Carl Barus.

The diffusion of gases through liquids and allied experiments online

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Hazard Professor of Physics and Dean of the Graduate Department
in Brown University


Published by the Carnegie Institution of Washington






Hazard Professor of Physics and Dean of the Graduate Department
in Brown University





Published by the Carnegie Institution of Washington




Observing that the Cartesian diver used in my lectures since 1895 grew
heavier from year to year, I resolved in 1900 to make definite measurements
of the rate of loss of buoyancy, believing that these would be fruitful ; they
would bear directly on the coefficient of diffusion of the imprisoned gas
through the liquid in which the diver is floating; it would be easily possible
to vary the liquids and gases, within and without, under conditions of a
determinable diffusion gradient. Ultimately the transfer of single mole-
cules of a gas through the intermolecular pores of the liquid is in question,
so that the experiment might throw definite light on the size of physical
pores and on the other molecular relations involved.

The experiments in Chapter I, made during a period of eleven years, with
an ordinary glass balloon-shaped Cartesian diver with a small aperture,
culminated in a plausible value of the diffusion coefficient (i. e. t grams of
gas or standard volume of gas transpiring per second across an orthogonal
square centimeter, in case of a unit pressure gradient) of the imprisoned air
through water, together with suggestive relations of the mean viscosity of
the imaginary medium within the molecular pores of the liquid through
which a single molecule of the gas virtually transpires. The investigation
was therefore taken up on a more extended scale, for different pairs of gases.

In Chapter II the diver is modified in form and the endeavor is made to
obtain equal areas in the section of the cylindrical swimmer and the annular
space without, in order to conform more closely to the equation of diffusion.
The theory of the phenomenon and the errors involved are discussed. It
appears that, even for mixed gases, the volumes diffusing (if not the masses)
are fully determinable. The accuracy essentially depends on the measure-
ment of absolute temperature and of barometric pressure and should there-
fore be of an order below 1/2730 per 0.1 C. or 1/7600 per o.i centimeter of
mercury. As the masses of gas contained are as a rule much less than io~ 2
gram, even in case of air, the weight less than 0.000004 gram is determinable,
showing the remarkable sensitiveness of the method. Moreover, in the region
of constant temperature, the limit of sensitiveness is immensely greater.

In order to elucidate the phenomenon, experiments were begun with the
transpiration of imprisoned hydrogen into air, in which the resultant diffu-
sion is always unidirectional, outward from the diver. Initially rates as
large as 5 mg. per day were obtained, which eventually decreased to a con-
stant value, equivalent to a fixed diffusion coefficient which indicated the
diffusion of air only. The case of air into air through water showed a
definite mean rate throughout the two or three months of observation ; but
the daily march of the loss by diffusion was remarkably irregular, a result
finally referred to the change of solubility of the gases in water with tern-


perature. The result of this is absorption and release of gas as temperature
falls or rises, respectively, during the occurrence of the otherwise steady
diffusion. In the long series the temperature effect was eliminated by the
method of least squares.

Much more striking were the phenomena encountered in endeavoring to
find the coefficient of diffusion of hydrogen through water into hydrogen in
which, however, the ultimate daily loss of weight of the diver became con-
stant, corresponding to the diffusion coefficient of hydrogen alone. Referred
to molecular conditions, the molecule can be regarded as moving through a
medium about 1 5 times as viscous as ordinary hydrogen, whereas in case of
air the medium would be about 13 times as viscous as air. The daily march
of results in the hydrogen observations was most striking, inasmuch as the
diver first lost weight at an initially enormous rate for two days, then
rapidly gained weight at a decreasing rate during the ensuing ten days, and
thereafter assumed the steady rate of loss for months. Changes of this
nature are, as a rule, abrupt. It was found that a similar doubly inflected
progression of results usually occurs unless all manipulations at the outset
are conducted not in air, but in a medium of hydrogen, or in general of the
identical gas within the diver. Otherwise the imprisoned gas is at once con-
taminated by diffusion of the surrounding gas into it.

