EXPERIMENTAL STUDY OF THE VELOCITY DISTRIBUTION OF GAS MOLECULES

G. Razorenov, Moscow.

At the foundation of the kinetic theory of gases lies the Maxwell–Boltzmann law of distribution of molecular velocities. Despite the great importance of an experimental verification of this law, until recently only indirect indications—for example, measurements of the width of spectral lines—confirmed its validity.

In 1908 Richardson [1] investigated the velocity distribution of electrons emitted from an incandescent body, and found good agreement with the Maxwell–Boltzmann law. However, this work gave only an indirect indication of the character of the distribution of molecular velocities.

It is no accident that the first work was carried out on electrons. The charge which they bear removes the difficulties connected with separating neutral molecules, moving with some definite velocity, from the total number of uncharged molecules. This problem was first solved experimentally by Stern [2] in 1920, using the “aberration” of a narrow beam of moving silver atoms. Stern’s work established the value of the mean velocity of the atoms, coinciding with the conclusions of the kinetic theory. The insufficient accuracy of the results did not allow him to draw a conclusion concerning the law of distribution of velocities.

Subsequently Stern, together with Gerlach, applied a narrow beam of atoms to the investigation of the magnetic moments of atoms. The works of Gerlach and Stern [3] led to the experimental confirmation of space quantization and to the determination of the magnetic moments of atoms of a whole series of elements. Both Stern’s work and the works of Gerlach and Stern were described by N. N. Semenov [4] in the pages of this journal.

However, the accuracy of the determination of the magnetic moments of atoms in the works of Gerlach and Stern suffered from the fact that the beam consisted of atoms moving with different velocities. In order to eliminate this shortcoming, Tykocinski-Tykociner [5] in 1927 proposed a method of selecting a beam of atoms whose velocities lie within known limits.

The scheme of the apparatus by means of which Tykocinski-Tykociner proposes to measure the magnetic moment of hydrogen atoms, using

shown in Fig. 1. The entire apparatus is in a high vacuum, which is constantly maintained by pumps. \(K\) is some source of atoms; \(R\) is a plate on which the trace of a beam of flying hydrogen atoms is marked by a chemical reaction (similarly to what was done in the work of Wrede \([^{6}]\) and in the work of Fips and Taylor \([^{7}]\). The works of Wrede, Fips, and Taylor have been described by G. S. Landsberg \([^{8}]\) in the pages of this journal).

On the path of the atoms from the source \(K\) to the plate \(R\), three screens \(P_1\), \(P_2\), and \(P_3\) are placed perpendicular to \(KR\). The distance between the two latter screens is equal to \(D\). The screens \(P_2\) and \(P_3\) are fastened to flexible bronze strips \(BB\), so that they can oscillate in a plane perpendicular to \(KR\). Along these strips an alternating electric

Fig. 1.

current with period \(T\), generated by the cathode lamp \(V\), is passed. To excite oscillations of the screens, a magnetic field \(H\) is produced in the apparatus, directed parallel to \(KR\). In the screens \(P_1\), \(P_2\), and \(P_3\), slits \(S_1\), \(S_2\), and \(S_3\) are cut, perpendicular to the direction of oscillation of the screens \(P_2\) and \(P_3\). The slit \(S_1\) serves to select a narrow beam of atoms moving with all possible velocities in the direction \(KR\).

When the screens \(P_2\) and \(P_3\) are at rest, this beam passes entirely through the slits \(S_2\) and \(S_3\) and reaches the plate \(R\). Immediately in front of the plate there is installed, just as in the experiments of Gerlach and Stern, an electromagnet \(M\), producing a nonuniform magnetic field with a large gradient. In the field of this magnet the beam splits, and on the plate two bands can be recorded, corresponding to the orientation of the atoms along and against the field. Owing to the difference in the velocities of the individual atoms, these bands are somewhat blurred.

