Diffusion, Thermal Conductivity, and Viscous Flow of Liquids R. E. POWELL, W. E. ROSEVEARE, HENRY EYRING
A general theory of viscosity and diffusion is developed in terms of the statistical mechanical theory of reaction rate. Quantitative expressions are obtained which express these processes in terms of measurable thermodynamic properties of liquids. Variations in the theory when molecules of different sizes are interdiffusing are treated. Equations for thermal conductivity and ionic conductance are likewise developed.
AND
Princeton University, Princeton, N. J.
A
POINT of view is presented with respect to rate processes in liquids. I n the now widely applied statisticalmechanical approach to thermodynamic properties, one starts with a known or assumed molecular model, and from that is able to predict the properties of matter in bulk. Similarly in the study of dynamic processes in liquids we may start from a model of how single molecules behave, and from that we are able to predict the relations of such processes to one another and to the thermodynamic properties of the liquids, and to predict the functional dependence of such processes on different variables.
Absolute Reaction Rates The method of absolute reaction rates, which has been applied with success to many examples of chemical kinetics ( 4 ) , provides a simple approach to a study of viscosity, diffusion, and other rate processes in pure liquids. The underlying equations are developed as follows: When a system is undergoing any chemical reaction or other rate process, the reactants in the initial state first approach each other to form a quasi molecule or “activated complex”, which further decomposes to yield the final products. The situation is represented schematically in Figure 1, in which the initial system has to gain enough activation energy to pass over the energy barrier before it can reach the finalstate; The rate of reaction will be given by the concentration of activated complexes multiplied into the rate a t which they are crossing the top of the barrier: rate = coU/G where U is the mean velocity along the reaction coordinate. The mean velocity is given by /-m
dynamic notation, AFi. The factor ( 2 ~ m k T ) ’ / ~ 6which /h has been taken out of the equilibrium constant is the partition function for the one degree of translational freedom of the activated complex along the reaction coordinate. The rate of reaction is then given by i%T rate = co - f E - ~ t =
kT co-e h
(4)
-AF~/RT
,etc.
and the specific reaction rate, k‘, is given by the same expressions without the factor CO. This simple result may be stated in these words: The concentration of activated cornplexes is determined by an equilibrium constant, K t ; the activated complex decomposes for all reactions a t the same rate which is the universal frequency, IcT/h, or numerically -1O’a per second at room temperature.
Viscosity An explicit formula for viscosity of liquids is now readily derived (6). The model for the viscous flow of a liquid is illustrated in Figure 2. A shearing force, f, is applied across the two layers of molecules, and flow takes place when a single molecule squeezes past its neighbors and drops into a vacant equilibrium position (a hole) a t a distance, A, from its original position. We shall discuss later the evidence that a single molecule is the flowing unit and return also to the question of whether the activation necessary to surmount the barrier or the presence of holes is the major requirement for flOW.
Fluidity, or reciprocal viscosity, is defined as the difference in velocity per unit shear:
To find the concentration of activated complexes, we need know only the equilibrium constant of activation connecting it with the initial state:
5
=
K , = K f (Z?rrnkT)‘/%/h
CO
Fa
= - =
Fa
FFo ,‘ ( 2 ~ r n k T ) ~ / ~ G / h
= ,-Apt /RT(2prnhT)1/2G/h =
,-AH:
(3)
(5)
The difference in velocity is given by the distance a molecule moves in one jump multiplied into the net rate of jumping, or AU
:RT+Ast/R(2?rrnkT)l/2g/h
A(k,’
-
kb’)
(6)
If the free energy of activation for a molecule jumping under no external force is A F i , the forward process is helped by the work due to a force f acting on an area X& through a distance
where the last five expressions give alternative ways of expressing the equilibrium constant, either in terms of the partition functions of statistical mechanics, F , or in thermo430
..
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1941
A/2, and the reverse process is hindered by the same amount, so that kT - A F * / R T ifX2X3X/2kT k,) = e (e 1
(744)
h
Upon combining Equations 5, 6, and 7, and expanding the second exponentials, the viscosity equation is obtained in its final form:
6
=
17 = 'M a e-ARSIRT Xih
431
cosity at their melting points. This is in fact approximately true, the viscosity being about 0.02 poise for many liquids. Figure 4 illustrates this for the series of normal paraffins (7). For the higher members, factors other than holes are coming into play. It has been pointed out (9)that the electrical conductivities of many molten salts, extrapolated to their melting points, are approximately equal; since our model formulates viscosity, diffusion, and electrical conductivity as exactly the same process, this provides another confirmation for the theory.