It is not, perhaps, fully appreciated by chemists that gases, otherwise
pure, if stored over water, at once lose purity in consequence of air by
diffusion. In fact a gas, A, in the swimmer, in presence of gases, B, C, etc.,
can not escape by diffusion until the sum of the partial pressures, B, C, etc.,
is equal to or greater than the pressure equivalent of the head of water under
which the gas A is submerged. Before that the gas of the environment will
diffuse into the diver against the hydrostatic pressure of the head of water,
i. e., apparently up hill. The same explanation accounts for the enormous
inflation of the microscopic air bubbles, for instance, in the liquid, when the
surrounding atmosphere is some other gas, like hydrogen; also for the
bubbles which still appear and grow at rough points of a surface after the
effervescence of a compound gas has ceased.

Other diffusion experiments, air into hydrogen, oxygen into hydrogen,
hydrogen into air, etc., were eventually pursued through months and com-
pleted in a similar manner and with similar results. The graphs obtained
are throughout striking. It is feasible to derive the differential equation
for these phenomena, but, as might be expected from the complications in could not be integrated. Finally, it is interesting to note that
if the diffusion coefficients are given, the densities of the gases diffusing at a
constant rate may be computed; or, from another point of view, the degree
of purity of the gas so diffusing may be ascertained.

The sensitiveness of weighing in case of the Cartesian diver, where the
whole apparatus is quite submerged in water or some other liquid and capil-
lary forces are out of the question, naturally suggested the application of this
method for the measurement of high potentials in case of the absolute elec-


trometer. For this purpose the whole condenser, as described in Chapter
III, is submerged in a clear non-conducting paraffin oil, while the mov-
able disk of the electrometer is floated on a Cartesian diver, or the circular
top of a cylindrical diver is itself the disk. The difference of weight of a
charged and uncharged condenser is determinable, the former in view of the
electrical pressures being less. It may then be shown that the absolute
difference of potential of the plates, cat. par., varies as their distance apart
and as the square root of the difference of the manometer pressures which
are just compatible with flotation, in the case of the charged and uncharged
condensers, respectively. By keeping the difference in question constant,
potentials may be absolutely measured in terms of the distance apart of the
plates from about 50 volts to indefinitely large magnitudes.

These experiments suggested a variety of other methods. Thus the disk
of the absolute electrometer, now kept in air, was buoyed up and held in
place on a hydrometer, with its body submerged in water or in oil, where the
capillary forces are small. Particularly interesting results were obtained
when the hydrometer was a very thin, straight aluminum tube, at right
angles to the light aluminum plate of the condenser, the aluminum tube
being submerged in a glass tube which is one shank of a U-tube. It is shown
that for a difference of potential of the disks (supposed horizontal), not too
large, there is a stable and an unstable position of the movable disk, the
former below the latter. The disk therefore rises from its fiducial position in
the uncharged condenser to a definite height. As the difference of potential
increases this height increases until at a transitional height both stable and
unstable positions coincide. For greater differences of potential the disk
passes without intermission from the lower plate (guard ring) to the upper
plate of the condenser. If the difference of potential is constant, the same
phenomena may be evoked on diminishing the distance apart of the plates
of the condenser, by lowering the upper plate on a micrometer screw.
Potentials may then be absolutely measured in terms of the distance apart
of the plates at which the continuous rise of the disk first occurs.

Other similar experiments were devised, such as the treatment of Cou-
lomb's law when one of the repelling bodies is a Cartesian diver, the repe-
tition of Mayer's experiments when the charged metallic bodies are floated
in oil in a charged guard ring, etc.

Finally, the experience gained in Chapter III, in relation to methods of
filling the diver with a gas in an environment of the same gas, a condition
rigorously necessary if the gases are to remain adequately pure for diffusion
measurements, suggested the further development of certain of the experi-
ments in Chapter II. These are given in Chapter IV. In addition to this,
the chapter begins the work of treating the diffusion of gases through solu-
tions systematically and at length. It contains the effect produced on the
diffusion coefficient of air by dissolving in water different quantities of KC1,
NaCl, CaCl 2 , BaCl 2 , SrCl 2 , K 2 SO 4 , Na 2 SO 4 , FeCl 3 , A1C1 3 , etc. The purpose
here is at present chiefly the gathering of data. The work is so laborious,


so essentially slow, and so full of pitfalls, that the serious attempt to draw
conclusions from the data in hand must be deferred. It appears, however,
that in all cases the physical pores of a solvent like water are effectively
closed by a solute, but that the amount of closure is dependent on the char-
acter of the salt and the density of the solution in a way not to be easily
surveyed. Thus a dilute solution may show greater cloture than a concen-
trated solution of the same salt, due no doubt to the formation of hydrates
effective in this respect. It appears also that the diffusion coefficients
obtained from direct manometer experiments in the lapse of years are not
at once comparable with the results for the divers in the lapse of months,
all of which disparities will need long-continued observation.

