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Continuing this process we find, ultimately,
X= a/b Uoe-a (1+ a/b + a2/b2+...)
Y2 = a/c Uoe-at (1+a/b+a/c + a2/b2 + a2/bc + a2/c2 + ...) (23) In this example the condition for the convergence of the successive approximations is immediately apparent. We must have
that is to say, the parent substance A must have a smaller decay constant than any of the succeeding members of the series. For obvious reasons this condition is always satisfied in natural radioactive mixtures."
The series (22), (23) bring out the relation between the uncorrected equilibrium, as commonly computed on the assumption of constancy of mass of the parent substance, and the true equilibrium. The firstmentioned (for which Rutherford has suggested the term "secular equilibrium") is represented by the first term of the series. As Rutherford points out, the error of the first approximation, i.e. the difference between the secular and the true equilibrium, amounts, in some cases, to nearly 1% though in others the error is quite negligible.
The series are easily summed, and then lead to the well-known expressions obtained by other methods (for the equations of radioactive change are readily integrable in finite terms, while the method here developed is applicable also in more refractory cases).
The case of radioactive equilibrium was here selected as an illustration, primarily because the functions involved are known and of simple form. But the same example will serve very aptly to illustrate also some other points.
In the first place we observe that moving equilibria might be divided into three classes, according as their progress is determined by a change in the P's, the Q's or the A's. As has been shown, the radioactive equilibrium is of the type in which the pace is set by a parameter of the class A, namely the mass of one of the links in the chain, which thus acts as a brake, or a limiting factor checking the series of transformations. Such limiting factors play an important rôle also in the highly complex network of interlocking cycles upon which the continuance of abundant life upon the earth depends. For life processes are energy transformation processes carried out by the agency of material energy transformers. Such transformers, if they are to work continuously and indefinitely must perforce work in closed transformation chains or cycles (such as the cycle CO2→→→ Plant Animal CO2). The moving equilibria engendered in such systems of cycles by a slow change in a limiting factor, in a parameter of class A, invite further study. The influence of man upon the world's events seems to have been largely to accelerate the circulation of matter and energy through such cycles, either by "enlarging the wheel", i.e., increasing the mass taking part in certain cycles, or else by causing it to
In either case Whether, in this
"spin faster," i.e., increasing the velocity of the circulation, decreasing the time required for a given mass to complete the cycle. he has increased the energy turn-over per unit of time. he has been unconsciously fulfilling one of those laws of nature according to which certain quantities tend toward a maximum, is a question well deserving of our attention.
* Papers from the Department of Biometry and Vital Statistics, School of Hygiene and Public Health, Johns Hopkins University, No. 44.
'Lotka, A. J., Physic. Rev., 24, 1912 (235-238); J. Washington Acad. Sci., 2, 1912 (2, 49, 66); Science Progress, 55, 1920 (406-417); Proc. Am. Acad. Arts Sci., 55, 1920 (237-153); these PROCEEDINGS, Sci. 6, 1910 (410–415).
See for example Picard, Traité d'Analyse; H. Bateman, Differential Equations, 1918, p. 245.
'Spencer, First Principles, Chapter XXII; Winiarskie, “Essai sur la Mécanique Sociale," Revue Philosophique, 44, 1900 (113).
At the time of reading proof this project is partially realized. A discussion of the applicability of the Le Chatelier principle to systems of the general character here considered will appear in a forthcoming issue of the Proceedings of the American Academy of Arts and Sciences
"Lotka, A. J., London, Phil. Mag., Aug., 1911, p. 353.
A FORMULA FOR THE VISCOSITY OF LIQUIDS'
By H. B. PHILLIPS
Department of MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated by A. G. Webster, March 22, 1921
1. In this paper I obtain for the viscosity of a liquid the formula
n Nh 2M (v-8)'
(1) where N is the number of molecules in a mol, h is Planck's constant, M is the molecular weight of the liquid in the gas phase, v its volume per gram, and n an integer. The quantity & is the co-volume as used in the equation of state of Keyes1
In all the cases to which I have applied the formula, n=6 and so (1) takes the form
(3) It is to be noted that 3N/M is the number of translational degrees of freedom of the molecules in the volume v of the liquid.
2. To prove equation (1), let x, y, z be rectangular coördinates and suppose the liquid to flow parallel to the x-axis in such a way that, u。 being
the velocity at the point (x, y, z),
duo/dz = 1.