By setting screens \(P_2\) and \(P_3\) into oscillation, one can select a beam of atoms moving with certain discrete velocities

\[ v_0,\ v_1,\ v_2,\ldots,\ v_n,\ldots\quad (v_0=\infty). \]

Indeed, if we assume that the slits \(S_1\), \(S_2\), and \(S_3\) are infinitely narrow, then atoms will be able to pass through \(S_2\) and \(S_3\) only at definite instants of time

\[ t_0,\ t_0+\frac{T}{2},\ t_1+\frac{T}{2}\cdot 2,\quad t_0+\frac{T}{2}\cdot 3,\ldots,\ t_0+\frac{T}{2}\cdot n,\ldots, \]

since at all other times screens \(P_2\) and \(P_3\) are displaced to the side. Therefore only those atoms will reach the plate \(R\) whose velocities are determined by the formula

\[ v_n=\frac{D}{\frac{T}{2}\cdot n};\quad (n=0,1,2,\ldots), \tag{1} \]

since all other atoms that have flown through slit \(S_2\) will encounter screen \(P_3\) in a deflected position and will be reflected back.

In reality the width of slit \(S_2\), as well as of slit \(S_3\), is equal to \(w\). The time during which slit \(S_2\) or \(S_3\) crosses the beam of atoms is equal to \(\tau\):

\[ \tau=\frac{wT}{2\pi a}, \tag{2} \]

where \(a\) is the amplitude of the oscillations of screens \(P_2\) and \(P_3\). Therefore, not only atoms whose velocities are determined by formula (1) will pass through both slits \(S_2\) and \(S_3\), but also all atoms whose velocities lie within the limits

\[ v_n' < v_n < v_n'';\quad (n=1,2,\ldots) \]

Tykocinski-Tykociner gives the following expressions for \(v_n'\) and \(v_n''\):

\[ \left. \begin{aligned} v_n'&=\frac{D}{\frac{T}{2}\cdot n+\frac{\tau}{2}}\\[6pt] v_n''&=\frac{D}{\frac{T}{2}\cdot n-\frac{\tau}{2}} \end{aligned} \right\} \quad (n=1,2,\ldots) \tag{3} \]

With a suitable choice of \(w\) and \(a\), these limits can be made as narrow as desired.

As a result of such selection of a beam of atoms whose velocities lie within known limits, the plate \(R\) will record, in place of the two blurred bands of Gerlach and Stern, a number of considerably narrower bands.

G. L. RAZORENOV

Tykocinsky-Tykociner calculates the width and position of these bands as a function of the magnetic moment and mass of the atom, the gradient of the magnetic field, the dimensions of the magnet, and the velocity limits \(v'_n\), \(v''_n\). At the end of the work numerical data are given, obtained on the basis of these calculations, as applied to the apparatus on which Tykocinsky-Tykociner proposes to make an accurate measurement of the magnetic moment of the hydrogen atom.

The calculations carried out by Tykocinsky-Tykociner with respect to determining the magnetic moment of an atom are directly connected with the study of the velocity distribution of gas molecules. His method of selecting a beam of atoms moving with definite velocities is analogous to Fizeau’s method for determining the speed of light. This analogy appears still more vividly in the work of Costa, Smith, and Compton [9], who set themselves the aim of constructing a “spectrometer” of the velocities of gas molecules. The scheme of their apparatus is shown in Fig. 2.

Fig. 2.

\(S_1\) and \(S_2\) are two parallel slits, between which a high vacuum is constantly maintained by pumps. On the rotating axis \(xx'\) two disks \(D_1\) and \(D_2\) with radial slots \(s, s\) and \(s', s'\) are fixed. Disk \(D_1\) is situated behind slit \(S_1\), disk \(D_2\)—in front of slit \(S_2\). The disks can be turned so that the slots \(s, s'\) will come opposite the slits \(S_1, S_2\). The gas under investigation enters through slit \(S_1\). Some of the molecules of this gas move with different velocities in the direction \(S_1R\). If the slots stand opposite the slits, these molecules will pass successively through \(S_1, s, s'\), \(S_2\) and will strike the vane of the radiometer \(R\), installed behind slit \(S_2\).

With uniform rotation of the disks, molecules moving with definite discrete velocities will strike the vane of the radiometer

\[ v_1,\ v_2,\ldots\ v_n\ldots;\qquad v_n=\frac{v_1}{n}, \tag{1 bis} \]

and also, owing to the finite width of the slots, molecules whose velocities are contained within the limits

\[ v'_n < v_n < v''_n. \]

All the remaining molecules will be stopped by disk \(D_2\). The velocities \(v_n\) and the limits \(v'_n\) and \(v''_n\) are determined, depending on the dimensions of the apparatus, on the number of slits in the disks, and on the angular velocity of rotation of the disks, by formulas analogous to formulas (1), (2), (3).

