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5. The External Friction of Gases;

by W. Gaede.



Contents:  1.  Introduction .  -- 2. Preliminary tests.  -- 3. Hydrogen current/flow through a narrow slit. -- 4. Current/flow of a gas mixture through a narrow slit.  -- 5. Current/flow by hydrogen and nitrogen through a capillary with very low pressures.  -- 6. Influence of the gas surface on molecule movement.  -- 7.  Summary.


1 . Introduction.


  According to kinetic gas theory the external friction is independent of pressure.  This law is affirmed through numerous experiments within large pressure/compression intervals.  At very low pressures Kundt and Warburg 1) found prominent deviations to vibration methods and this attributed to the slip of the gases on the wall.   Warburg 2) found the same result in flow-method tubes.  The external friction of the wall gases wane with decreasing pressures, so the more the gases slide on the walls the lower the pressure.  H. Eger 3) believed the flow method observations he received are ascribed to the changing internal friction at low pressure.  J. L Hogg 4) reconfirmed the vibration method results at very low pressures observed by Kundt and Warburg.  M. Knudsen 5) came to the theoretical conclusion for which external friction gives a specific value and found confirmation of his theory through careful observations at very low pressures.


1) A. Kundt u. E. Warburg, Pogg. Ann. 155. p. 337, 525. 1875.

2) E. Warburg, Wied. Ann. 159. p. 399. 1876.

3) H. Eger, Ann. d. Phys. 27. p. 819. 1908.

4) J. L. Hogg, Philosophical Magazine p. 376. 1910.

5) M. Knudsen, Ann. d. Phys. 28. p. 75. 1909.

Annalen dar Phyaik. IV. Fo]ge. 41.                                     19

*Original text was "betätigt" which translates to active.


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W. Gaede.


Knudsen assumes that the gas molecules fixed on the wall are reflected in an angle, which is independent of the direction of incidence.  The swarm of molecules spread in absolute disorder in all directions, according to the well-known law of cosines just like the light from a glowing plate.  According to this theory computations are only relevant in the outermost vacuum of the pipe through which the mass of gases flow.

Knudsen derives an empirical interpolation formula in order to be able to indicate the higher flow-through pressures for the large quantities of gas.  The constant in his formula is evidenced through his particular theoretical molecular assumptions.  M. v. Smoluchowski 1) criticizes the Knudsen theory and there is evidence that there are similar special cases of the more general Maxwellian theory, according to which a component f of the molecules emanate independently from the angle of incidence.  The component 1 f  is reflected with unchanged velocity as a ray of light.  For f = 1 the Maxwellian is transferred to the Knudsen theory.  On some points v. Smoluchowski further supplemented the Knudsen theory. 

In order to answer to the charge that the mercury vapor of the MacLeod pressure gauge could have affected the result of its measurements, Knudsen 2) implements some regulations of the flow velocity with a hot wire pressure gauge.  In conclusion it should be mentioned that Knudsen and S. Weber 3), and Keehan 4) researched the external friction between spheres and attenuated gases. 


Preliminary tests.


In order to be able to employ a comparison with the Knudsen interpolation formula, I derived an interpolation formula by hydrodynamic methods.


1) M. v. Smoluchowski, Ann. d. Phys. 33. p. 1559. 1910.

2)   M. Knudsen, Ann. d. Phys. 35. p. 389. 1911.

3) M. Knudsen u. S. Weber, Ann. d. Phys. 36. P. 981. 1911.

4)  L. Mc Keehan, Physik. Zeitschr. 12. p. 707. 1911.


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The External Friction of Gases.


Sliding the wall the neighboring gas layers slide on the wall with the velocity u, so we then designate the force exerted on the unit area in the direction of motion by e u, whereby e is the coefficient of the external friction.  This force must equal, the transmitted force of the gas flow on the outermost gas layer h (du /dx), when the coefficient of the external friction is h and x is the coordinate perpendicular to the wall.  If we then introduce this equation in observance of Boyle’s law as the boundary condition into the equations for the Poiseuille current, we then receive:




when n0 is measured near the pressure p0, the volume flowing through the tube of the radius R and length per second is L. 

p1 and p2 are the pressures dominant at the ends.  We give the variable e the value:




the Knudsen tests at very low gas pressures are equivalent, when j1 the pressure is reduced to dynes per square centimeter, the existing density of gas is in grams per cubic centimeter at prevailing temperatures, and when we use this value in equation (1), we then obtain: 




when G is the unit per second pressure difference through the flowing pipe and the mass of gas is measured by the product of pressure and volume, and when the quantities a and b are introduced for simplification.  Knudsen gives the Interpolation formula:


(4)                                                                                                                                                           19*