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T. W. Patzek, Fick’s Diffusion Experiments Revisited, 6/22/06 1

Fick’s Diffusion Experiments Revisited T. W. Patzek

Department of Civil and Environmental Engineering

University of California at Berkeley

591 Evans Hall

Berkeley, CA 94720

[email protected]

Abstract

In this paper, we revisit Fick’s original diffusion experiments and reconstruct the

geometry of his inverted funnel. We show that Fick’s experimental approach was sound

and measurements were accurate despite his own claims to the contrary. Using the

standard modern approach, we predict Fick’s cylindrical tube measurements with a high

degree of accuracy. We calculate that the salt reservoir at the bottom of the inverted

funnel must have been about 5 cm in height and the unreported depth of the deepest salt

concentration measurement by Fick was yet another 3 cm above the reservoir top. We

verify the latter calculation by using Fick’s own calculated concentration profiles and

show that the modern diffusion theory predicts the inverted funnel measurements almost

as well as those in the cylindrical tube. We also append a translation of Fick’s discussion

of diffusion in liquids in the first edition of his three-volume monograph on Medical

Physics published in 1856, one year after his seminal Pogendorff’s Annalen paper On

Diffusion.

1 Introduction

Fick’s classical diffusion equation is so much a part of the scientific folklore that I

decided to revisit its original derivation and the underlying experiments in a graduate

course taught at Cal. To my surprise, Fick’s own experiments turned out to be remarkably

accurate and the geometry of his funnel experiments could be adequately reconstructed.

While searching for Fick’s original publications, a student in the class found on the Web

an antique book dealer who had the first edition of Fick’s handbook on medical physics,

mailto:[email protected]


which I quickly purchased. I then asked my neighbor who is German by birth and

education to help me in the translation of the relevant sections of the book and she kindly

agreed. The book was written by Fick as an account of physics applicable to medicine

and it brings interesting insights into Fick’s way of thinking about diffusion and osmosis,

see Appendix A. The book section on the diffusion in liquids is more personal and

somewhat different from the two famous papers published one year earlier (Fick 1855;

Fick 1855).

This paper is a simple evaluation of Fick’s original experiments based on the

theory of diffusion presented so well in (Hirschfelder 1954) and (Bird 1960). For an

exquisite and brief discussion of the various theories of diffusion developed over the

decades by Maxwell, Stefan, Onsager, Chapman and Enskog, Eckart and Meixner, and

many others, one may refer to (Truesdell 1962).

Fick’s theory of diffusion is almost as old as the theories of heat conduction by

Fourier (Fourier 1807; Fourier 1822) and viscosity by Newton. While originally it was

too primitive to reveal any ideas of principle, Maxwell soon1 gave it a rational basis in

his kinetic theory of gas mixtures and Stefan (Stefan 1871) cleared away the specifically

kinetic details to achieve an inclusive phenomenological theory, see (Truesdell 1966) for

further details.

2 Background

For convenience, a few pertinent equations describing diffusion will be listed

here. A full derivation may be found in, e.g., (Bird 1960). Let Bv denote the velocity of

constituent2 B of a fluid mixture with respect to a stationary coordinate system, and

define this velocity as by taking a snapshot of instantaneous velocities of the molecules of

B. For a mixture of constituents, we define the local mass-average velocity as cN

1 Maxwell published four important papers on gas theory, the first of which, “Illustrations of the Dynamical Theory of Gases,” appeared in 1860, and contained Maxwell’s first theory of diffusion. See (Garber 1986) for details. 2 “Constituent” denotes a chemical species or a pseudo-component.


1 1

1

c c

c

N N

B B B BB B

N

BB

v vv

ρ ρ

ρρ

= =

=

= =∑ ∑

∑ (1)

Thus vρ is the local rate at which mass passes through a unit area perpendicular to v .

Similarly, we may define a local molar average velocity as

* 11

1

cc

c

NN

B BB BiB

N

BB

c vc vv

cc

==

=

= =∑∑

∑ (2)

Thus is the local rate at which moles pass through a unit area perpendicular to *cv *v .

In flow systems one is often interested in the velocity of a given constituent with

respect to or , rather than with respect to a stationary coordinate system. This leads

to the following definition of the diffusion velocities:

v *v

(3) ( )

* *( )

diffusion velocity of to

diffusion velocity of to D B B

D B B

v v v B v

v v v B

≡ − =

≡ − =

relative

relative *v

These velocities measure the motion of constituent B in a fluid relative to the local mean

motion of the fluid.

Now we can define the different mass and molar fluxes. The mass or molar flux

of constituent B is defined as a vector whose magnitude is equal to the mass or moles of

constituent B that pass through a unit area per unit time. The motion may be referred to

stationary coordinates, to the local mass-average velocity, v , or to the local molar-

average velocity, *v . Thus the mass and molar flux densities relative to stationary

coordinates are

( ) ( )mass, molarm B B B n B B Bj v j c vρ= = (4)


The mass and molar flux densities relative to the mass-average velocity are v

(5) ( | ) ( ) ( | ) ( )mass, molarm v B b D B n v B B D Bj v j c vρ= =

and the mass and molar flux densities relative to the molar-average velocity are *v

(6) * ** *( ) ( )( | ) ( | )

mass, molarB D B B D Bm v B n v Bj v j c vρ= =

Again, one should remember that the definition of a mass flux density is incomplete until

both the units and the frame of reference are given.

By analogy with the flux of energy in one-dimensional systems:

( ) (Fourier's law for constant ),y pdq c Tdy

α ρ ρ= − pc (7)

one may define the mass flux of constituent 1 in a binary system 12 as

1 12 1( ) (Fick's law for constant )ydj Ddy

ρ ρ= − (8)

Note that the total mass density, ρ , of the mixture is uniform, i.e., constant given a

constant temperature. Here α is the thermal diffusivity, is the heat capacity, is

the diffusion coefficient, and

pc 12D

1ρ is the mass density of constituent 1.

If the fluid density is not uniform, one defines the mass diffusivity in a

binary system in an analogous fashion:

12 21D D=

*( | )1 12 1 12 1( | )1mass, molarm v n v

j D w j cD xρ= ∇ = ∇ (9)


Note that Eq. (9) is purely kinematic, i.e., it involves only the dimensions of length and

time, and it must be justified from an independent dynamic theory that invokes forces.

It may easily be shown (Bird 1960) that the mass or molar diffusion fluxes in Eq.

(9) are also given by

*

2

( | )1 ( )1 1 ( ) 12 11

2

( )1 1 ( ) 12 1( | )11

m v m m BB

n n Bn vB

j j w j D w

j j x j cD x

ρ=

=

= − = − ∇

= − = − ∇

∑

∑ (10)

These two relationships are of special importance for this paper.

3 Fick’s cylindrical tube experiment

Fick’s experiment (Fick 1855) in a cylindrical tube is shown in Figure 1. At

, salt (1) diffuses into fresh water (2), while solid salt in a salt reservoir dissolves,

replenishing the diffusing salt, and maintaining the saturated salt concentration at

z L=

0z = .

A stream of fresh water sweeps the salt emerging from the tube, keeping the zero salt

concentration at z L= . The whole system is at constant (albeit unknown) temperature

and pressure. The mixture of salt and water is assumed ideal, i.e., each component

activity is equal to its mole fraction.


0z =

z L=

Saturated salt solutionand solid salt

Liquid stream of water (2) and salt (1) from tube

dz

( )1( )nj z

( )1( )nj z dz+

0z =

z L=


Liquid stream of water (2) and salt (1) from tube

dz

( )1( )nj z

( )1( )nj z dz+

Figure 1 – Fick’s diffusion experiment in a cylindrical tube with a salt reservoir at the bottom and flowing fresh water at the top.

When this dissolving salt system attains a steady state, there is a net motion of salt

away from the salt reservoir and the water in the tube is stationary. Hence we can use

expression (10)2 for the molar flux of salt relative to a stationary frame of reference:

* ( )1 1 ( )1 1 12 1( | )1

12 1( )1

1

( ) ( ) ( ) 0

( )1

n nn v

n

j z j z x j z x cD x

cD dxj zx dz

= − − = −

= −−

∇

(11)

The salt mass balance over an incremental column height, dz states that at steady state

( )1( )0ndj z

dz= (12)

Substitution of Eq. (11) into Eq. (12) gives


12 1

1

01

d cD dxdz x dz

⎛ ⎞=⎜ ⎟−⎝ ⎠

(13)

We shall assume that is nearly independent of concentration. Indeed the literature

data show that for 0.1-1 normal salt solutions. The saturated salt

concentration is about 36g NaCl/100g of water or 26g NaCl/100g solution at room

temperatures of 10-20 degrees C. From Appendix B, we can calculate the total

concentration of water and salt at 20 deg C. This calculation is summarized in

12D

212 1 cm /dayD ≈

Table 1.