My thanks are as usual due to Miss Ada I. Burton for most efficient
assistance through the whole of this work, both in its experimental and
editorial parts. The data of Chapter IV, requiring a high order of patience
and accuracy, both as to observations and computation, have been largely
contributed by her.



Providence, Rhode Island.




1 . Molecular transpiration of a gas i

2. Apparatus. Fig. i i

3. Barometer i

4. Equations. Manipulation. Fig. 2 2

5. Data. Table i 2

6. Conditions of flow 3

7. Coefficients of transpiration 4

8. Values of the coefficients 5

9. Conclusion 6


10. Introductory. Apparatus. Figs. 3 and 4 7

1 1 . Imprisoned hydrogen diffusing into free air. Preliminary data. Fig. 5,

table 2 8

12. Continued. Coefficients depending upon water heads only 10

13. Continued. Apparent frictional resistance per molecule. Virtual viscosity. n

14. Continued. Transpiration depending upon barometric pressure 12

15. Continued. Influx of air into the imprisoned hydrogen 13

1 6. Continued. Coefficients depending on diffusion gradients. Transpiration.. 13

1 7. Continued. Flotation 14

1 8. Continued. Potential energy of the gas mixture 17

19. Transpiration of air into air through water. Fig. 6; table 3 18

20. Transpiration of hydrogen into hydrogen through water. Fig. 7; table 4. . . 22

21. Transpiration of imprisoned air into hydrogen through water. Fig. 8 ; table 5 . 25

22. Transpiration of oxygen into hydrogen through water. Fig. 9; table 6 28

23. Transpiration of hydrogen into air through water. Fig. 10; table 7 31

24. Correction for density of the glass. Table 8 32

25. Summary. Relatively slow diffusion of mixed gases. Tables 9 and 10 33


I. Hydrometer Methods.

26. Introduction 39

27. Absolute electrometer. Figs, n A, n B, 12, 13 39

28. Equations for the tubular float 42

29. Constants of the tubular float. Fig. 14 43

30. Constants of the conical float (capsule) 44

31. Experiments with the tubular float. Table 1 1 45

II. Absolute Electrometry by Aid of the Cartesian Diver.

32. Introductory 46

33. Apparatus. Fig. 15 47

34. Equations. Table 12 48

35. Measurements. Tables 13, 14 50


36. Purpose 55

37. Apparatus. Fig. 16; table 15 55

38. Equations 57

39. Diffusion of air into air through water. Fig. 17; table 16 58

40. The same, continued. Fig. 18; table 17 61

41. Diffusion of air into air through water; further experiments. Figs. 19 A, B, c,

20; tables 18, 19, 20, 21, 22 62



42. Diffusion of hydrogen into hydrogen through water. Fig. 21 ; table 23 66

43. Diffusion of ah- into air through KC1 solution. Fig. 22; table 24 67

44. The same, continued. Fig. 23 ; table 25 69

45. The same, continued. Fig 24; table 26 70

46. The same, continued. Fig. 25 ; table 27 71

47. Diffusion of air into air through NaCl solution. Fig. 26 A; table 28 72

48. The same, continued. Fig. 263; table 29 73

49. 5o, 51, 52. Diffusion of air into air through CaCl 2 solution. Figs. 27, 2 8 A,

288, 29; tables 30, 31, 32, 33 73

53. 54. 55- Diffusion of air into air through Bad, solution. Figs. 30, 31 ; tables

34. 35. 36 76

56, 57, 58. Diffusion of air into air through K 2 SO 4 solution. Figs. 32, 33; tables

37, 38, 39 78

59, 60. Diffusion of air into air through Na 2 SO 4 solution. Figs. 34 A, 34 B ; tables 40, 41 80

61,62. Diffusion of air into air through FeCl, solution. Figs.35A, 353; tables 42, 43. 81