Starting from some given curve of the distribution of molecular velocities, on the basis of these formulas one can calculate the number and velocities of the molecules striking the wing of the radiometer as a function of the angular velocity of rotation of the disks.

It remains to compute the magnitude of the deflection of the radiometer under the action of the molecules falling on it. The authors of the work indicate two different methods of calculation. It may be assumed, first, that the molecules are reflected from the wing of the radiometer with velocities proportional to the velocities of incidence, and, second, that the velocities of the reflected molecules are distributed according to the Maxwell–Boltzmann law corresponding to the temperature of the wing. In one way or another, the amount of motion imparted by the molecules to the wing of the radiometer is determined as a function of the number and velocities of these molecules. The deflection of the radiometer is calculated by integrating the obtained expression over all the molecules falling on its wing.

Fig. 3.

Calculation by the first and second methods under the assumption of a Maxwellian distribution of the velocities of the incident molecules shows that in this case the results differ little from one another.

Thus, starting from some given curve of the distribution of molecular velocities, one can obtain the curve of deflection of the radiometer as a function of the angular velocity of rotation of the disks. On the other hand, on Kosta’s apparatus,

Smith and Compton radiometer deflections can be observed directly. By comparing the observed and calculated curves one can judge the degree of suitability of the particular law of distribution of molecular velocities on the basis of which the calculation was made.

The apparatus of Costa, Smith and Compton is shown in Fig. 3.

A bronze cylindrical vessel \(B\) is divided by partitions \(F_1\) and \(F_2\) into three parts. Slits \(S_1\) and \(S_2\) are cut in the partitions. Gas enters the vessel through the glass tube \(l\) and is pumped out by a powerful diffusion pump through the opening \(O\), so that a constant pressure difference is maintained between \(l\) and \(O\). The axis on which the disks \(D_1\) and \(D_2\) are mounted is fixed in sapphire bearings in the partitions \(F_1\) and \(F_2\). With the aid of an electric motor, whose magnets \(M\) are outside and whose armature is inside the vessel in vacuum, the disks can be rotated at a speed from 500 to 6000 revolutions per minute. The distance between the disks is 8 cm. The radiometer \(E\) consists of a system \(K\), suspended on a thin thread \(g\). The weight of the system \(K\) is 1.2 mg. In the cylinder lids \(C_1\) and \(C_2\) two windows are cut, closed with thick glasses \(W_1\) and \(W_2\). The first window serves for the stroboscopic determination of the number of revolutions of the disks, the second for observing the deflections of the radiometer by the zero method. The system \(K\) was brought to zero at each measurement by means of the screw \(N\).

Fig. 4.

Costa, Smith and Compton investigated hydrogen, nitrogen, and carbon tetrachloride with this apparatus. In Fig. 4 are shown the curves observed by them and the curves calculated on the assumption of a distribution of molecular velocities according to the Maxwell–Boltzmann law. The calculation of the radiometer deflections was carried out by the first of the two methods indicated above. The second method of calculation somewhat smooths the maximum of the upper theoretical curve.

The authors of the work consider that the results obtained by them very well confirm the validity of the Maxwell–Boltzmann law.

of Maxwell’s. The deviations of the observed curve from the calculated one, which are noticeable in Fig. 4, are fully covered by the possible errors of the experiment. Thus, thanks to the high perfection of modern vacuum technique, it proved possible to construct Maxwell’s “mechanical demon,” selecting from the gas molecules moving with definite velocities. At present, however, the apparatus of Costa, Smith, and Compton cannot be called a “spectrometer” of the velocities of gas molecules, as the authors had hoped at the beginning of their work. The chief difficulty lies in the fact that the radiometer is already operating at the limit of its sensitivity, and also in the unavoidable vibrations of the apparatus at a large number of revolutions of the disks.

Eldridge [10] succeeded in obtaining considerably more accurate results.