The percent coefficient of variation is generally less than 1%; therefore, the total

concentration can be assumed constant.

Table 1 – Total concentration of aqueous salt solutions, see Appendix B.

c1-water c5-water Total c C.V.

99 55.2915 55.4635 0.5408

98 55.1087 55.4551 0.5255

96 54.7360 55.4386 0.4956

94 54.3790 55.4478 0.5123

92 53.9682 55.4133 0.4498

90 53.5350 55.3667 0.3653

88 53.0787 55.3075 0.2579

86 52.5986 55.2353 0.1270

84 52.0893 55.1446 -0.0372

82 51.5643 55.0498 -0.2091

80 51.0133 54.9405 -0.4072

78 50.4400 54.8208 -0.6243

76 49.8391 54.6856 -0.8693

74 49.2182 54.5433 -1.1273

The coefficient of variation is defined as C.V. 100%c c

c−

=

Thus, we can recast the steady state diffusion mass balance of salt, Eq. (13), as


1

1

1 1

1 01

(0) 0.0976, ( ) 0

d dxdz x dzx x

⎛ ⎞=⎜ ⎟−⎝ ⎠

L= =

(14)

This equation can be easily solved (Bird 1960) and the result is

1 1

1 1

1 1 (1 (0) 1 (0)

z

) Lx x Lx x

⎛− −= ⎜− −⎝ ⎠

⎞⎟ (15)

The density-mole concentration data from Table 1 and Table 3 are summarized in Table

2. The latter table can be used in conjunction with the excess gravities reported by Fick to

convert his gravity versus depth results into the salt mole fraction versus depth. This was

implemented as linear interpolation.

Table 2 – Salt solution density at 20 deg C and the mole fraction of salt

g/cc x1

1.0053 0.0031

1.0122 0.0062

1.0263 0.0127

1.0413 0.0193

1.0559 0.0261

1.0707 0.0331

1.0857 0.0403

1.1009 0.0477

1.1162 0.0554

1.1319 0.0633

1.1478 0.0715

1.1640 0.0799

1.1804 0.0886

1.1972 0.0976

The results are shown in Figure 2. Agreement between the theory and experiment is

excellent.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

5

10

15

20

25Fick's Cylindrical Tube Experiment

Ele

vatio

n ab

ove

salt

rese

rvoi

r, cm

Mole fraction of salt, x1

Figure 2 – Fick’s cylindrical tube excess gravities (+) have been converted to mole fractions of salt versus depth by linear interpolation of data in Table 2. The theoretical curve given by Eq. (15) is the solid line. Note excellent agreement between theory and experiment.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25Fick's Cylindrical Tube Experiment

Ele

vatio

n ab

ove

salt

rese

rvoi

r, cm

Excess gravity, g/cc

Figure 3 – This is how Fick would interpret his cylindrical tube experiment. Note the systematic deviation between the data and the straight line drawn by Fick.


3.1 Remarks

Figure 3 shows the profile of excess gravity that would have been calculated by

Fick for the cylindrical tube experiment. Note that there is a systematic deviation of the

calculation from the experimental data. Fick himself said (Fick 1855): “That the degrees

of concentration in the lower layers decrease a little more slowly than in the upper ones,

is easily explained by the consideration, that the stationary condition had not been

perfectly attained.” Similar excuses have been used by experimentalists ever since.

4 Fick’s conical funnel experiment

The conical funnel data obtained by Fick (Fick 1855) are more difficult to

decipher because of the incomplete reporting of the experiment. The most likely

experiment geometry may be inferred from Figure 6. In an inverted conical funnel, and in

the absence of gravity, the salt concentration contours would be sections of concentric

spheres centered on the salt reservoir at the funnel tip. One may argue that in the

gravitational field the spherical concentration profiles of salt will be flattened vigorously

by buoyancy force3. The denser salt solution near the funnel axis will sink, while the less

dense solution near the walls will be buoyed. The concentration profiles will then become

almost perfectly horizontal. With help of Figure 4, this assertion can be proven as

follows. For the salt profile flattening to happen, the ratio of the characteristic time of

diffusion, Dτ spreading the salt radially, and convection, Cτ , tumbling the more

concentrated salt solution down a “hill” of height h, must be much more than one:

Fick's Number Fi 1D

C

ττ

= >> , (16)

i.e., buoyancy flattens the constant salt concentration profiles before diffusion propagates

them as sections of concentric spheres.

3 Fick was aware of this phenomenon, see the underlined sentences in Appendix A, Section 24.



Liquid stream of water (2) and salt (1) from cone

h

r

2L

ααSaturated salt solutionand solid salt

Liquid stream of water (2) and salt (1) from cone

h

r

2L

αα

Figure 4 – Geometry of flattening of diffusion front.

The characteristic diffusion time scale can be obtained from the diffusion

coefficient, , and the distance, h: 12D

2

12D

hD

τ ≈ (17)

Assuming no viscous dissipation, the shortest convection time scale can be obtained from

the balance of potential and kinetic energy:

2,max

,max

,max

122 /

/ 2

C

C

CC

gh v

vL

v

ρ ρ

ghρ ρ

τ

Δ =

≈ Δ

≈

(18)

From elementary geometry, it also follows that


sin ,(1 cos )

L rh r

αα

== −

(19)

By combining Eqs. (16) - (19), one obtains the following criterion

2 3/ 2 5/ 212

12

3/ 2

125

2/ (1 cos )Fi 1,2sin2 / 2

2 sin2 (1 cos )

crit

gh D r g gDL g h

Dr rg

α ρα ρ

αα

′− Δ′≈ = >> ≡′

⎡ ⎤>> ≡ ⎢ ⎥

′ −⎢ ⎥⎣ ⎦

(20)

The plot of the critical distance from the salt reservoir with the most conservative

is given in 0.01g′ = g Figure 5. It is obvious that at distances of more than 1 mm above

the salt reservoir, the salt concentration profiles are essentially flat.

5 10 15 20 25 3010

-5

10-4

10-3

10-2

Funnel half-angle, degrees

Crit

ical

dis

tanc

e, c

m

Figure 5 – The critical distance from the salt reservoir at the tip of an inverted conical funnel versus the funnel half-angle. At distances much greater than the critical distance, the salt concentration profiles will be essentially flat.


The geometry of Fick’s cemented funnel experiments is shown in Figure 6. The

junction between the funnel and the salt reservoir is at an unknown elevation at which

the salt solution is saturated. Because the deepest reported salt density is far from the

density of the saturated salt solution, there is an unknown offset

1z

zδ between and the

deepest measurement.

1z

z

( )A z dz+

( )A zdz

( )1( )mj z

( )1( )mj z dz+

1 1( , ) satz z tρ ρ= =

1 2( , ) 0z z tρ = =

0

1

α

zδOffset of deepestmeasurement

z

( )A z dz+

( )A zdz

( )1( )mj z

( )1( )mj z dz+

1 1( , ) satz z tρ ρ= =

1 2( , ) 0z z tρ = =

0

1

α

zδOffset of deepestmeasurement

Figure 6 – Geometry of Fick’s funnel experiment.

In the cylindrical-polar coordinate system in Figure 6, the vertical diffusion flux is

* ( )1 1 ( )1 1 12 1( | )1

12 1( )1

1

( ) ( ) ( ) 0

( )1

n nn v

n

j z j z x j z x cD x

cD dxj zx dz

= − − = −

= −−

∇

(21)

The salt mass balance over a cylinder with the base of and height dz , states that

at steady state

2 ( )r zπ


( )

( )

2( )1

2( )1

( ) ( ) 0,

( ) 0

n

n

d r z j zdzd z j zdz

=

=. (22)

Substitution of Eq. (21) into Eq. (22) and assumption of constant give 12cD

2 1

1

1 1 1 2

1 0,1

( ) 0.0976, ( ) 0.

d dxzdz x dzx z x z

⎛ ⎞=⎜ ⎟−⎝ ⎠

= =

(23)

Equation (23) can be readily solved (Bird 1960) and the result is

1

1 2

1/ 1/1/ 1/

1 1 2

1 1 1 1

1 ( ) 1 ( )1 ( ) 1 ( )

z zz zx z x z

x z x z

−−⎛ ⎞− −

= ⎜ ⎟− −⎝ ⎠ (24)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.058

10

12

14

16

18

20

22Fick's Conical Funnel Experiment

Rad

ial d

ista

nce

from

sal

t res

ervo

ir, c

m

Mole fraction of salt, x1

Figure 7 – Fick’s conical funnel excess gravities (+) have been converted to mole fractions of salt versus by linear interpolation of data in Table 2. The theoretical curve given by Eq. (24) is the solid line. Note excellent agreement between the theory and Fick’s experiment.