Let the components of velocity of a molecule parallel to the x and z axes be u, w respectively. When a molecule of mass m moves in the positive direction of the z-axis across the xy-plane, the x-component of momentum transferred across that plane is mu. When it crosses in the negative direction the transfer is -mu. Under these conditions the viscosity ʼn of the liquid is defined as the total x-component of momentum transferred across unit area of the xy-plane in unit time. To find the viscosity we therefore find the number of molecules of each velocity crossing unit area and multiply by the average momentum transferred by each.
Let f (w) dw be the number of molecules per cubic centimeter with components of velocity parallel to the z-axis between w and w+dw. If the molecules did not influence each other, the number of these that would cross unit area of the xy-plane in unit time would be
Because of the interference of molecules with each other, this must be replaced by
To see this we note that if the molecules moved independently, the zcomponent of momentum transferred across unit area of the xy-plane in both directions in unit time would be the pressure
From equation (2), in the real liquid this is replaced by
Now the mass of the molecule and the temperature (average kinetic energy) are the same in both cases. Hence the increased momentum (8) compared with (7), can only be due to an increase in the number of molecules crossing in unit time. Since the same law of distribution of velocities (Maxwell's Law) is assumed in both cases, the ratio in which the numbers are increased must be the same v/(v−d) for all velocities. Thus the number of molecules with velocity components between w and w + dw crossing unit area of the xy-plane in unit time is given by equation (6).
The velocity u of a given molecule parallel to the x-axis will not always be equal to the average velocity u。 of the liquid at that point. There will, however, be a series of instants (which we may call collisions) when u will be equal to u。. I assume that the interval between two such instants will be a half period (interval between libration limits) in the sense of the quantum theory, and that the molecules can be treated as moving with constant velocity between consecutive collisions. Suppose then a molecule reaching the xy-plane has been moving a time, t, since its last collision.
From (4) its momentum parallel to the x-axis due to the motion of the liquid at that point will be m wt.
From (6) the total x-component of momentum transferred in the positive direction across unit area of the xy-plane in unit time will then be
w't f(w) dw.
There will be an equal transfer due to molecules moving in the negative direction. Hence
The time t can be considered as the interval between collisions at times1 to, ti. Since the molecule is assumed to move with constant velocity between collisions,
where, according to Sommerfeld's theory, n is an integer. Also
2m w2 t 2 ti mw2 dt
since it is the number of molecules per gram for which w is positive. Combining (9), (10), and (11), we get (1) which was to be proved.
3. Owing to lack of data on the equation of state, the only substances on which the formula can at present be tested are carbon dioxide, ether, and mercury. Using the values of N and h given by Birge,5
and so equation (3) takes the form
Values of d for carbon dioxide and ether, tables I and II, were supplied by Professor Keyes. At low temperatures the measured viscosities and those calculated by equation (12) do not differ by more than the experimental error. In case of carbon dioxide the temperature 30° is too near the critical point (t=31°) for satisfactory use of the equation of state. Above 10° the calculated viscosity of ether is too large, the difference increasing with the temperature. This may be due to the fact that ether is a complex of more than one type of molecule. The equation of state was determined on the assumption that each liquid molecule is formed by the combination of two gas molecules. This may be substantially true at 10° and not at 100°.
To obtain values of v―d for mercury, I make use of the fact that monatomic substances seem to have constant co-volumes 8. Since mercury is monatomic in the gas phase, I assume that 8 is constant or nearly constant in the liquid phase. Also the term
in case of mercury is very large (more than 15000 atmospheres). Hence at atmospheric pressures we may neglect p and so write (2) in the form
where c is constant. Differentiating this equation with respect to v and then with respect to T, assuming & constant, we get
V=1+1805553 (1) 10+12444 (1)*10*+2539 (1) 10 (14)
is the ratio of the volume of mercury at t° to its volume at 0°. The values of v-8 in table III are obtained by calculating the derivatives of v from (14) and substituting in (13). Below 100° the difference of the calculated viscosities and those measured is not greater than the difference between the values obtained by different observers. At higher temperatures the calculated viscosities are consistently smaller than those measured. The observations, however, differ greatly, owing probably to oxidation of the mercury in contact with the air. The result may be to increase the viscosity by almost any amount. This work should be repeated with mercury of the greatest purity in a vacuum or in contact with some inactive gas.