Eldridge’s apparatus (Fig. 5) consists of a cylindrical glass vessel inside which there is placed a system of disks fixed on a common axis. A high vacuum is continuously maintained inside the vessel.

The lower disk serves as the rotor of a two-phase electric motor, whose stator is placed outside the vessel (in the atmosphere).

The system of the remaining disks constitutes a “filter,” which in a definite direction lets through atoms with definite discrete velocities (analogously to the apparatus of Costa, Smith, and Compton).

The system of disks can rotate more or less synchronously with an angular velocity of up to 7200 revolutions per minute.

Fig. 5.

Cadmium vapor was obtained by heating metallic cadmium in an electric furnace to approximately 400° C. From the furnace the cadmium vapor enters the vessel through an aluminum tube, closed at the end with aluminum foil in which a slit 0.1–0.2 mm wide is cut. This slit is located opposite the first (lower) of the disks constituting the filter.

Radial slots are made in each disk, through which cadmium atoms can pass in the direction along the cylindrical vessel.

Beyond the last disk there is placed a small cylindrical glass vessel with a flat bottom, filled with liquid air. Part of the atoms, having passed through the filter, strikes the plate forming the bottom of this vessel and condenses on it, forming a deposit of metallic cadmium.

The basic calculation of the action of the filter is analogous to the calculation of Tykoczyński–Tykociner.

Let us suppose that the filter consists only of two end disks, and that in each of them there is only one slit, the slits being situated opposite one another.

Cadmium atoms fly out of the target in all possible directions. If the disks rotate, atoms can pass through the slit in the first disk only at those instants when this slit is opposite the target. The atoms that have passed through the slit in the first disk form, under this disk, a strongly diverging beam.

A portion of the atoms will fly in a direction parallel to the axis of the cylindrical vessel. Of these, only atoms with the discrete velocities

\[ v_0,\ \frac{v_1}{1},\ \frac{v_1}{2},\ \frac{v_1}{3},\ldots (v_0=\infty). \]

will pass through the slit in the last disk.

The number of revolutions and the distance between the disks can be chosen so that \(v_1\) is small in comparison with the velocities of cadmium atoms at the given temperature.

Then atoms moving parallel to the axis of the vessel will not pass at all through the slit in the last disk (except for a very small number of atoms, which may be neglected). Atoms moving with definite velocities at an angle to this direction meet the slit in the last disk on one side somewhat earlier, and on the other side somewhat later, than atoms moving parallel to the axis of the vessel.

The former may be neglected, since, in order for them to pass through the slit in the last disk, their velocities must be considerably smaller than \(v_1\) (a few greater than \(\frac{v_1}{2}\)), and, by the condition of the choice of \(V_1\), the number of such atoms is very small.

The latter will be able to pass through the filter in a comparatively large number, since for this it is sufficient that their velocities be contained within the limits between \(v_1\) and \(\infty\).

The greater the angle between the direction of motion of the atoms and the axis of the vessel toward the rotation of the disks, the smaller the velocity required in order that atoms with this velocity might pass through the slit in the second disk.

With a constant angular velocity of rotation of the disks, on a plate cooled by liquid air there will be obtained a blurred trace of the beam of atoms that have passed through the filter.

Initially the “undisplaced line,” situated opposite the target, was marked. The greater the angle between the direction of motion of the atoms and the axis of the vessel, the greater the distance between the “undisplaced line” and the trace of the beam of these atoms. The “undisplaced line” corresponds to an infinite velocity of the atoms. The various parts of the blurred cadmium deposit correspond to different velocities of the atoms.

In order to reduce the required angular velocity of rotation of the filter, instead of one slit, 100 slits were made in each disk. The principle of operation of the filter is not changed by this.

Atoms reflected from the disks form a “counter-flow.” In order to eliminate these atoms, three more disks are mounted on the same axis between the first and the last disk. For the same purpose, fixed partitions are placed inside the vessel (not shown in Fig. 1).

The accuracy of the results depends very strongly on the degree of rarefaction inside the vessel. To increase the vacuum, the vessel is surrounded by a jacket with liquid air.