To match Fick’s results with the theoretical curve (24), one needs to find the

lower elevation and the measurement offset 1z zδ . Then

2 1 (155.4 27.7) /10 cmz z zδ= + + − . The resulting two-parameter optimization problem

has been solved using the Nelder-Mead amoeba algorithm and the solution is:

1

2

4.985 5 cm,3.0848 3 cm,20.84 21 cm.

zz

zδ= ≈= ≈= ≈

(25)

Without invoking a nonzero offset, Fick’s experimental results cannot be matched. Thus,

the funnel-salt reservoir junction corresponds to the elevation of 5 cm above the funnel

tip and the deepest gravity measurement was preformed 3 cm above that junction. The

resulting fit of Fick’s funnel data is shown in Figure 7. A little bit of scientific sleuthing

has resulted in excellent agreement with Fick’s experimental data – a handsome reward

indeed!

5 Discussion of Fick’s equation

On no basis beyond an asserted analogy to Fourier’s analysis of the flow of heat,

Fick proposed that in a binary mixture the diffusion mass flux density vector is

proportional to the gradient of mass density:

( | ) ( ) , 1,2, 0,m v B B D B Bj v D B Dρ ρ≡ = − ∇ = > (26)

where D is a phenomenological coefficient which may depend upon the densities and the

temperature.

From Eq. (9) we now know that Eq. (26) holds only if the solution density is

constant:

( | ) ( ) ( / ) ,1 (salt), 2 (water), 0, const

m v B B D B B B Bj v D w D DB B D

ρ ρ ρ ρ ρ ρ

ρ

≡ = − ∇ = − ∇ = − ∇

= = > = (27)


Because densities of Fick’s salt solutions varied by almost 20%, Fick in fact was

mistaken in asserting Eq. (26). His subsequent diffusion equation was derived from a

shell mass balance between two horizontal planes dz apart.

Based on the mass balance shown in Figure 6, Fick derived the following

equation:

( )

1( )1 ( )1

1( )1

( ) ( , ) ( ) ( , ) ( )

1 ( ) ( , )( )

m m

m

A z dz j z t A z j z dz t A z dzt

A z j z tt A z z

ρ

ρ

∂= − +

∂∂ ∂

= −∂ ∂

+ (28)

Instead of inserting the salt mass flux density from Eq. (10) (he did not know this

equation), Fick asserted Eq. (26), and obtained the following equation:

2

1 12

1( )

dADt z A z dz

1

zρ ρ⎛∂ ∂ ∂

= +⎜∂ ∂ ∂⎝ ⎠

ρ ⎞⎟ . (29)

At steady state, and in the conical funnel whose radius is given as ( ) (tan )r z zα= ,

Fick simplified equation (29) and solved it:

21 1

2

21 1

2 0

( )

d ddz z dz

Cz Cz

ρ ρ

ρ

+ =

= −, (30)

The two constants and “…are to be so determined, that for a certain z

(where the cone is cut off and rests upon the salt reservoir

1C 2C4) 1ρ is equal to perfect

4 Fick used y instead of 1ρ and x instead of z. So his “certain distance x“ becomes in our notation. 1z z=


saturation; and for a certain value of z which corresponds to the base of the funnel5, 1ρ

becomes =0.” Thus

1 2 1 1( ) 0, ( ) 0.196 g/cc.satz z z zρ ρ ρ= = = = = (31)

If Fick knew about Eq. (10), then instead of Eq. (30) he would have probably

written:

21

1 1

1 1( ) 0,1 1

d dx d dxcD A z zdz x dz dz x dz

⎛ ⎞ ⎛ 1 ⎞= =⎜ ⎟ ⎜− −⎝ ⎠ ⎝⎟⎠

(32)

and he would have obtained Eq. (23).

0 0.02 0.04 0.06 0.08 0.1 0.128

10

12

14

16

18

20

22Fick's Conical Funnel Experiment

Ele

vatio

n ov

er c

one

tip, c

m

Excess gravity, g/cc

Fick's data Fick's calculation TWP's calculation with r1 and δr

Figure 8 – Fick’s excess gravity data (+) in the conical funnel, his calculated salt density profile (upper curve), and the profile reconstructed here with use of the previously obtained values of and 1z zδ .

5 Equal to the outer elevation in our notation. 2z


Because Fick does not report these “certain values of z,” to obtain the two

constants we must use the elevation of the funnel base, and the offset of the deepest

measurement calculated above. The two constants in Eq. (30) are then readily calculated

as . In the paper, however, Fick lists his calculated salt

density profile! The raw data, Fick’s calculation and the present calculation are shown in

1 2-0.0616, C 1.2843C = =

Figure 8. Agreement between the reconstructed solution and Fick’s ‘exact’ calculation is

very good indeed. The latter comparison confirms validity of the elevations (or axial

distances) listed in Eq. (25). Of course, because of the incorrect form of the governing

equation, the solutions shown in Figure 8 are nowhere nearly as good as the solution

shown in Figure 7.

6 Conclusions

In general, Fick’s equation (29) is not entirely correct. For the concentrated salt

solutions, one may not assert that mass density 1ρ is the appropriate unit of

concentration, and that constρ = . As it stands, after the sign correction, Fick’s original

equation can only be applied to dilute solutions and to such tubes of variable cross-

section whose walls are orthogonal to the surfaces of constant concentration. This does

not detract however from Fick’s great physical insight, and from his creative and quite

accurate6 experimental technique!

7 Acknowledgements

The financial support for this work was provided by PV Technologies, Inc. I

would like to thank Ms. Karin Lloyd-Mumford for her patience and tenacity in translating

Fick’s difficult prose. I am grateful to Prof. T. N. Narasimhan for sharing his insights into

the theories of diffusion and copies of the out-of-print books and papers. I would also like

to thank Dr. Mark Jellinek for sketching the gravitational equilibration equations (16)-

(18). Finally, I would like to thank the Spring 2000, E241 Class for the excellent

questions about diffusion.

6 Fick remained modest about his approach: ”…This method (of weighing an immersed glass bulb to determine density) creates little confidence at first sight, nevertheless preliminary experiments showed it to be sufficiently accurate.”


8 References

Achmatowicz, O., et al., Ed. (1954). Chemical Calendar, Vol. I (in Polish). Warsaw,

PWN.

Achmatowicz, O., et al., Ed. (1954). Chemical Calendar, Vol. 2, Part II (in Polish).

Warsaw, PWN.

Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960). Transport Phenomena. New

York, John Wiley & Sons, Inc.

Fick, A. (1855). “On Liquid Diffusion.” The London, Edinburgh, and Dublin

Philosophical Magazine and Journal of Science 10 - Fourth Series(July-September): 30-

39.

Fick, A. (1855). “Ueber Diffusion (On Diffusion).” Annalen der Physik und Chemie von

J. C. Pogendorff 94: 59-86.

Fourier, J. B. J. (1807). “Théorie des mouvements de la chaleur dans le corps solides.”

French Academy.

Fourier, J. B. J. (1822). Theory of Heat (Théorie Analytique del la Chaleur). Paris, F.

Didot.

Garber, E., Brush, S. G., and Everitt, C. W. F., Ed. (1986). Maxwell on Molecules and

Gases. Cambridge, Massachusetts, MIT Press.

Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. (1954). Molecular Theory of Gases and

Liquids. New York, John Wiley & Sons, Inc.

Stefan, J. (1871). “Über das Gleichgewicht und die Bewegung, insbesondere die

Diffusion von Gasmengen.” Wiener Sitzungsberichte 63: 63-124.

Truesdell, C. (1962). “Mechanical basis of diffusion.” J. Chem. Phys. 37.

Truesdell, C. (1966). The Elements of Continuum Mechanics. New York, Springer-

Verlag.


Appendix A – Laws of Diffusions

Section 1, Molecular Physics, Chapter 2, in Medical Physics, Parts I and II, by Dr. Adolf Fick,

Anatomy Demonstrator in Zürich, Printing Press and Publishing Company of Friedrich Vieweg and Son,

Braunschweig, 1856.

This translation of Fick’s discourse on laws

of diffusion in liquid mixtures in direct contact and

separated by permeable membranes (Articles 23-39)

was performed by Karin Mumford7 with help from

Tad W. Patzek. The figure numbers follow Fick’s

book. The book was typeset in German gothic, Figure

9, and Fick’s painfully long compound sentences did

not make the translation any easier.