(8)
To a close approximation, the factor involving A's is simply the volume of a molecule, and Equation 8 takes the more usable form, + =
=
- ve - ~ ~ . f IRT : Nh
@A)
Vh , A B e - A~ fit B /HT N
(9B)
FIGURE 2. MODEL ILLUSTRATING
FIGURE1. ENERGY PROFILEOF REACTING SYSTEM
A
CORRELATION WITH THERMODYNAMIC PROPERTIES. ACcording to our model of the flow process, some of the same bonds are broken in flow as in vaporization; however, the flowing molecule does not gain the entropy of the expansion into the vapor, neither does it have charged against it the work RT per mole of expansion against the atmosphere. The thermodynamic property with which the free energy of activation is to be correlated is then
- RT
AFvsp. $. TL?LSYBp.
AEvctp.
and Figure 3 shows values of AFt plotted against AEw. for a large number of liauids (12). The sauares a t the left are the liquefied gases nitrogen, oxygen, argon; the circles in the intermediate portion are various typical organic liquids, and the triangles are water and various alcohols; the triangle a t the far right is glycerol. The experimental points fall well along a straight line, and the result may be used as an empirical formula to predict viscosities from heats of vaporization:
-
,
I
FLUIDFLOW
Equation 9B, which shows EFFECT OF TEMPERATURE. that the viscosity should vary exponentially with the reciprocal of temperature, is in excellent accord with the experimental results. It has been found (8)that the heat of activation, A H t , is from one fourth to one third the heat of vaporization for typical organic liquids but is more for associated liquids. The range of validity of this relation is indicated by Figure 5. For surface films the heat of activation is found to be much greater than for the same substances in bulk, the increase being ascribed to lateral association of molecules in the film (11). The heat of activation for liquid metals is found to be one third-the heat of vaporization multiplied by the ratio (volume of ion/volume of atom), and this has been interpreted to mean that the metallic ion is the unit of flow (3). We now return to the question of whether the activation energy of surmounting a barrier or the presence of holes is the determining factor. Batchinski's relation, 9 = k ( V - VO), and the work of Bingham would suggest that holes are the all-important factor. If this were so, the heat of activation at constant volume should be zero. Experimental data on this point are plotted in Figure 6 ( l a ) . Three different cases are to be noted: (a) For typical organic liquids the heat of activation a t constant volume is small (of the order of 0.5 kg.-cal.), so that holes control the flow in this case. (b) For associated liquids the heat of activation a t constant volume is almost all of the total heat of activation, so that in this
a
Fully as important from a point of view of mechanism, this result confirms the close relation between flow and vaporization. Now both vaporization and melting have been treated by a theory of structure in which liquids are considered as binary mixtures of molecules and holes. According to this theory, all simple liquids should have the same vis-
in FIGURB 3. RELATION BETWEEN AFS
AND
ABvap.
INDUSTRIAL AND ENGINEERING CHEMISTRY
432
case the breaking of a hydrogen bond is the important factor. (c) In acetone, as in water, the heat of activation a t constant volume is negative, and this may be interpreted as a change of structure with increasing pressure. I n general, then, holes in liquids are found to be the flow-determining fartor, and the exceptional cases in which this is not so may readily be predicted from a conbideration of molecular structure.
Vol. 33, No. 4
pressure, the slopes of which increase with increasing molecular weight ( 2 ) . From Equation 11 the equation of the plots should be &=1+-log 7 0
PAVt
AF;
+ RTlog ( N h / V )
(12)
so the line should be linear with P , and its slope give a measure of A V * . Since the activation volume should increase with tlie increasing volume of the unit of flow, a line of steeper slope is to be predicted for larger molecules. EFFECTOF COMPOSITIO~T;. I n mixtures of siniple liquids, where the flow-determining factor is the presence of holes, the readiness of a particular molecule to flow is determined not so much by its own properties, as by the readiness with which the solvent contributes holes for i t to flow into-in other words, the properties of the whole solution. I n explicit form,
FIGURE4. VISCOSITY OF NORMAL PARAFFISS AT THEIR MELTING POINTS
PRESSURE EFFECT.At high pressures the viscosity equation may be written AE of By considering the pressure effect, the activation volume is found to be about one sixth the molecular volume for simple liquids, or about one twentieth the molecular volume for liquid metals. These numerical values represent the size of a hole necessary for flow. If the expansion of substances on melting is considered to be due to the formation of holes, the holes are again found to be about one sixth the size of a nonmetallic molecule, or one twentieth the size of a metallic molecule. Thus a study of melting provides an independent experimental check of the hole theory of viscous flow. A careful study was recently made by Dow of tlie pressure effect on the viscosity of Russian and Rumanian oils, and the values of log vp/log 7 0 give excellent linear plots against
FIGURE 6. EFFECT OF PRESSURE Opeu ciioles and triangles = A H $ a t constant pressuie, f i o n i Figure 5 ; filled circles a n d triangles = A H : a t constant volume.
