parallel to P and d cm. above it. On applying an alternating potential difference of half period T between P and E the ions starting from P will alternately be driven from, and drawn back to P, for periods of T seconds. If u is the velocity of the ion in unit field, then for given values of T and d there will be no current to E until V reaches a value V。 such

=

that uTV. d. At V. then a current to E will become manifest which will increase with increasing V to a saturation value. On plotting the current i as ordinate against the voltage V as abscissa we will get a curve of the type of curve I, figure II. Such a curve will be termed a mobility curve for if the value of the intercept V. of such an experimental curve be substituted in the equation

the mobility u of the carrier is at once obtained if d and T are known. When the pressures are reduced the curves remain the same but the value of the mobility u of the ion is found to increase inversely as the pressure. In other words up/760 = K, where K, the mobility constant of the ions, is independent of pressure.

Now let us assume with J. J. Thomson' that an electron does not attach to form an ion on its first encounter with a gas molecule, but that the attachment is a chance phenomenon occurring on the average in one out of n impacts with molecules. Let us further assume that for air n is large, say, about 250,000. Under ordinary conditions the velocity of thermal agitation of the electron is great compared with its velocity acquired in the electric field. The electron, therefore, covers a zigzag path drifting slowly all the while in the direction of the electric field. Consequently while making the 250,000 impacts necessary for attachment the majority of the electrons will move a distance, A, in the field, which may be estimated. Since A is covered by the electron in 1/100 of the time that an ion would take, one may practically assume that the majority of the ions start at a point distant A cm from P instead of from P. For appreciable values of A the values of V required to drive the majority of the carriers to E may become notably lower than they are for ions starting at P. The values of u and hence of K obtained from experiments where A becomes significant will then be high. In air at atmospheric pressure the time taken for an electron to make 250,000 impacts is about 1.2 × 10−6 sec. In a field of 40 volts with d 15 mm. the electron would cover a distance of 6.3 X 10-2 mm. towards E in this time. This small value of

=

the distance A covered as an electron would produce no measurable effect on V. and the values of K would be normal. Now let the pressure be reduced to 152 mm. The mobility of the electrons will be increased fivefold. Furthermore as the density of the gas is reduced to one-fifth the electrons will take five times as long to make the 250,000 impacts required to attach. The distance A will, therefore, become 1.5 mm. which is an appreciable fraction of the 15.0 mm. between the plates. The intercept V. as determined from the portion of the curve corresponding to the majority of the carriers will, therefore, be but 0.9 of the V. required for ions, and the value of K will be about 10% higher than for normal ions. In other words assuming the Thomson mode of attachment the mobility constant of the ions will appear to increase with decreasing pressures below about 152 mm.2 Since the number of impacts required to form an attachment is a chance phenomenon there will be an appreciable number of ions reaching E that have traversed distances greater than A before attaching. These will reach E at values of V. below those for the majority of carriers. As a result the otherwise sharp intercepts of mobility curves with the voltage axis observed for ions will be masked by asymptotic feet of exponential form rising at values of V well below the V. required for the majority of the ions. These will become rapidly more pronounced

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

as the pressures decrease. Such curves are seen in the experimental curves II, III, IV, V and VI, figure II, obtained in air under conditions indicated in the legend. To estimate the hybrid mobilities of such carriers. the asymptotic feet must naturally be ignored, as has been done by previous observers. 28

With the method of analysis outlined above it would be possible to pursue the evolution of the mobility curves as a function of p and T in a qualitative manner indefinitely. Fortunately a much simpler and more accurate study of the theory is made possible by the application of the mathematical analysis of Thomson. On the basis of his theory Thomson has shown that out of Q. electrons starting from P the number that can travel x cm. through the air without combining to form ions is given by e-(Wx)/(nK'LV/d)

=

(2)

where W is the velocity of thermal agitation of the electron, K' is the mobility of the electron, L is its mean free path, V is the voltage, d is the distance between the plates, and ʼn is the chance of ion formation per impact.

Let us assume that when the voltage V is equal to V. all the Q. carriers starting from P at atmospheric pressures can reach E. This simplifying assumption is contrary to fact for at V. the carriers actually only begin to reach E. We may with this assumption impose the conditions implied in equation (1) on the equation (2). This will lead to the conclusion that out of a maximum possible current Io, the real current I which reaches E as a function of the frequency N = 1/T, the pressure p, the plate distance d, and the voltage V, is given by the equation

I =

Ioe K'L

W (d2(p/760)?
V

K(p/760))

(3)

where p is in mm. of mercury, and K is the mobility of the normal ion. Now it is possible to evaluate W from the mean kinetic energy of the molecules, for it is assumed in the theory that the electrons move in the electric field with a velocity small compared to W. Let us further assume K' to be constant and equal to 200 cm./sec., 7.3 while we take L as 4√2 times the mean free path of the molecules, and K as about 2.5. We thus have the equation

I = I。e

9.9 X 108 (d2 (p/760); _ (2.5p/760))

n

V

(4)

This equation is open to experimental verification for it contains but one unknown quantity n, as I/I。 can be determined experimentally under known conditions of N, p, V and d.