Figure 9 – Cover of Fick’s Medical Physics

23 Now all that’s left are those molecular

movements that occur on contact with

heterogeneous liquids. The essence of this is

given the name liquid diffusion. At this point

one has to remember the commonly known

fact that in contrast to gases in each case of

contact of heterogeneous molecular

aggregates, diffusion currents will occur

regardless of hydrostatic equilibrium. Liquid diffusion operates in much more

heterogeneous liquids, which on contact are absolutely indifferent against each other, so

that in no location flow will prevail as soon as each of the liquids by itself is in a state of

hydrostatic equilibrium. Examples of these indifferent characteristics are combinations of

fatty oils with water, or mercury with most other liquids. To explain this one cannot

assume that the heterogeneous molecules in question reject each other entirely, rather it is

sufficient to assume that the attraction between each two heterogeneous molecules is less

than that between two homogeneous of one as well as of the other aggregate. If attraction

between the heterogeneous molecules is stronger than between a pair of homogeneous

7 6638 Longwalk, Oakland, CA 94611, [email protected].


molecules, a molecular movement happens immediately at the interface, which develops

the diffusion current in a manner analogous to the gas process. Therefore, both processes

have in common that none of them comes to rest and attains final equilibrium until a

uniform equilibrium distribution is produced in the entire space that is occupied by both

diffusing substances, meaning until in each part of this space the same condition is

obtained for each of the substances.

24 In the currently still inadequate understanding of the structure of molecules, it

would be useless to try to demonstrate this course of events. We have to restrict ourselves

to just demonstrating here the known laws from our experience. Until now, from all

possible cases, only the ones where the touching liquid solutions of the same solid in the

same solvent, which only differed in their concentrations, were subject to a specific test.

This is the reason for us to start this case with the governing law, but not without

mentioning immediately that other cases with a probability bordering certainty are

subject to the same law, if one only interprets or modifies terms of their mathematical

description. To avoid unnecessary notation and at the same time deal with fixed concepts,

we will simply name the dissolved solid salt and the solvent water. The governing law for

the spreading of salt in a water mass is, to say it very briefly, the same as the one that

rules the spreading of heat in a heat-conducting body, on which Fourier based his famous

mathematical theory of heat, and that was extended by Ohm, as we know with great

success, to the flow of electricity in a conductor. Avoiding the shorthand language of

mathematical analysis, we must express the same as follows. One has to think of two

very thin horizontal solvent layers on top of each other, the lower one contains more salt

and its concentration is = k; in the top one we will assume the lesser concentration k ′ . If

now the separation of both layers is = d, all these quantities will somehow retain the same

values during a certain time interval ϑ . The diffusion current, caused by the

heterogeneity (concentration difference) between both layers during the imagined time

interval, conveys a quantity of salt through the unit area of the dividing surface from the

concentrated lower layer to the thinner upper one that is k kCd

ϑ′− . Here C is a quantity

depending on the chemical nature of the relevant salt and the temperature alone, and


therefore for the present consideration it is considered constant. If Q were the total

surface area of both layers, the total amount of salt delivered to the upper would be =

A

C

M

P

m

B

N

QA

C

M

P

m

B

N

Q Fig. 7.

. k kC Qd

ϑ′− . At the same time, a water mass will be added from the upper layer to the

bottom layer, which occupies the same volume so that the volumes of both layers

together will not be altered during the process. The special assumption of horizontal

layers with constant concentration was necessary to make, because only in this case a

hydrostatic equilibrium remains in the liquid mass. It also had to be assumed, as we did

just now, that the change of the concentration from one horizontal layer to the other one

only takes place if the upper layer is more diluted than the lower layer. Only then,

meaning that always when a less dense layer rests upon a heavier one, the hydrostatic

equilibrium is stabilized. Hydrostatic equilibrium must also hold in the liquid mass if the

diffusion current shall be achieved untroubled by hydrodynamic currents8. One can

derive through integration of the just formulated law that directly represents the

elementary process, how the concentrations in a finite mass will change after elapse of a

finite time, if initially they were distributed in a given manner. By comparing the thus

achieved results with the observational facts, one can recognize the accuracy or

inaccuracy of the assumed governing law.

25 At this point, we shall discuss in more detail a case that can be perceived without

application of the higher analysis. We think namely of a cylinder filled with liquid whose

density from bottom to top in elevation

above the ground is proportionally

reduced in such a way that if one looked

at the concentration of a given layer as

ordinate of a curve of which the abscise

would be the height of the same above

the lower end surface, this curve would

make a straight line9. In this geometry, of

8 Text underlined by TWP. 9 Hence, Fick wrote Eq. (30), text underlined by TWP.


course, the given concentration difference of two arbitrarily chosen layers is always

proportional to their distance, and the constant proportion of these two quantities can

namely be expressed through the proportion of the concentration difference between the

lower- and upper-end cross-section to the length of the whole cylinder. Figure 7 further

makes this assertion graphically clear. It means namely, AB is the length of the whole

cylinder and CB is a line, the ordinates of which denote the concentrations of the layers,

so that for example over a distance AP from the lower end cross-section the concentration

PM occurs. At Q, in comparison, the concentration QN is located. By looking at the

figure, one recognizes immediately that PM-QN or Mm has a constant proportion to the

distance of the layers PQ or mN. This distance is arbitrary, and specifically the latter

proportion is the same as the proportion of the difference of the concentration A (AC)

and B (zero is the assumed value of the liquid concentration at the upper end of the

cylinder) to the full length of the cylinder, therefore = ACAB

. The same ratio has to hold

between the infinitely small differences of concentrations in two immediately adjacent

layers at the just as infinitely small distance from each other. One may take the same

proportion at whichever section of the cylinder one chooses. As to how from this

proportion the salt and water exchange between consecutive layers depends on the

assumed governing law. This exchange is just as intense in all parts of the cylinder; that

means that each layer receives as much salt from the previous one during a given small

time interval as it distributes to the following one. Thus, the concentration (in each layer)

remains the same. If therefore the concentration AC of the lower-end cross-section and

the zero concentration of the upper end during a finite time interval could be kept

constant by any means, the concentration in all of the cylinder would remain constant

during the entire time. This way a stationary diffusion current is formed, which moves the

same amount of salt from bottom to top in each time instant through all cross-sections,

which according to our law has to be . ACC QAB

= . In due time, the stationary distribution

of the concentration in the cylinder evolves, it could have been in the beginning or

otherwise, one only needs to keep the concentration of the two end cross-sections

constant. This condition can be accomplished easily in an experiment. One keeps the


lower-end cross-section of the cylinder temporarily filled with any saturated solution in

contact with solid salt, the upper one though in contact with (fresh) water. This way the

lowest layer always maintains total saturation by constantly being able to dissolve salt,

the top one constantly delivers each trace of salt it receives from the bottom through

diffusion immediately to the surrounding water and its concentration remains constant =

zero.

Temperature

during the

processes10

C calculated from the amount of salt delivered

The longest

tube

The medium

tube

The shortest

tube

15.8-14.8 9.67 9.7 9.30

15.5-16 9.57

16-16.5 9.94

17.5-18.5 10.79

18-19 10.71 11.08 10.50

20 11.14 11.02

19-22 11.44 11.33

20-21 11.89 11.12

In fact, experiments took place11 in part to prove the law, and in part to determine

the constant C. So far, though, only experiments with kitchen salt and water are at hand.

The experimental setup is as follows. A cylinder filled with a salt solution was cemented

into a salt reservoir, and the whole unit was sunk into a huge water container. The same

amount of salt, which passed through each cross-section of the cylinder after achievement

of the stationary condition, also had to pass through the end cross-section and pass over

into the water container. The difference in the density of both end cross-sections was the

value of the total saturation as the subtrahend, the concentration on the upper end was

=zero. If the cylinder, the length of which was likewise known, would deliver a salt

amount m during the given time interval, all remaining quantities will be known in the

10 In centigrades.


equation . ACm C QAB

= , so that C can be found from it. The numeric results for the

combination of cooking salt and water are compiled in the following table. We remark

that each time three tubes of different length were set up at the same time and from the

delivered salt in each one C was calculated. The three values of C of course had to be the

same and in fact corresponded so perfectly as is to be expected in similar experiments

anywhere.

The basis of consideration for these units are: the time, one day (24 hours); the length,

1mm; the area, a circle of 10mm radius; and the concentration, complete saturation. One

also recognizes from the above table that the quantity C is defined henceforth as the

amount of salt that moves through the unit cross section during the unit time interval in a

stationary diffusion current. This current owes its existence to a uniform concentration

decrease, which, expanded over the whole length, will reduce the concentration from 1 to

zero under controlled temperature increase.

