilibrium with

5 This is a special case of a general law that if all the exponents d are real and negative, the final stages of the process of evolution are characterized by constancy in the ratios of the Variables z. Compare p. 261, Special Case; also, Lotka, A. J., Proc. Am. Acad. Sci., vol. 55, 1920. It should be noted, however, that in the general case, 7; denotes not mass of S;, but excess of that mass over the equilibrium mass of S;. In the radioactive equilibrium there is no distinction between X and 2, since the ultimate value of both is zero.

6 For the second and higher approximation, applied to the radioactive equilibrium, the reader may be referred to A. J. Lotka, Proceedings Natl. its own products of disintegration, the error of the first approximation may become appreciable.? This is shown, for example in table 29, which exhibits the amounts of radon gas (radium emanation) and its several products of disintegration in radioactive equilibrium. It will be observed that in the case of radium C there is a discrepancy of about 1 per cent between the amount computed by first approximation and the true amount.

Radioactive Chains as Cosmic Clocks. It must be noted that all that has been said above regarding the amounts of the substances present in radioactive equilibrium does not apply to the last link in the chain, the end product. This does not, of course, take part in the equilibrium, but accumulates, if non-volatile, as in the case of lead, or, it may in part escape and be lost to observation, as in the case of helium. Radioactive equilibrium of radon (radium emanation) in contact with its dis-

If the amounts of parent substance and end product are large as compared with the amount of intermediates, the amount of any one end product formed is evidently simply proportional to the amount of parent substance lost by disintegration in a given time. The chain of substances in transformation behaves, in fact, much like a sandglass clock having a number of bulbs and from the accumulation in the end bulb we can obtain an indication of the age of the system, on the assumption that originally all was in the top bulb, that initially only the parent substance was present. The application of this principle to radioactive mineral deposits has given us a quantitative time scale in historical geology where before we had to

rest satisfied with a crude chronology recognizing only order of precedence, or at best dealing in exceedingly uncertain lestimates of lapse of time. Thus the investigation of radioactivity, remote as it seems from the field of life phenomena, has nevertheless contributed to biology essential information regarding the time that has been available for the evolution of the earth and itsinhabitants. The estimate reached upon this basis is that the age of the radium-bearing rocks (uranium ore) examined is at least eight million years, and at most seventeen hundred million years old. For a résumé of various estimates of the age of the earth the reader may be referred to G.

SE IIVERI Te aaetatire: chee ctaioiaic are eel iwi le © sienSe 6) Dalsie =e 5 4 POM AOI ee och oi oop cienieoicrslevasic pleut a mins +s 12 11 ICAO ADEE elec ors aie shapes vests 28 30 BUAUL Vale LOGEROZONG. clr x ie s:sisinsn ac else piesa A 15 55 COME ZOLG wise a ess aie) .cpeieoe is tas. <8 visto vas © sia\e<* 25 } Schuchert, The Evolution of the Earth and its Inhabitants, 1919’ pp. 56 et seq.; 80; and to the Proceedings of the American Philosophical Society, 1922, vol. 61, pp. 247-288. See also E. Rutherford, Radioactive Substances, 1913. Schuchert’s estimate is that “geologic time endured about eight hundred million years,” distributed among the several geological eras as indicated in table 30.

The Origin of the Elements and the Ultimate Genesis of the Organism. ‘The case of radioactive equilibrium has here been introduced primarily by the way of illustration, as probably the most typical example in nature of a moving equilibrium in a system in the course of evolution. But the matter is also of more material interest to us in our survey of the evolution of the earth as the abode of life. For, as has already been emphasized, we are not only on the earth but of it; we have thus a two-fold interest in the evolution of its substance —first, as providing the stage upon which our life drama is set; and second, as furnishing the material of our bodies. Of these same ele-

ments that make up the earth’s crust we also are composed: their genesis is therefore also the first, remote chapter in the genesis of our own bodies. Through the discovery of radioactive chains of elements we hold a clue regarding the fundamental influences that have determined the quantitative chemical composition of our world, and have thus appointed the measure of the supplies available for our needs. Those elements whose genesis is known to us came into being in perfectly definite proportions. Presumably the same is true also of those whose precise mode origin is still unknown. There is good evidence to support this view. We have at present no detailed quantitative knowledge of the laws which determine the value of the decay coefficients \ of the radioactive elements, and which thus ultimately fix their relative abundance in equilibrium. Buta significant qualitative relation has been pointed out by W. D. Harkins. When the elements in a radioactive chain are arranged in order of their atomic numbers,’ and are separated into two groups, those of odd and those of even number, it is found that each even-numbered element is more abundant than the adjacent odd-numbered elements. And, what is of particular significance, the law of relative abundance of odd and even-numbered elements extends also to those elements, regarding whose precise mode of origin we have not, as yet, that sure knowledge which is gained by direct observation within the four walls of the physical laboratory. (See fig. 57.)