63. Diffusion of ah" into air through A1C1 S solution. Fig 36; table 44 83

64. Diffusion of a gas through a manometer tube. Fig. 37; table 45 83

65. Summary. Fig. 38 ; table 46 85



Hazard Professor of Physics and Dean of the Graduate Department
in Brown University



1. Molecular Transpiration of a Gas. Ever since 1895 I have observed
that the Cartesian diver used in my lectures grew regularly heavier from
year to year. The possibility of such an occurrence is at hand; for the
imprisoned air is under a slight pressure-excess as compared with the
external atmospheric air. But this pressure gradient is apparently so insig-
nificant as compared with the long column of water through which the
flow must take place that opportunities of obtaining quantitative evidence
in favor of such transpiration seem remote. If, however, this evidence is
here actually forthcoming, then the experiment is of unusual interest, as it
will probably indicate the nature of the passage of a gas molecularly through
the intermolecular pores of a liquid. It should be possible, for instance, to
obtain comparisons between the dimensions of the molecules transferred
and the channels of transfer involved.

2. Apparatus. Hence on February 27, 1900, I made a series of definite
experiments* sufficiently sensitive so that in the lapse of years one might
expect to obtain an issue. The swimmer was a small, light, balloon-shaped
glass vessel, vd, fig. i, unfortunately with a very

narrow mouth 2 mm. in diameter at d, in the long
column of water A . The small opening, however,
gave assurance that the air would not be acci-
dentally spilled in the intervening years. For this
reason it was temporarily retained, the purpose
being that of getting a safe estimate of the con-
ditions under which flow takes place.

In fig. i, ab is a rubber hose filled with water,
terminating in the receiver R. Here the lower
level of water may be read off. Moreover, R is FlG T . Cartesian diver ad-
provided with an open hose C, through which pres- justed for diffusion meas-
sure or suction may be applied by the mouth, for

the purpose of raising or lowering the swimmer, vd, in the column A. In
this way constancy of temperature is secured throughout the column.

3. Barometer. The apparatus is obviously useful for ordinary baro-
metric purposes, and provided the temperature, t, of the air at v is known
to 0.025 C., the barometric height should be determinable as far as o.i mm.
Apart from this the sensitiveness of the apparatus is surprising. Great care
must be taken to avoid adiabatic changes of temperature, so that slow
manipulation is essential. These and other precautions were pointed out
in the original paper. The apparatus labors under one fundamental diffi-
culty, as the diffusion of a compound gas like air is a complicated discrep-

*Am. Journ. Sci., ix, 1900, pp. 397-400.


ancy which will be felt in the lapse of time. The question will be discussed
in the next chapter.

4. Equations. Manipulation. Let h be the difference of level of the impris-
oned water and the free surface in the reservoir R. Then it follows easily that


Pw gM(i+m/M}-p w /p a

where H is the corrected height of the barometer (from which the mercury
head equivalent to the vapor pressure of water is to be deducted) , p m , p w , p a ,
the densities of mercury (o C.), water (t C.), and glass,
respectively, m the mass of the imprisoned air at v, R its
gas constant, and r = / + 273 its absolute temperature.
M is the mass of the glass of the swimmer and g the acceler-
ation of gravity.

The equilibrium position of the swimmer is unstable.
To find it R may be raised and lowered for a fixed level of
the swimmer ; or R may be clamped and the proper level of
FIG. 2. Cylin- the swimmer determined by suction and release at C. The
dropping of the swimmer throughout the column of water
may occasion adiabatic change of temperature of 0.23. It was my practice
in the present experiments to use the latter method and to indicate the
equilibrium position of the swimmer by an elastic steel ring encircling A .
In this way the correct level may be found to about i mm. and afterwards
read off on the cathetometer.

After making the observations, the hose ab is to be separated at a, so that
the swimmer falls to a support some distance above the bottom, admitting
of free passage for diffusion. Clearly this diffusion is due to the difference
of level, h", between the water in v and at the free surface of the liquid
(see fig. 2). Increase of barometric pressure has no differential effect. A
large head h'", however, means a longer column for diffusion.