The constant C is the only true constant for a given solid combination and temperature,

which can be comfortably drawn from experiments in the just described order. It is

quantity independent from all other conditions, which one can also fittingly name “the

rule of the diffusivity of the given solid combination at the given temperature. Graham12,

in his generally well-known work about the subject with which we are dealing here did

not proceed to the analysis of the process. He would derive the measure of the diffusivity

directly from different experiments which took place in the following manner: He filled

two bottles of the same size and shape up to the neck (which has a radius of 1.25 English

inches) with the solutions of the solid to be tested. The neck itself was still filled with

clear water and then the whole receptacle was sunk into a water container; he determined

the amount of the removed solid, which in the course of a certain time was diffused from

the bottle into the water of the container. He was convinced that in this formula he had

discovered the measure of diffusivity. Following these advanced discussions one may by

no means expect that these numbers must be proportional to our C. In the neck of the

11 “Ueber Diffusion,” von Dr. Adolph Fick, Annalen der Physik und Chemie, Herausgegeben zu Berlin von J. C. Pogendorff, Vol. CXIV (Vier und Neunzigster Band), Leipzig, 1855, pages 59-86. 12 Annalen b. Chem. u. Pharm., Vol. 77, p. 56.


bottle, a stationary diffusion current obviously does not develop, instead a process that

would be very difficult to discuss analytically. Nevertheless, these numbers are very

interesting whilst they at least grow simultaneously with the diffusivity, though not

proportionally, so that in Graham’s experiments a solid described by a larger number is

more diffusive than one described by a smaller one. Therefore, I will here now list

Graham’s numerical results, especially that a comparable overall view of different solids

has not been reached yet in a different way.

Solution of 20 parts of

dehydrated salt per

100 parts of water

Specific

gravity

of solution

The amount

diffused

during 8 days

in Grains

Sodium chloride 1.1265 58.5

Magnesium sulfate 1.1265 58.87

Sodium sulfate 1.185 27.42

Aqueous sulfuric acid 1.120 52.1

1.120 51.02

1.108 68.79

1.108 69.86

Cane sugar 1.070 27.74

Melted cane sugar 1.066 26.21

Starch sugar 1.061 26.94

Syrup of cane sugar 1.069 32.55

Arabian gum 1.060 13.24

Albumen 1.053 3.08

The temperature for all experiments was kept between the limits of 15.5 and 17 degrees

Celsius. The diffusion capability of albumen will be increased by the addition of acetic

acid. The adding of egg white does not reduce the diffusivity of the salts, just as it does

little to that of urea, which incidentally stands very close to that of sodium chloride.

27 Graham also conducted very remarkable experiments with combinations of

different salt solutions, which he filled into his bottles, retaining all other arrangements,

and let them diffuse. It was namely found that with the simultaneous presence of the


other salt the diffusivity of the less soluble one became reduced. If bi-sulfuric acid potash

was filled to the bottle, not only this salt diffused out but also still free sulfuric acid, so

that therefore a part of this salt decomposed through the diffusion. Under these

circumstances, alum will be partly decomposed just as well, whilst more sulfuric acid

potash diffuses into the surrounding water than alum is built up by the simultaneously

diffusing sulfuric acid aluminum oxide. If Graham mixed sulfuric acid potash (in a very

diluted solution to not develop immediate concentration gradient), potassium chloride or

sodium chloride with calcium water and let the mixture diffuse against calcium water, the

freed alkali would diffuse and the acid would combine with the calcium, which would

precipitate if it was in sulfuric acid.

28 It also still seems to follow from Graham’s experiments that weak diffusion

currents can cross through each other without any disturbance; because it was found that

the diffusion of a 4-percent solution of sodium carbonate took place in the same manner

whether the surrounding liquid was pure water or a 4-percent solution of sodium chloride.

In addition, with several other different combinations of two salts, the same behavior was

observed.

29 The equilibration of heterogeneous liquids still also takes place, just alike the

gases, if they are separated by a porous partition wall of suitable composition, and with it

still occur very strange phenomena, which are known under the names of “endosmose13”

and “exosmose.” Without a doubt (the membrane) is the last reason for the equilibration

or the driving forces are here the same as in the simpler case of diffusion without a

membrane, only the effect is modified through the physical conditions of the liquid

molecules in the membrane. Therefore, above all, the attention has to be directed to these

if the reason is to discover the diffusion of liquids through membranes. That means one

has mainly to examine the way in which the liquids penetrate into those bodies, which

used as a partition wall can cause diffusion. Actually, the idea that a substance penetrates

13 The general term osmose (now osmosis) was introduced in 1854 by a British chemist, Thomas Graham. Here we shall use the modern term endosmosis, which is synonymous with motion through, passage, transmission; permeation; penetration, interpenetration; infiltration; endosmose, exosmose (obs); endosmosis (Chem) (Roget’s Thesaurus, Project Gutenberg, http://promo.net/pg/).


into a space that is occupied by another substance cannot create any problems for us. We

assume from the atomic picture that each space occupied by an ever so dense substance

still leaves enough empty space to grant room for the atoms of another substance, which

will store themselves among the atoms of the first one. The penetration of a liquid into

certain solids, known by the name of “imbibition,” can also be thought of in a

considerably different way though. One can, 1) imagine the molecules of the imbibed

liquid are among themselves in the uniform molecular interstices of the solid body

distributed in the same manner. Were this opinion correct, then one might understandably

assume very radical differences between the constitution of “imbibable” and not

imbibable bodies; nevertheless, the same chemical substance (i.e., clay) can now have the

one and then the other characteristic depending on the mechanical formation of its small

(not smallest) particles. Admittedly, the imbibition of a piece of porous clay may still be

different in nature from that in a piece of coagulated protein. The difference shows itself

immediately if one changes the expression and says “swelling” instead of “imbibition.”

From a piece of clay, one cannot say that it swells. 2) It also can be thought that a body

capable to imbibe is similar to a network or a spongy tissue that besides the molecular

interstices also has other gaps that in mechanical sense stand far apart from the

interstices, namely much surpassing them in size. These gaps, one could think then, fill

themselves during the swelling with the imbibed liquid; meanwhile, no atom penetrates

into the space between the molecules that constitute the solid part of the network. This

idea offers the immense formal convenience that, if one takes it as a basis of the

explanation of the swelling appearances as well as the one of the diffusion through

separating walls, one becomes independent of each molecular hypothesis.

In fact, most of the researchers in these fields have more or less explicitly adopted the

just discussed second theory. By the way, it is still to consider that these two theories do

not exclude each other. In particular, Ludwig tends to connect them when he states14: “As

much as we find ourselves in the dark about the special kind of fusion (of the solid

substance and the imbibed liquid particles), we may yet still assert that in most of the

swollen substances the absorbed liquids partly are present in larger pores, which are

14 Lehrbuch der Physiologie, Vol. I, p 60.


existing between more or less large piles of molecules, but partly are nestling between the

molecules themselves.”

30 The appearance of the swelling itself is now essentially that a substance capable

of it absorbs a certain quantity from the surrounding liquid in the course of a more or less

long time. When this happens, the process reaches equilibrium, whilst no other liquid

molecule penetrates into it, may by the way, as much or as little liquid be in store.

Through the comparison of the amount of the solid substance and the maximum absorbed

liquid (or in the equilibrium after completion of the process), one obtains the swelling

ratio. It (this ratio) obviously depends on the nature of the solid body as well as the

liquid, so that the same solid body can absorb different amounts of different liquids, and

that different amounts of the same liquid penetrate into different solids. Besides that, the

swelling ratio varies also with temperature. No extended quantitative series of

experiments are available yet on all these aspects.

Particularly important for our purposes is the imbibition of saline solutions. First, it has

been shown that as far as the imbibition in the swelling body, thus the swelling ratio, is

concerned, regularly less liquid is imbibed from the saline solution than from pure water,

and from different solutions of the same salts even less, the more concentrated they are.

After Liebig, for example, 100 weight parts of dry ox bladder take in 310 weight parts of

pure water, 288 parts of a 9% sodium chloride solution, 235 parts of a 13.5%, and 219

parts of 18% solution. After Cloetta,15 the swelling ratio (the imbibed weight of liquid

divided by the weight of the swelling body) of the ox heart bag is 1.35 for 5.4% and 1.01

for 24.3% saline solution, respectively; it is 1.15 for a solution of 3.5% Glauber’s salt16,

and 0.86 for a 11.7% solution of the same salt. –

Much more remarkable though is the difference in the concentration of the solution

imbibed into the swelling body and that of the surrounding liquid from which it

originated. Bruecke, relying on other appearances, with which we will soon become

acquainted, has proposed the hypothesis that a substance that can cause diffusion of a

certain saline solution and consequently is swelling-capable for the same, would exert a

15 Experiments with diffusion through membranes with two salts, 1851. 16 Na2SO4

.10H2O - crystalline hydrated sodium sulfate.


larger attraction towards the molecules of water than towards those of the dissolved salt.