*'The chemical elements, arranged in ascending order of atomic weights, beginning at hydrogen = 1, may be given ordinal numbers 1, 2, 3, ete., indicating their position in the series. These ordinal numbers have been found to have important relation to the atomic architecture. They have been termed the atomic numbers. The definition here given is not quite exact; in certain places allowance must be made, gaps left for unknown elements, and the several isotopes of one element receive the same atomic number, though differing in their atomic weights. A more precise definition is the following: The atomic number of an element represents the excess of positive over negative charges in the constitution of the atomic nucleus. Each atomic number also represents the place occupied by the element in Mendeléef’s table (Jour. Am. Chem. Soc., 1923, vol. 45, p. 868). For further information the reader must be referred to the special literature; of compre-

hensive works the following may be mentioned: Bragg, X-rays and Crystals; F. W. Aston, Isotopes. Each element of even atomic number is more plentiful than the adjacent elements of odd stomic numbers. Diagram according to W. D. Harkins, based on analysis of meteorites. (Jour. Am. Chem. Soc., 1916, p. 863.) But the laboratory is not a prison, and the eye of the physicist is free to sweep the sky, where nature’s great smelteries gleam at night. With the aid of the spectroscope he has studied the multitudes of the stars, and has recognized in them a number of distinct stages of evolution. Life’s day is far too short to give the observer any opportunity to study directly the evolutionary changes in any one star. But by piecing together the observations made upon the mixed population of stars of different ages, it has been possible to construct with considerable certainty the main stages in stellar evolution, just as the stages of human life could be gathered from a single observation of a mixed population comprising persons of all ages. The evidence points clearly that the elements, such as we know them, are the product of “‘the general brewing of material which occurs under the intense heat in the interior of the stars.’”’ Out of such foundry came our own abode, if we accept the well-considered views of Eddington :!” “T do not say that the earth was a gaseous body when it first became recognizable as an independent planet, but I am convinced that its material was at one time merged in a completely gaseous sun.” And since we are of earth, ours also is the same origin. The hand that writes these words and the eye that reads them alike are composed of the selfsame atoms that came into being, ages and ages ago, in the young sun. Far, far more wonderful than any dream of old

19 A. 8. Eddington, The Borderland of Astronomy and Geology, Nature, 1923, p. 18, also, The Interior of a Star, Supplt. to Nature, May 12, 1923. The reader who wishes to acquaint himself in greater detail on this subject may refer to Eddingtons’s work Stellar Evolution. In the interest of unbiassed presentation it must be noted here that T. C. Chamberlin (The Origin of the Earth, 1916) has put forward a theory of the origin of the earth and the planets which is at variance with that sustained by Eddington. On the other hand it is also proper to mention a fundamental objection to theories of cosmogony of the type of that of Chamberlin and Moulton, which is based on the supposition that the luminous stars are formed by the collision of dead suns. ‘‘The distances separating the stars are enormous compared with their own dimensions. Sir Frank Dyson once used the illustration of twenty tennis balls distributed at random throughout the whole interior of the earth, to give a model of the density of distribution of the stars. a Taking a very liberal view of the kind of approach that can be held to constitute a collision, it is estimated that a star would suffer a collision about once in a hundred million million years” (Eddington). For a survey of the modern views on this subject see J. Barrell, The Evolution of the Earth (Yale University Press, 1919).