I have recently made a series of determinations of the mobilities of the carriers produced in air at different pressures under essentially the simple conditions in the foregoing discussion. These determinations yielded mobility curves of which the set of curves shown in figure II are typical. As is seen at once the form of the curves resembles the curves to be expected from the qualitative discussion above. The values of I/I。 may be determined in such curves from the ratio of the current to E caused by a given alternating potential between P and E and that caused by an equal fixed negative potential on P. By substituting this value of I/I。 in the equation with the corresponding values of N, P, V and d one may solve for n. As the result of a large number of determinations the value of n obtained under conditions best conforming to the theoretical assumptions was 250,000. With n determined the theory may be further tested by computing the curves for I/I。 as a function of V for different values of d, N and p. A large number of curves were thus computed. A typical comparison of the theoretical curves so obtained and the curves actually observed under the same conditions may be seen in figures III, IV and V.

ones.

The dotted curves are the observed and the full curves are the computed The experimental data are given in the legend. The shapes of the curves at atmospheric pressures or at low values of N, where the carriers are all ions, should not agree. For in deducing the theory we assumed that at V. all carriers succeeded in reaching E, which is not the case in fact. Also at the lower pressures close agreement is not to be expected, for the velocity acquired in the field becomes commensurable with the velocity of agitation W. Barring these points the general shapes of the theoretical and observed curves, and the changes in shape of these curves with p and N, are quite similar. It is also evident that the points of in

flection, or the asymptotic feet, of the corresponding observed and computed curves fall on values of V which lie close together. This means that the values of the hybrid, or abnormal, mobilities estimated from the two sets of curves are nearly the same. There is, therefore, sufficient similarity in the two sets of curves to permit one to assert that the curves are represented by the same type of equation. We may then conclude that the Thomson theory is in good qualitative agreement with the results observed.

With the evidence before us I believe we are justified in concluding that the mechanism of negative ion formation in air consists in the electron attaching itself to a molecule to form a negative ion on the average in one out of 250,000** molecular impacts. Now it has been shown that in pure nitrogen34 the electron does not attach to form an ion to any appreciable extent. We must, therefore, assume that the electron attaches to the oxygen molecule in air. Measurements made in pure oxygen give n as about 50,000. As there are four molecules of nitrogen in air to one of oxygen this agrees quite well with our conclusion. What the significance. of n is, whether it depends on the electronic ring in the molecule struck, whether it depends on some particular state of the molecule struck, or whether it depends on the energy conditions of the impact remains for the future to say.

In conclusion I desire to express my thanks to Professor R. A. Millikan for his kind advice and criticism.

* A more detailed description of these experiments will later appear in the Physical Review.

† NATIONAL Research FELLOW of the NATIONAL RESEARCH COUNCIL.

Such mobilities have been obtained below 100 mm pressures in air by all the experimenters in this field up to the time of Wellisch. Until Thomson proposed his theory they had never been adequately explained.

** This value depends on the correctness of the assumptions as to the numerical values of K' and L for the electron.

1 Thomson, J. J., London Phil. Mag., Sept., 1915.

2 Kovarik, A. F., Physic. Rev., Ithaca, 30, 1910 (415).

3 Franck, J., Verh. deuts. physik. Ges., 12, 1910 (613).

4 Loeb, L. B., These PROCEEDINGS, June, 1920.

5 Loeb, L. B., Physic. Rev., 8, 1916 (6).

6 Yen, K. L., Ibid., 9, 1918 (5).

7 Wellisch, E. M., Amer. J. Sci., New Haven, (Ser. 4) 44, 1917 (1); Phil. Mag., 34, July, 1917.

THE BASAL METABOLISM OF GIRLS 12 TO 17 YEARS OF AGE BY FRANCIS G. BENEDICT, MARY F. HENDRY AND MARION L. BAKER NUTRITION LABORATORY, CArnegie INSTITUTION OF WASHINGTON, BOSTON

Read before the Academy, November 17, 1920

The Nutrition Laboratory's task of charting the field of basal metabolism of humans from birth to old age has resulted in a reasonable completion of the study of boys and girls from birth to puberty, of both sexes, from the college age to 35 years, and of women about 50 years of age. The metabolism during the important age-range from 12 to 17 years, representing as it does a period of rapid growth as well as the period of the establishment of puberty, has recently been studied, so far as girls are concerned, in a large respiration chamber permitting the simultaneous measurement of the carbon-dioxide production of a dozen or more subjects. Groups of twelve Girl Scouts each volunteered as subjects, and a typical experiment involved their entering the respiration chamber after a light standard supper and sleeping quietly throughout the night. The entire carbon-dioxide production during the period of "bed rest," as well as the "minimum" carbon-dioxide production found throughout the night, are the bases for the computations of the energy needs for "bed rest" and for the basal metabolism. The quiet, resting morning pulse rate was obtained with all groups and the insensible perspiration was also measured with most of the groups.

Special interest centres around the gaseous metabolism or, more particularly, the energy computations therefrom. It was found that 55.0 calories represented the average hourly heat production per individual, with very little difference due to either average age or average weight.