If this assumption were correct, in each of the pores in which the imbibed liquid is

contained, there would be a less concentrated solution against the walls than in the center,

whilst the predominant attraction of the wall molecules against the water simply in the

proximity of the same has to draw together more water. Ludwig, based on this thought,

and also considering that even in the most central part of the pore in any case no larger

concentration could occur than in the surrounding liquid, assumed that the concentration

in all of the imbibed liquid should be lower than the one in the surrounding liquid from

which the imbibed one is taken. It had to be an average result of the lower concentrations

in the wall layers and the higher ones in the center layers, the last of which can at best

equal the concentration of the surrounding liquid. One therefore understands in one word

as the concentration on the imbibed liquid the figure, which expresses the proportion of

the imbibed salt amount to the total imbibed liquid mass, regardless of how water and salt

are distributed in the pores of the solid body. Ludwig saw himself not deceived by his

theory but confirmed it through compelling experiments. If he namely put an ox bladder

in a 7.2% solution of Glauber’s salt, it imbibed a liquid that contained only 4.4% salt. Did

he put the same substance in a 19% kitchen salt solution, it so imbibed a 16.5% solution

in its pores.- Cloetta, who continued this line of experiments, ascertained the ratio of the

densities of liquids imbibed in the bodies capable of swelling (animal membrane) and the

surrounding ones is for kitchen salt constant, in fact =0.84:1, for Glauber’s salt though it

depends on the solution density of the surrounding liquid.

Experiments with kitchen salt:

Percent content

of surrounding

liquid

Percent content

of solution inside

swollen solid

Experiment

time in hrs

Temperature in

deg Reaumur

Density ratio

of inner and

outer liquid

24.002 20.022 76 10 -14 0.83

24.288 20.427 78 12 -15 0.84

6.005 4.679 48 9 -14 0.77

5.540 4.545 76 9 -15 0.82

5.493 4.512 76 10 -15 0.82


Experiments with Glauber’s salt:

11.692 4.623 70 8 -13 0.39

6.500 3.578 70 10 -16 0.55

4.831 2.744 48 10 -13 0.56

4.803 2.755 48 8 -12 0.57

In Cloetta’s experiments with kitchen salt, the density of the surrounding liquid

fluctuated between 24 and 5 percent and always turned out in the same proportion. I will

place here several numeric results of Cloetta’s investigation so that the reader can build

his own opinion about the degree of accuracy. If one squeezes out the imbibed liquid, as

far as this is possible, one receives according to Ludwig’s experiments, not a solution of

average concentration of the solution that was contained in the pores, but one receives

one that corresponds to the density of the original surrounding liquid. Therefore, if the

body swelled by dipping it into a 10 % kitchen salt solution, one also squeezes out a 10%

kitchen salt solution although if one uses a different way to determine the kitchen salt

content in the liquid contained in the pores it results in less than 10%. This at first sight

very surprising appearance has absolutely no illogicality if one only remembers that

through squeezing the central liquid, mass can only be removed from those pores in

which the concentration agrees with the surrounding solution. Whereas the wall layers

with less concentration are held tightly by the force of attraction of the wall molecules, so

that they cannot be removed by the mechanical compressive force.

31 If a solution containing two salts next to each other imbibes, the presence of one

will modify the imbibing capacity of the other one noticeably. So in particular Cloetta

discovered that a piece of ox pericardium has less capability of imbibing Glauber’s salt if

at the same time the solution contains kitchen salt; and the more kitchen salt it contains

the less it will get Glauber’s salt. The following table contains the numerical pieces of

evidence according to Cloetta’s experiments.


Percent content

of the outer liquid

Percent content

of the inner liquid

Duration of

experiment

Temperature

in deg

Reaumur

Specific gravity

of inner liquid

(outer = 1)

Relation of

imbibed NaS to

NaCl

NaCl=100

10.81 7.838 NaCl

2.972 Na S⎧⎨⎩

7.7086.629 NaCl

1.079 Na S⎧⎨⎩

72 10 -13 0.71 38

10.817 7.832 NaCl

2.985 Na S⎧⎨⎩

7.8566.776 NaCl

1.079 Na S⎧⎨⎩

72 10 -15 0.7 38

15.636 10.514 NaCl

5.122 Na S⎧⎨⎩

9.073 NaCl

1.841 Na S⎧⎨⎩

10.904 48 11 -14 0.69 48

10.799 5.813 NaCl

4.968 Na S⎧⎨⎩

7.0304.646 NaCl

2.384 Na S⎧⎨⎩

48 9 -12 0.65 85

9.93 5.312 NaCl

4.618 Na S⎧⎨⎩

6.2754.093 NaCl

2.182 Na S⎧⎨⎩

46 10 -13 0.63 86

10.026 5.326 NaCl

4.700 Na S⎧⎨⎩

6.51 4.295 NaCl

2.215 Na S⎧⎨⎩

48 9 -13 0.64 88

32 Now we will imagine two different liquids, which in direct contact would actually

balance out their differences through diffusion currents, separated from each other

through a partition wall that dipped into each of the two would swell, which means that

its pores could accommodate each of the two liquids. Then we would have all the

conditions under which the phenomenon of the endosmosis appears. It is furthermore

tacitly assumed that the pores are so tight that filtration is avoided through the attraction

between the membrane and the liquid in connection with the cohesion. In addition, even

if both sides of the partition wall are moistened, equilibration of the hydrostatic pressure

if at all is only possible over a disproportionately long time interval, so that finally within

a pore mixing currents through specific gravity differences cannot be produced. When

these conditions are met, the heterogeneous solutions will obviously come in direct

contact with each other within the partition wall pores and equilibrating diffusion currents

will occur immediately. These, of course, cannot come to a stop until each inhomogeneity

of the solutions is balanced and so we then see in fact that even with the inclusion of a


partition wall between the two solutions the process will not end until all differences are

balanced out. During the process, though, fundamental modifications will be caused by

the existence of the partition wall. The important difference between the free diffusion

and the diffusion through partition walls is mainly because in the former the equal

volumes of the same substances are exchanged between two consecutive layers without

any difficulty, so that two equally strong currents in opposite directions are constantly

moving. In the latter, as a rule, both currents are not equally strong so that after some

time the volume on one side of the partition wall has become larger and on the other side

it has become smaller. A theory of the endosmosis has therefore first to be based on

explaining how the modification of the diffusion current can be created by the presence

of the partition wall. Bruecke carried out the first experiment in this direction, which is

already based on the aforementioned idea of the distribution of removed substances in the

pores of a body capable of swelling. By going into that in more detail now, it has to be

remarked beforehand, that here again we can only rely on the case where solutions of the

same salt of different density balance their differences through diffusion. The case is that

different mixtures of originally liquid masses (for example alcohol and water) diffuse in

one another. However, without hesitation, one can go back to the case that a salt in

solution is to be regarded as a liquid itself; its molecules are undoubtfully just as easy to

rearrange as the ones of a certain amount of alcohol distributed in water. It therefore

actually will be a shortcut expression when we talk about salt and water. If now, in fact,

the material of the partition wall has a stronger force of attraction to the water than to the

salt, and as a result of that a wall layer of pure water surrounds a central liquid thread

within the pores, which can adopt the density of the surrounding solution, the latter one

will cause a diffusion current the same way as in the pores which we have imagined

previously (see #25). That will distribute as much salt to one side as water to the other

side, whereas salt will never be able to penetrate into the wall layer, which can be brought

into motion by the attraction of the salt solutions with which it is in touch on both sides.

From the side of the concentrated solution, however, works an obviously stronger

attraction than from the other side, as here in the same area more attracting salt molecules

are facing it. Through the continuing currents, the dilute layer at the wall therefore will be

in the process of moving towards the concentrated solution, by replenishing itself


constantly from the less concentrated one. Thus, to the amount of water, reaching through

the central thread the concentrated solution (which is the same as the amount of salt

traveling to the other side), an additional amount of water is added to the concentrated

solution through the peripheral layer for which no salt equivalent travels to the other side.