mythology is the story of our creation. Thus was the birth of man prepared in the grey dawn of time; thus the metal of his frame compounded in the flaming furnace of a star. Geophysics and Geochemistry. For the last stages in the evolution of the elements and their chemical combinations we do not look to the stars. We can study them at close quarters in the field and in the laboratory. In this way, with the application of physics and chemistry to general problems of geology, have grown up the sciences of geophysics and geochemisty. Indeed, it naturally might be supposed that on the terrestrial phases of inorganic evolution we should be altogether better informed than on those prior stages, far remote in time and space, which run their course in distant suns. But this is true, at best, only in restricted measure. It is a singular circumstance that, in some ways, we are better informed regarding the physics and chemistry of the stars, of which the nearest, outside the solar system, is twenty-five million million miles distant, than regarding that of our own planet. To say that the earth’s surface layers accessible to our direct observation are comparable, scale for scale, to the shell of an egg, is to err on the side of liberality. The deepest burrow into the earth made by human agency, the mine shaft at Morro Velho, Brazil,!! is 14 miles (6400 feet) deep or only about aky of the earth’s diameter. Direct observation can therefore give us at best only the most uncertain information regarding theconditions at even moderate depths. Where the crust has been creased and thrown into folds, subsequent denundation may have exposed layers of some 50 or 60 miles aggregated thickness.” But, though this gives us an invaluable record of some of the most significant chapters in the earth’s history, it adds little or nothing to our knowledge of conditions of temperature and pressure even at comparatively trivial depths, and regarding chemical composition also it gives us, after all mere surface indications.

Indeed, our information on all these points is very largely of a negative character. We know from the earth’s average density that the composition of the core must be very different from that of the shell. We know that the temperature gradient observed in boreholes near the surface averages an increase of about 1°F. for every 60 feet (1°C. for every 35 meters) descent, but of the further course of temperature at greater depths we know with reasonable certainty only that it cannot continue at this gradient, which would give a phantastic temperature of 300,000°F. (180,000°C.) at the center. Lord Kelvin’s calculations, based on the rate of cooling of the earth, and the more recent figures of the same character given by Van Orstrand, are rendered uncertain in their application owing to the presence of radium, in unknown amounts, evolving heat in its distintegration. Strutt has made the tentative estimate that the temperature rises uniformly to a depth of about 30 miles, and after that remains sensibly constant at 2700°F.

A little more definite is our information regarding the pressure in the earth’s interior. A first approximation of its value is found by considering the earth asa fluid. It is thus found" that the pressure at the center would be three million atmospheres. In any case there can be no doubt whatever that the pressures reached vastly exceed anything at our command in the laboratory, where a pressure of twenty-four thousand atmospheres, employed by P. W. Bridgman in his researches, stands out as a record achievement, though it corresponds to a depth of rock of only 56 miles.

It must be clear from what has been said above, that all conjectures as to the physical, chemical and subatomic transformations going on in the earth’s interior are subject to a very large margin of uncertainty. In this connection it is well to recall the words of F. W. Clarke:15 The chemistry of great pressures and concurrently high temperatures is entirely unknown, and its problems are not likely to be unravelled by any experiments within the range of our resources. The temperatures we can command, but the pressures are beyond our reach. . . . . We may devise mathematical formulae to fit determinable conditions; but the moment we seek to apply them to the phenomena displayed at great depths, we are forced to employ the dangerous method of extrapolation, and our conclusions are not verified.

8 This supposition, in calculating pressures, probably does not err far from the truth. The researches of F. D. Adams (Journal of Geology, February, 1912), P. W. Bridgeman and others have shown that at depths of some 30 miles rocks probably give way like butter to the pressure of the layers above them. : In these circumstances we can feel but little confidence in inferences based upon our laboratory observations, relating to radioactive and other possible atomic transformations going on at greater depths within the earth. If the degree and character of radioactivity which we observe in the accessible surface layers were to continue throughout the mass of the earth, the amount of heat developed would be much in excess of the observed losses by radiation.'® Unless therefore, we are to draw the highly improbable inference that the temperature of the earth’s mass is steadily rising, we are forced to one of two assumptions. Either the radioactive elements are segregated chiefly in the earth’s crust; or, the present rate of heat loss by radiation from the earth does not represent its average rate. This latter is the alternative elected by J. Joly!’ in an original conception. Ac- cording to this there are alternate periods of accumulation of heat in the solid rocks, followed by periods in which these rocks, having finally become melted, well to the surface in a death-dealing flood of fire. Thus, by convection, a process far speedier than conduction, through the solid rock mass, heat is dissipated until, after sufficient cooling, a second period of quiescence, with a solid earth’s crust, is ushered in. And so, in long waves of perhaps some thirty million years duration, the planet alternates between periods of hospitable clemency and periods intolerant of life.