Equation 13 reduces, for not too imperfect solutions, to log 7 = N1 log q 1
S1
1
'5'
AE
"
'
IO' '
of Vaporization
'
FIGURE 5 , RELATION BETWEEN AHX AND AEvSp.
+
N 2
log
(14)
VI
which ie the best empirical mixture law previously proposed. The third term in the exponential is a small correction for the excess free energy of mixing, obtainable from vapor-pressure data. The application of this mixture law is illustrated in Figure 7. " VISCOSITYOF LONGMOLECULES.In long molecules i t is probable that flow takes place by segments rather than by the molecule as a whole ( 7 ) . Definite evidence for this is that observed heats of activation are only about 10 kg.-cal., and that heats of activation are independent of chain length for homologous I polymers ( 6 ) . If the heat of activation for viscous flow for the normal paraffins is plotted against chain length (Figure 8), it approaches a limiting value. If AEv,,./4 is assumed to give the proper heat of activation, the flow unit for hydrocarbons is 20 to 25 carbon atoms in length. Similar calculations for some other molecules give the following values:
1
d,-
Vaporization
'
Substance Hydrocarbons Polyesters Plastic sulfur Rubber
Unit of Flow, Atoms 20-26 28-34
20 40
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
April, 1941
Diffusion The model for diffusion is precisely the same as that for viscous flow-a single molecule jumping from one lattice position to another through the liquid (6). Thus if we have a theory for the one process, the theory for the other follows automatically.
I50
Lt
10
N
0100
x
c 0 G
50
0
centration, is the driving force for diffusion. If Equation 18 is derived, taking activity into account, the result (13) is =
hlkT d log a1 xIx31 [ d log ivJ
where the factor in brackets is to be calculated from vap& pressure data. Equation 19 was computed on the basis of a single molecule diffusing; if two or more molecules were involved, the activity correction would enter to a higher power. Thus confirmation of Equation 19 will simultaneously verify our model in which molecules move as individuals rather than as clusters. As an experimental test of the relation, the product DT was plotted against molal composition, with and without correcting for activity, in the two upper graphs of Figure 9. The correction brings the experimental points into a straight line. The product Dq may not be estimated for any composition simply by reversing the procedure. It is necessary only to have the vapor pressure data and the values of Dq for two points. The lower graph of Figure 9 gives an example of this treatment. DIFFUSION COEFFICIENT FOR LARGEAND SMALLMOLECULES. The temperature coefficient for the diffusion of large molecules into a solvent composed of small ones is the same as that for the diffusion of the solvent molecules themselves. This may be seen by inspection of the familiar Stokes-Einstein equation for diffusion:
where 7 ienzene
= viscosity of solvent r = radius of large molecule
Phenc
As we have seen,
FIGURE7. FLUIDITY OF A MIXTURE OF BENZENE AND PHENOL 1.
433
+ considered additive
~
considered additive 3. Equation 13 4. Equation 14 Circles are experimental points. 2.