More water therefore has to travel each time from the more dilute solution to the denser

one than salt travels in the opposite direction. The volume of the concentrated solution

has to grow through the endosmosis. In fact, the success has been observed regularly. In

one case, Bruecke was able to substantiate this theoretical analysis by an unambiguous

experiment. Turpentine oil has a substantially stronger attraction to glass than tree oil as

it forces out completely the latter from a glass surface; therefore a wall layer of turpentine

oil and a central layer of a mixture of both liquids will be found in a glass capillary if

both can force their way into it. Bruecke manufactured such a capillary surface, which on

one side bordered turpentine oil and on the other side tree oil, and discovered according

to the theory that more turpentine oil diffused through the capillary to the tree oil than

tree oil in the opposite direction.

α α′

β β ′

a′a

b′b

α α′

β β ′

a′a

b′b Fig. 8.

33 The idea that Bruecke had can be displayed a little farther so that several

conclusions of quantitative nature can be

drawn out of this. We think about the pores

as being cylindrical, and the rectangular

cross-section of the separating wall straight

through the axial plane would be (see

Fig. 8). If now the separating wall were

surrounded by a totally saturated salt

solution, according to the above findings, an

also saturated solution would only be found

in the proximity of the axis of the inner pore

(if the pore were extremely tight, probably

not even there). Towards the wall, the concentration would be reduced so that directly at

the wall it would be zero. Of course, one concentration would dominate due to the

symmetry in a concentric cylindrical layer. The cross-section of such in the plane of the

aa bb′ ′


figure should be, e.g., αβ and α β′ ′ . The saturated solution beneath the separating wall

(thought as being horizontal) would now be replaced with pure water, diffusion would

start, and we could observe what must happen in particular in the cylindrical volume with

the cross-section αβ and α β′ ′ . At its lower end, the concentration is kept at the zero

level through constant contact with the water. At its upper end, though, the concentration

maintains the highest possible level through the contact with the saturated solution.

Immediately the stationary concentration distribution occurs in this space. The

concentration will grow in proportion to the height above the lower end area ( ββ ′). This

distribution of the concentration is indicated in the figure by the hatching. Due to the

smallness of this volume, stationary condition occurs almost immediately; thus, we can

restrict ourselves to its consideration. The concentration distribution now necessitates a

stationary diffusion current that moves just as much salt to the bottom as water to the top.

One should even be able to calculate the amounts following the above principles (see 26)

if the dimensions of the small volume and the maximum possible concentration in it were

known. Now, though, the water passing to the top demands a special consideration as

well. If the concentration in the small volume (what yet in general has to be

presupposed), on which we now have our eyes, has to be lower by a finite amount than

the absolute saturation, then a density jump will occur at α and α′ from the possible

maximum in the small volume to the absolute saturation above the separation wall. This

jump would cause an almost infinitely strong diffusion current on the spot, which means

almost infinitely large amounts of salt would be driven into the pores and just as large

amounts of water would be expelled. Let us suppose for a moment that this has actually

happened. The salt amounts could not penetrate because the highest concentration in this

particular area already exists at the upper end; they must somehow slip off sidewise.

Larger amounts of water though can be delivered from the bottom than the diffusion

current, already thought of as established in the cylindrical volume, delivers. A certain

water amount is in a way simultaneously being drawn through. This changes the solution

density, whilst the water distributes itself generally in the upper liquid, up to a certain

distance noticeable from the upper pore opening. In this way, water creates an upward

extending conically widening zone (its two cross-sections are shown in simplification in

Figure 8 above α and α′ ), in which the concentration grows from bottom to top up to the


total saturation of the upper solution. Immediately, also here a stationary condition comes

into effect and causes a diffusion current of the same strength, which brings just as much

water to the top as can be distributed in the same time from the upper end of the imagined

volume into the reservoir of saturated solution without altering the concentration.

Because obviously the imagined zone would extend itself upwards (and through that the

intensity of the diffusion current would be reduced), as soon as more water climbed up in

this way, the concentration at the upper end of the zone could still be altered. On the

other hand, if less water went up, a jump in the concentration increase would occur at

some point, which immediately would cause an infinitely strong diffusion current and

this way again absorb the relevant water amount.

34 It is now apparent that through all those cylindrical elementary layers, which are

so close to the pore wall that totally saturated solution can no longer exist within them,

more water moves to one side than salt moves to other side. As now in a tighter pore

proportionally more parts are lying closer to the wall than in a larger one, considerably

more of water will be flowing in a tighter pore than in a wider one. One can hardly hope

to strongly substantiate this assertion through experiment, as one will hardly find two

separation walls that only distinguish themselves from each other by the width of the

pores. In any case, though, the following fact speaks for the just established assertion.

One has obviously some reason to assume that in the glassy colloidal membrane the pores

are unproportionally tighter than in one produced from animal connective tissue (heart

pericardium or bladder), so that all other differences against this one are only

insignificantly effective. However, it is shown that in fact with a colloidal separation

wall, the quantity of water that diffuses towards the concentrated solution is much larger

than with an animal bladder. Whilst with the bladder wall approximately 3 to 6 times as

much water moves into the solution as salt into the water, approximately 20 times as

much water moves through a colloidal separation wall as salt. In most of my experiments

only traces of salt had moved through the colloid, while substantial water quantities made

it through in the opposite direction.

According to the aforementioned idea, the excess of the water above the salt diffusing in

the opposite direction must still grow with the increasing mobility of the particles in the


saturated solution above the membrane. The amount of absorbed water, which could

spread itself in the reservoir without alteration of the saturation, will be even larger the

easier movable the particles are. To prove this theory, I mixed solid particles with the

upper solution, which interfered with the mobility; received a positive result though,

while the proportion of the diffusing salt and water amounts was not changed.

35 Let us now imagine that instead of the pure water, a solution of the same salt is

located underneath the membrane. Above it there is a completely saturated solution at a

certain density c. Counting from the pore wall a number of cylindrical elementary layers

will immediately inactive; namely all those in which no higher concentration than that c

is possible. Naturally they fill themselves with a uniformly saturated solution, each one

with possibly the most concentrated, and on both their ends the absorbing power will be,

proportionally to the difference, between the concentration in the solution contained in it

and the respective solutions above and under it. As the upper solution is totally saturated,

the difference at the upper end and consequently the absorbency is higher, that is why

now only water moves from bottom to top through all these characterized elementary

layers. The more central layers in which a larger concentration than c is possible are

making way for a bilateral diffusion current which will produce itself in the afore

described manner even though in less intensity. As in doing this, to the contrary of the

first described case, some elementary layers have become passive (dormant) for the salt

movement, the surplus of the continuing water flow has to grow. If c were larger than the

possible maximum of concentration in the central liquid thread of the pore, the

endosmotic current would become unilateral and now only water would be able to move

from the thinner solution to the thicker one. In fact, I found out that 11.05 times as much,

in a different experiment even 17.05 times as much, water as salt went through a

membrane, if the same separated absolutely saturated kitchen salt solution from 22 %

one. Only 5 or 6 times as much water as salt went through if the membrane separated the

saturated kitchen salt solution from pure water.

(Other pieces of evidence in the following table.)


36 Finally we will imagine that there is again pure water below the membrane, above

it though, instead of a saturated solution, there is a solution of the concentration c.

Assume that c is not as large as the highest concentration, which is still possible at the

axis of the pore. Starting from here a series of elementary layers will now show at their

upper ends the concentration c; namely, all up to the one lying so close to a wall that in it

the concentration c is no longer possible. In the indicated concentric elementary layers

now a completely free two-way diffusion current will develop which will carry the same

amounts of salt and water to opposite sides. At the upper end the density jump no longer

develops and therefore no reason exists to absorb more of the water. Parts of the pores,

which in the first case delivered more water, do not deliver it in this one. One can

therefore expect that if an unsaturated solution is present above the membrane, the water

surplus will be smaller and even much smaller the more diluted the solution is. Ludwig’s

experiments on the diffusion of kitchen salt confirm this forecast. Things stand different

though with his experiments performed with Glauber’s salt. In the following table, the

numerical results of his experiments have been put together.