The “velocity spectrum” obtained in this way was photometered with a microphotometer and, by comparison with photometric data for plates coated with a cadmium deposit of previously known thickness, the thickness of the deposit in different parts of the “spectrum” was determined; thereby the curve of the distribution of velocities of the atoms deposited on the plate was found.

Fig. 6.

In the graph (Fig. 6), along the abscissa axis are plotted the experimentally obtained linear dimensions of the “spectrum” in millimeters, counted from the “undisplaced line”; along the ordinate axis—the thickness of the deposit in different parts of the “spectrum.” According to the conditions of the experiment, the distance from the “undisplaced” line must be directly proportional to \(\lambda\left(\lambda=\frac{1}{v}\right)\).

Maxwell’s formula is reduced to the form

\[ dN = - C \frac{1}{\lambda^5} e^{-\frac{1}{\lambda^2 x^2}}\, d\lambda, \tag{4} \]

where \(dN\) is the number of atoms that have settled on the plate and whose velocities lie within the limits \(\lambda\) and \(\lambda+d\lambda\) \(\left(\lambda=\frac{1}{v}\right)\), \(a\) is the most probable velocity, and \(C\) is a constant coefficient. The experimental curve (Fig. 6) was obtained by Eldridge in good agreement with formula (4).

At a temperature of cadmium vapor of \(400^\circ\mathrm{C}\), the mean velocity of the atoms is

\[ \overline{v}=388\ m/sec, \]

the most probable velocity is

\[ a=\overline{v}\sqrt{\frac{2}{3}}=317\ m/sec \]

and the maximum value of \(\lambda\) is

\[ \lambda_m=\frac{1}{a}\sqrt{\frac{2}{5}}=\frac{1}{2000}\ sec/m. \]

On the other hand, the radius of the disks was \(3.15\ cm\), the length of the filter \(12.7\ cm\), and the number of revolutions \(85\) per second. With these values, each millimeter of the “undisplaced line” corresponds approximately to

\[ -\lambda=\frac{1}{2140}\ sec/m. \]

According to the photometric data for the “spectrum” (Fig. 6), the maximum value of \(\lambda\) is obtained as nearly equal to

\[ \lambda_m=\frac{1}{2000}\ sec/m. \]

In exactly the same way, within the possible errors of the experiment, all points of the theoretical and experimental distribution curves coincide with one another.

The works of Stern, Tykocinski-Tykociner, Costa, Smith and Compton, and Eldridge in the years 1920 to 1927 outlined the ways toward the direct study of the velocity distribution of gas molecules and, in concrete examples, demonstrated the validity of the Maxwell–Boltzmann law.

LITERATURE

[¹] Richardson. Phil. Mag. 16, p. 353, 890, 1908; 18, p. 681, 1909. Richardson, “The Electron Theory of Matter,” p. 442, Cambridge 1914, Richardson, “The Emission of Electricity from Hot Bodies,” London 1916. P. p. 150.

[²] O. Stern, ZS. f. Phys. 2, 49, 1920. O. Stern, ZS. f. Phis. 3, 417, 1920.

[³] O. Stern, ZS. f. Phys. 7, 149, 1921. W. Gerlach und O. Stern, ZS. f. Phis. 8, 110, 1921. W. Gerlach und O. Stern, ZS. f. Phys. 9, 349, 1922. W. Gerlach und O. Stern, Ann. d. Phis., 1924.

[⁴] H. N. Semenov, “On the Molecular Beam.” U. F. N. 5, p. 57, 1925.

[⁵] I. Tykocinski-Tykociner, “Velocity selektor for atomic rays.” Journal of the Optical Society of America, 14, 5, 423. May 1927.

[6] E. Werde, ZS. f. Phys. 41, 560, 1927.

[7] T. Phipps and J. Taylor, Phys. Rev. 29, 409, 1927.

[8] G. S. Landsberg, “New experiments with a molecular beam by O. Stern’s method,” Uspekhi Fizicheskikh Nauk 7, p. 494, 1927.

[9] J. L. Costa, H. D. Smyth, and K. T. Compton, “A Mechanical Maxwell demon,” Phys. Rev. 30, 349, September 1927.

[10] John A. Eldridge, “Experimental Test of Maxwell’s Distribution Law,” Physical Review 30, 931, December 1927, No. 6.