There is, for us living inhabitants of this globe, a certain wildly romantic element, a feature of calamitous tragedy, in the hypothetical picture of the world’s history thus summoned up before our imagination. In its biological aspect how great and wonderful it all is! The living being working out his destiny on this poor raft, unknowing of the fiery ocean upon which this world is floating: unknowing of the inevitable sinking and uplifting which in truth largely controls the destinies of his race. Deathdealing forces all around, and yet the light of life shining age after age upon the earth.

17 J. Joly, Movements of the Earth’s Crust; lecture under the auspices of the Royal Dublin Society, published in Nature, 1923, p. 603. A third possibility is that under the extreme conditions of temperature and pressure prevailing in the earth’s interior reversal of the familiar radioactive disintegrations, or similar endothermic processes may go on. (Jour. Natl. Acad. Sci., 1924, p. 89. Such conceptions as this, stimulating as they are to the imagination in reconstructing for us an image of the remote past and distant future, must be entertained with reserve, remembering the words of caution quoted above from Clarke’s classic work. We must be prepared to consider the possibility that under the extreme conditions of temperature and pressure prevailing at great depths other subatomic transformations than those known to us in the laboratory may occur. Perhaps some evidence of this is seen in the evolution of helium as a component of natural gas, in amounts (up to 1.5 per cent!’) in excess of anything readily accounted for on the basis of the observed radioactivity of the rocks. And, while the subatomic transformations known to us are exothermic, accompanied by liberation of heat, others undoubtedly are endothermic, associated with absorption of heat. We seem to have carte blanche, in the present state of knowledge, in our speculations regarding the net heat balance of the elemental transformations that may be going on under our feet.

On surer ground rest our conceptions regarding the organization of matter, especially in the more superficial layers of the earth, under the action of ordinary physical and chemical influences. That the prime factor effecting the first and fundamental segregation of the lighter elements is flotation under gravity can hardly be doubted; this statement would in fact, be little more than a platitude if we were assured that the elements themselves remain unchanged under the extreme conditions of temperature and pressure prevailing in the earth’s interior. Beyond this prime factor the study of mineral and rock formation becomes a complex chapter in applied physical chemistry, the consideration of which is not within the plan of this work. The reader who wishes to follow out further this phase of the subject will find a comprehensive survey of the field in an article Der Stoffwechsal der Erde, by V. Moritz, in the Zeitschrift fir Elektrochemice, 1922, pp. 411-421; and in the memoir The Chemistry of the Earth’s Crust by H. C. Washington, which has already been quoted.

Organic Moving Equilibria. Of moving equilibria in the organic world, data are most readily available for the system comprising man and his domestic animals. Here the human race acts as the controlling factor, drawing its dependents after it in its growth. The equilibrium polygon for the principal items of animal husbandry in the United States is shown in figure 58. The geometric similarity of successive polygons is in this case only approximate, the proportion of the several components varies somewhat; except in the case of the sheep population, however, the variation is moderate over the halfcentury from 1871 to 1921. (Compare fig. 56 on p. 265.)

Aside from the features for the express illustration of which the diagram figure 58 was drawn, it also serves to point once more to the Fig. 58. EquitipriuM PoLyGon ror THE HuMAN SPECIES AND SOME OF THE Species ON wuHicu It Deppnps ror Irs Foop Suppiy. ScaLzs Reap IN MILLIONS fact already emphasized, that the concept of evolution, to serve us in its full utility, must be applied, not to an individual species, but to groups of species which evolve in mutual interdependence; and further, to the system as a whole, of which such groups form inseparable part.

It would be conveying a false impression in a very essential respect, to exhibit the example illustrated in figure 58, without a comment in emphatic reservation. Although, in a roughly approximate way, it is true, as shown by the polygon diagram, that in its relation suoy aa: spunod pole Ferae = see Sek [TSM oer] sec | SE | Jounwrr| Row Jona] ‘hom | 208 | sil | as, | Sa" | a | Mott Sa eh ae a a et Oe ag ge oe | Dw ee ge *8I6T °F 0087 ‘aduawmumoy pup ‘abouuoz jassea ‘uoyonposg uoynjndod fo “uawdojaaap $ p).10 10) i} YOINPOL UOUYD) Ja0ep S$. P]40M