h e AS^ = N -
v
IR
e
+AH$ IRT
where A H t = heat of activation for viscous flow or for selfdiffusion of solvent molecules
The diffusion coefficient, D , is defined by the relation dn rate of diffusion = D nl -2
6
dx
The forward, backward, and net rates are given by the absolute rate method as
4
(16A) 2-
So that D becomes
0Number of Carbon Atoms
or combining with the viscosity Equation 8,
The close relation between the processes of viscous flow and diffusion, explicitly expressed by Equation 18, is well confirmed by the correspondence of the heats of activation for the two processes (14). EFFECTOF COMPOSITION. If a solute is partitioned between two immiscible solvents, there is obviously no net diffusion across the interface; hence activity, and not con-
FIGURE 8. AHS FOR VISCOUS FLOWAS A FUNCTION OF CHAINLENGTH
Thus the rate-determining mechanism for large molecules diffusing through a liquid composed of small molecules is the diffusion of the smaller molecules around the oncoming large ones by the same mechanism which these small molecules use in diffusing around other small molecules. I n Figure 10 a small molecule is indicated by A and the large molecule it must diffuse around, by B. B’ is dotted in for the alternative case of a molecule of the same kind as A . Now
INDUSTRIAL AND ENGINEERING CHEMISTRY
434
1.5
1.0
0.5 1.0
IChloroform
Ether
‘ I
’dater
’ n-Propyl olcohc
FIGURE 9. ACTIVITY EFFECT os DIFFUSION
B’ or B will each be advanced the same distance when A in one case flows around B’ and in the other around B. But 1.0 ! Aceionel the ratio of the Chloroform distances which A must travel in the two cases is X/aZr, where the undetermined number, a, will vary v i t h the path followed by A in passing around B or B’. We thus expect for the ratio of the diffusion coefficients: D h _ e _ _ D‘ a(2r) Substituting for the coefficient of self-diffusion as given by Equation 18, we obtain for the diffusion of molecules in general:
Vol. 33, No. 4
are plotted in Figure 11. A particle as large as gold sol obeys the Stokes-Einstein equation closely; but many small molecules show deviations in the direction of “flowing too fast”-i. e., toward a = 1. Since the cavity permitting diffusion around a large molecule may be formed in an environment quite different from that in the case of self-diffusion, the heat of activation AH may be correspondingly changed. This is especially true if the large molecule is an ion. Thus a part of the divergence may be due to the use of an uncorrected viscosity, and only a more detailed analysis into such factors as temperature coefficients would justify throwing the whole responsibility onto a as we have done in Figure 11. However, this short discussion should help to clarifv the problem of treating- the diffusion of large particles.
IONIC CONDUCTANCE. The conductance of elec-
U
%ee
h
FIGURE 10. MODELILLUSTRATING THE DIFFUSION OF LARGEMOLECULES
--LIP: ~ R T (23)
which becomes identical with the familiar equation derived and tested by Nernst : U = Dze/lcT
(24)
The conclusions of the preceding sections, then, may be applied a t once to problems of ionic conductance. For very large spherical molecules diffusing into small molecules, one can justify the hydrodynamic assumptions used in deriving the Stokes-E i n s t e i n equation, so that a comparison for this case gives a: = 3ir. The hydrodynamic arguments applied to large cylindrical molecules give a = 2 7 ~ ;and the value for self -diff usion, as we have seen, gives a = 1. The actual values found for a series of molecules (16)
M
UPPER LIMIT
F
FOR a 1
1
GENERAL EQUATION : kT D = a (dhameter) 7
0K+ IN WATER 0 ‘OLD
I
t: t4
0 cs+ 0 CI0 Br-
0 TETRABROMOETHANE
IN TETRACHLOROETHANE
0 BENZOTRIBROMIDE
IN ETHYL BENZOATE
ETHYL BENZOATE IN BENZOTRIBROMIDE HYDROGEN GAS IN WATER HEAVY WATER IN LIGHT WATER
0OH-
ETHYL ACETATE IN ETHYL BENZOATE NITROMETHANE IN ETHYL BENZOATE IN ETHYL BENZOATE
0 ACETONE
LOWER LIMIT F O R A
FIGURE
OF MOLECULAR SIZE 11. EFFECT
ON
DIFFUSION
April, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
Thermal Conductivity When the velocity of sound in liquids is compared to the kinetic theory velocity of the molecules, it is found to be greater by a factor of 5 t o 10. Figure 12 illustrates the mechanism of this process. The signal is transmitted from
K (liquid)
=
0.931
435
(27)
This formula is of the same form as that successfully employed by Bridgman ( I ) . A comparison of experimental and predicted thermal conductivities of liquids at 30” C. is given in Figure 13 (1, 8).