Experiments with kitchen salt:

Membrane

name

Temperature

deg Réaumur

Ratio of

transported

salt to

transported

water

Upper, % Lower, %

XIX 6 Sat. 14.329 1 : 4.2

XIX 8,25 Sat. 0.0 1 : 3.2

XIX 8 4.920 0.0 1 : 1.4

XX 6 Sat. 14.329 1 : 4.5

XX 8,25 Sat. 0.0 1 : 4.3

XX 8 4.920 0.0 1 : 1.4

XXI 8 Sat. 14.34 1 : 4.5

XXI 8 Sat. 0.0 1 : 3.5

Concentration of originally surrounding

liquids


XXI 8,25 4.920 0.0 1 : 1.6

X 9 Sat. 22.6 1 : 7.0

X 9,75 2.006 0.0 1 : 3.1

XVI 8 Sat. 5.0 1 : 3.6

XVI 8,75 2.006 0.0 1 : 1.1

Experiments with Glauber’s salt:

XI 6,5 Sat. 0.0 1 : 5.9

XI 8 5.358 0.0 1 : 8.2

XI 8,5 1.028 0.0 1 : 24.3

XII 6,5 Sat. 0.0 1 : 5.5

XII 8 5.359 0.0 1 : 8.0

XII 8,5 1.028 0.0 1 : 23.5

XVI 6 Sat. 0.0 1 : 5.8

XVI 7 5.084 0.0 1 : 10.5

XVI 8 1.028 0.0 1 : 21.6

37 Jolln believed to be able to prove that the proportion between the salt and water

amounts moved back and forth for one particular salt, one particular membrane, and one

particular temperature should be a certain constant, might the concentration of the

solution above and below the membrane be what it may. He called this proportion the

“endosmotic equivalent.”

Jolln has let the density of the solution vary within too narrow limits, therefore he could

easily fail to notice the variations of the endosmotic equivalent observed by Ludwig.

From the start, by the way, constancy of the endosmotic equivalent was not actually

probable. Apart from the particular picture we have drawn ourselves on the basis of

Bruecke’s hypothesis that we explained previously, it is yet this much without a doubt

that in endosmosis one does not have to deal with a fundamental phenomenon in which

the actual constants, which rely only on the innermost unchanging nature of the

participating substances, come out that uncomplicated. Endosmosis is in any case a


course of events fundamentally modified much more through complicated conditions,

through the fundamental strength of the matter, though in last instance mechanically

caused like all others.

I have noticed that for one particular membrane the equivalent came out differently if I

brought saturated kitchen salt solution above and pure water underneath it than if I

poured saturated kitchen salt solution underneath and pure water over the membrane. In

the last case, the endosmosis equivalent was smaller and more salt went through during

the time unit than in the first case. Whether the density plays a role in this case or

whether the different sides of the membrane which were facing the water were the actual

cause, I do not dare to decide after my not adequately numerous experiments. It is known

that already Cima and Matteucci have observed phenomena that favor the last

interpretation.

38 Very important (especially for the organic process) are the diffusions that occur

when solution mixtures take the place of simple solutions. Theoretically, one can hardly

suspect anything about this object. Contrary to that, though, we have an exact and

extensive experimental investigation from Cloetta on the diffusion of mixtures of kitchen

salt and Glauber’s salt solution. This investigation has resulted in the following laws:

First, the diffusion equivalent for each of these two salts is, if it diffuses as the mixture,

just as large as if it separately diffuses through the same membrane under the same

circumstances (particularly therefore in the same solution density). The deviations from

this law in the different experiments are insignificant. If for example n and were the

endosmosic equivalents of the two salts for a certain membrane and concentration, and if

it were found in an experiment with the mixture that during any given time the amount a

of the first salt, and the amount b of the second salt moved across. Then, for the first one

a water amount =na, and for the second one a water amount =nb must have moved

across. Therefore the equivalent of the mixture =

n′

na n ba b

′++

should result from it. Thus this

quotient should express the proportion of the entire water amount moved in this

experiment counter currently to the simultaneously moved entire salt amount.


Furthermore, it produced also certain relations between the diffusion velocity of the

different salt in itself and in a mixture. If one talks about diffusion velocity, for the

moment one must distinguish between the salt and the water current. The velocity of the

first would be measurable in each single case by the salt amount that moved through the

surface unit of the membrane during a given time. The velocity of the water current

would evidently find its measure in the related water amount moved to the opposite side.

During experiments with each of the two salts separately, it has become obvious that the

salt current for kitchen salt “ceteris paribus” has approximately twice the velocity as the

one for Glauber’s salt. The velocity of the water current is for both salts approximately

the same. It now became clear that within the mixture the velocity of the salt current

remained unchanged while that of Glauber’s salt was reduced. The velocity of the total

salt current was therefore larger than the one of a mere kitchen salt current, nevertheless

it not quite reached the value of the sum of the velocities of a kitchen salt current and a

Glauber’s salt current. The water current occurs for both salts in the mixture also with the

same velocity; understandably, though, the sum of each of the two speeds in itself cannot

combine to the velocity of the total water current because the diffusion velocity of the

Glauber’s salt is reduced in the mixture. When in this manner the diffusions of the two

mixed salts are independent of each other, so of course the salt endowed with a higher

velocity (or larger diffusion current against water) will remove itself from the container,

leaving behind the other one. From these conclusions can be founded about occurrences

in the animal body, about which one should read the quoted treatise by Cloetta.

39 In the chronological evolution of the diffusion phenomena, only very few

sentences can be expressed with certainty. Furthermore it is plausible that the

equilibration of chemical differences of two liquid masses must take place in a shorter

time frame when these masses are contacting each other directly than if they were

separated by a porous separation wall. The wall must due to the necessary tightness of the

pores, offer delayed resistance to the currents, which cause the equilibration. It can also

be asserted with the same evidence that everything else being equal, the time required for

equilibration increases with the thickness of the separation wall (length of the pores). –

Furthermore, the equilibration time depends on the size of the endosmosic equivalent. In


fact, the equilibration happens even faster, the smaller the value of the endosmosic

equivalent is (same velocity of the water current presupposed), in other words the more

its value approaches unity. If, for example, the equivalent is very large then this means

that the water current dominates the opposite salt current by far and will therefore quickly

dilute the thicker solution. But the reversed current will hardly thicken the thinner

solution; while this would be the case in a more similar measure if the equivalent would

not differ so much from unity, i.e., if the strength of both currents would not differ so

much. – Furthermore, as illuminated from our consideration about the nature of the

phenomena (but, apart from this, also would be concluded from every other theoretical

idea) and already determined through older experiments (Vierordt17): The liquid amounts

exchanged in the same time frame are larger, everything else set equal, the larger is the

chemical difference of the masses separated by the membrane. With application of water

on one and salt solution on the other side, the transported mass will grow with the salt

amount in the solution. Different salts contained in a solution mixture equilibrate

themselves with pure water in almost the same amount of time as if they were taken

separately with water. The equilibration period of the already less diffusive salt seems to

become larger by mixing it with the other one.

17 In the article “Endosmosis” in the Dictionary of Physiology (Handwörtenbuch der Physiologie).


Appendix B - Different units for salt concentrations

There are several ways of expressing salt concentrations (Achmatowicz 1954):

1. Number of grams of salt per 100 g of solution, c1

2. Number of grams of salt per 100 g of water, c2

3. Number of grams of salt per 1 liter of solution, mass concentration, c3

4. Number of moles of salt per 1000 moles of water, c4

5. Number of moles of salt per 1 liter of solution, molar concentration, c5

6. Number of moles of salt per 1 kg of water, molality, c6

7. Mole fraction of salt in water, c7

Let γ denote the specific gravity of solution, and M the molecular weight of salt:

3 2 31

2

1 32

1 3

23 1 5

2

24

3 15

26

47

4

10010 100 10

100 100100 1000 10

100010100

180.2

10

10

1000

c c c Mcc

c ccc c

cc c c Mc

ccM

c ccM M

ccM

ccc

γ γ

γγγ

γ

= = =+

= = =− −

= = =+

=

= =

=

=+

6c M

(33)


Table 3 – Density and concentrations of aqueous NaCl solutions at 20 deg C

1c % 204 g/lρ δ 3c 5c

1 1005.3 0.22 10.053 0.1720

2 1012.2 0.24 20.250 0.3464

4 1026.3 0.28 41.072 0.7026

6 1041.3 0.31 62.478 1.0688

8 1055.9 0.34 84.472 1.4451

10 1070.7 0.37 107.070 1.8317

12 1085.7 0.39 130.284 2.2288

14 1100.9 0.42 154.128 2.6367

16 1116.2 0.44 178.592 3.0553

18 1131.9 0.47 203.742 3.4855

20 1147.8 0.49 229.560 3.9272

22 1164.0 0.51 256.080 4.3808

24 1180.4 0.53 283.296 4.8465

26 1197.2 0.55 311.272 5.3251

The concentration units are explained above. The correction δ is as follows: if the

solution density is determined at deg C, then one adds 1T 2 1(T T )δ − to this density to

obtain the corresponding solution density at . The numeric value of specific gravity of

a solution relative to the specific gravity of water at 4 degrees C is identical to the

numeric value of the solution density in g/cc. Data source is (Achmatowicz 1954).

2T

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