5. Data. In table i a few of the data made in 1900 are inserted, chosen
at random.

In the intermediate time I did not return to the measurements until quite
recently (January 1911), when a second series of observations was made.
As much as one-fourth of the air contained in 1900 had now, however,
escaped, in consequence of which the above method had to be modified and
all heads measured in terms of mercury. Hence if H denotes the height
of the barometer (diminished by the head equivalent to the vapor pressure
of water) and if m/M be neglected in comparison with i (about 0.06 per
cent), the equation becomes

Mgp m H(i/ Pm -i/ Pg ) , .

~~X~ ~T~

in which the first factor of the right-hand member is constant. If the
observations are made at the instant the swimmer sinks from the free
surface in A, fig. 2, H must be increased by the mercury equivalent of the


height h" of v. The table contains all the data reduced to mercury heads.
A = Mgp m /R, Consequently 1 842 X io~ 6 grams of the imprisoned air escaped
in the intervening 10.92 years; i. e., 0.265 f the original mass of air. In
other words i68.7Xio~ 6 grams per year, 0.462 Xio~ 6 grams per day, or
5-35Xio~ 12 grams of dry air per second.

TABLE i. Weight m of the imprisoned air, v, fig. i. M=\o grams; p OT =i3.6;
p a =2.ST, mouth of diver, 2r = o.2 cm.; A =0.0465. Time interval 10.92 years.






Feb. 27, 1900




297. 1


Feb. 27, 1900





Jan. 27, 191 1


21 .02


51 10

6. Conditions of Flow. It is now necessary to analyze the above experi-
ment preparatory to the computation of constants. The mouth of the
swimmer had an area of but 0.0314 cm. 2 . When sunk, the head of water
above the surface v was h" = 24 cm. The column of water between v and d
was h"' = 8 cm. Hence the length of column within which transpiration
took place was 24+2X8 = 40 cm. The right section of this column is taken
as 0.0314 cm. 2 throughout. Naturally such an assumption, accepted in the
absence of a better one, is somewhat precarious ; but it may be admitted,
inasmuch as the pressure of the gas sinks in the same proportion in which
the breadth of the channel enlarges. Thus there must be at least an approxi-
mate compensation. In more definite experiments a cylindrical swimmer
whose internal area is the same as the annular area without will obviate
this difficulty (see fig. 2).

The pressure-difference urging the flow of air from v is
A/> = 24X0.997X981 =23,470 dynes/cm. 2
hence per dyne/cm. 2 per sec.

io~ 12 X5-346 = _-i6-

grams of air escape from the swimmer.

A few comparisons with a case of viscous flow may here be interesting.
Using Poiseuille's law in the form given by O. E. Meyer and Schumann's
data for the viscosity of air, it would follow that but O.I94X io~ 6 cm. 2 of the
0.0314 cm. 2 of right section at d is open to intermolecular transpiration.
The assumption of capillary transpiration is of course unwarrantable and
the comparison is made merely to show that relatively enormous resistances
are in question.

Again, the coefficient of viscosity




may be determined directly. In this equation m is the number of grams
of air transpiring in t seconds through the section irr 2 and in virtue of the


pressure gradient (Pp}/l, when rj is the viscosity and f the slip of the gas.
Hence the value TJ/(I + 4f/r) = 4-8 X io 6 would have to obtain, a resist-
ance which would still be enormously large relative to the viscosity of air
(17= i8oX io~ 6 ), even if the part of the section of the channel which is open
to capillary transpiration is a very small fraction.

7. Coefficients of Transpiration. To compute the constants under which
flow takes place the concentration gradient dc/dl may be replaced either by
a density gradient dp/dl or a pressure gradient dp/dl. If the coefficients
in question be k p and k p respectively


where the section a is equal to the area of the mouth of the swimmer, R is
the absolute gas constant, T the absolute temperature of the gas, and m the
loss of imprisoned air in grams per second. If v = mRr/p is the correspond-
ing loss of volume at T and p,

k= *!L = .^L_ ,_*

P RT aRrdp/dl

If in equation (3) the full value of m is inserted, and / denotes current time
or m = m/t; if

dl h"+2h" f

where p w is the density of water, h" and h'" the difference of level (see fig. 2)
of the surface in v below the free surface in A and above the mouth at d,
the relations are

M Pm Hi+2h'"/h"fi

Rt r ap w
k p = k p Rr (5)

The acceleration of gravity g has dropped from both equations; k p is inde-

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Online LibraryCarl BarusThe diffusion of gases through liquids and allied experiments → online text (page 1 of 10)