Nomenclature a a1
q,c, = concentration of reactant, of activated complex c,, cv = specific heat at constant ressure, constant volume C = average kinetic theory veyocity D = diffusion coefficient = base of natural logarithms e ABvap. = energy change of vaporization f = shearing force 5 = Faraday constant Fo,Fa,F,‘ = partition function for normal state, for activated
FIGURE12. MODELFOR THE CONDUCTION OF SOUND OR HEAT
This same model may be applied to the thermal conductivity The thermal conductivityof a gas is given by
I0
,
/’
=d o -
/.
340-
c
/’
/’ /
530-
/. 7206
810-
0’
7
4%&%
@ ,, I
Calculated
1
I
I
1
Conductivity
FIGURE 13. OBSERVED AND CALCULATED THERMAL CONDUCTIVITIES AT 30” C. 1. Methyl alaohol 2. E t h y l alcohol 3. Propyl alcohol 4 . Butyl alcohol 5. Isoamyl alcohol 6 . Ether
7.
8.
9. 10. 11.
state, and for activated state omitting degree of freedom corresponding to reaction AF,,,,, AF*, APE free energy change of va orization, of activation, and of mixing (in excess of ixeal) h = Planck constant A H t = heat change of activation IC = Boltzmann constant, R / N IC’ = specific resction rate kj’,kb’ = specific reaction rate in forward direction, in backward direction K, = equilibrium constant of activation = equilibrium constant of activation, omitting degree of Kt freedom corresponding to reaction K (gas, liquid) = thermal conductivity of gas, liquid L = mean free path m = mass N = Avogadro number nt, N1 = concentration of substance 1, mole fraction of substance 1 P = pressure R = gas constant r = radius of particle AS,., , AS: = entropy change of vaporization, of activation T = absolute temperature U = ionic mobility u,E, Au = velocity, average velocity, difference in velocity u (gas, liquid) = velocity of sound in gas, liquid v, V ,2rt = molecular volume, molal volume, molecular free volume AV. = volume change of activation x = distance = number of charges on an ion z = distance across top of barrier 6 E = specific electronic charge ?I, Q, = viscosity, viscosity as function of pressure & = fluidity A = ionic conductance = distance between equilibrium positions h hl, hz, h3 = intermolecular distances
-
molecule A to molecule B with the velocity of sound in the gas, then across molecule B almost instantaneously. In terms of the intermolecular distances,
60-
= a size parameter = activity of substance 1
Acetone Carbon disulfide Ethyl bromide Ethyl iodide Water
Literature Cited (1) Bridgman, P. W.,Proc. A m . Acud., 59, 109 (1923). (2) Dow, R. B., McCartney, J. S., and Fink, C. E., meeting of Soc. of Rheology, Oct. 18,1940. (3) Ewell, R., and Eyring, H., J . Chem. Phys., 5, 726 (1937). (4) Eyring, H., Ib;d., 3, 107 (1935),and a series of subsequent papers. (5) Ibid., 4, 283 (1936). (6) Flory, P.J., J . A m . Chem. SOC.,62, 1057 (1940). (7) Kauzmann, W., and Eyring, H., Ibid., 62, 3113 (1940). (8) Kincaid, J. F., and Eyring, H., J . Chem. Phys., 6, 620 (1938). (9) Lillie, H.R., meeting of Soc. of Rheology, Oct. 18, 1940. (10) Loeb, L. B., “Kinetic Theory of Gases”, New York, McGrawHill Book Co., 1927. (11) Moore, W., and Eyring, H., J . Chem. Phys., 6, 391 (1938). (12) Roseveare, W. E.,Powell, R. E., and Eyring, H., J . Applied Physics, 12, to be published, 1941. (13) Stearn, A. E.,and Eyring, H., J . Phys. Chem., 44, 955, 981
where N / V is the number of molecules per cc., c, is the specific heat per molecule, and the quantity in brackets is a correction factor given by Eucken (IO). In the liquid, N / V becomes the number of molecules per cc. of liquid, the average velocity C must be multiplied by the ratio ( ~ / z y ) l / ~ 11 0413) ,-”--,. for the reasons discussed above, the mean free path is replaced (14) Taylor, H. S.,J. Chem. Ph2/8., 6, 331 (1938). by ( V / N ) I P , and cv may be assigned the value of 3k per (15) Ulich, H., in Eucken-Wolf’s Hand- und Jahrbuch der chemisch molecule. The resulting formula reduces to: Physik, Vol. 6, Pt. 2,pp. 156-77 (1933).
