DIELECTRICS .ISD RHEOLOGY O F DISPCRSIOKS
1063
results frcm a structure in which vehicle constituents are immobilized between particles having minima of potential energy a t n distance.
I wish to express my appreciation to Louis R. Suriani for his able assistance in carrying out the expe1,irnents and to the ,T. 32. H u k r Corporation for their \\illingneas to rclcase the parts of the investigation v.hich are of more general intrrvst. REFERESCES (1) BRUGGEMAN, D. A . G . : .4nn. Physik 24, 636 (1935). E.: M ,Proc. Am. SOC.Testinghlaterials 21, (II), 1154 (1921). (2) B ~ C K I N G H A (3) D E B Y EP, . : Polare Nolekeln, 5. Hirzel, Leipsig (1929), for a general discussion of the physical basis of the Lorentz-Lorenz and the Clausius-Mosotti relations. (4) FRECNDLICH, H. : “Some Mechanical Properties of Sols and Gels and their Relation to Protoplasmic Structure”, in T h e Structure of Protoplasm, a monograph of the American Society of Plant Physiologists, Iowa State College Press, Ames, Iowa (1942). R. N . : in Alexander’s Colloid Chemistry, Vol. VI, p , 328. (5) GREEN,H., A N D WELTMAKN, Reinhold Publishing Corporation, New York (1946). R.: Ann. phys. 16, 205 (1941). (6) GUILLIEN, (7) LICHTENECKER, K . : Physik. %. 27, 115 (1926). A.: S a t u r e 166, 238 (1945). (8) PARTS, @) PRYCE-JOKES, J . : J. Oil Colour Chem. Assoc. 19, 295 (1936). (10) RAYLEIGH, J. W.: Phil. Mag. 1892, 481. (11) ROEDER,H. L . : Rheology of Suspensions. Paris, Amsterdam (1939). F . , A N D FRAXCESON, A , : Ilolloid-Z. 92, 158; 95, 234 (1940). (12) WACHHOLZ, R . S.: Ind. Eng. Chem., Anal. Ed. 16, 424 (1943). (13) WELTMANS, (14) WIENER,0.: -4bhandl. math.-phys. Klasse sachs. -4kad. Wiss. (Leipzig) 32, 509 (1912).
OK A GENERALIZATIOS IiY THE DIFFUSION THEORY OLE LAMM
Institute of Physical Chemzstry, University of Upsala, Upsala, Sweden, and Institute of Theoretical Chemistry, Royal Institute of Technology, Stockholm, Sweden Received February 14, 1947 I. SYMBOLS
a = n = N = y = f = M = t=
activity, concentration in mols per cubic centimeter, mol fraction, activity coefficient on the basis of concentration, activity coefficient on the basis of mol fraction, molecular weight, partial molar volume,]
LThe symbol V was used for this magnitude in earlier papers, except reference 10.
1064
OLE LAMM 812
molar volume of a binary solution (reference equations 4 and i ) ,
= mean
C = linear velocity, z = diffusion direction, t =
diffusion time,
B I QA12 , , QIZ= thermodynamic factors (equations 1, 17, 47), p = frictional coefficient per amount of substance contained in @ = D =
1 ec., frictional coefficient per mol, and diffusion coefficient.
11. GEiYERAL REFEREXCE EQUATIONS FOR TWO-COMPONENT
+ + + +
(la) Nld In a1 N2d In a2 = 0 (tb) nld In a1 nzd In a2 = 0 (IC) Nidiri Nzd8i = 0 (Id) nld81 n2dV2 = 0 (2) 12181 12282 = 1 (3) 121 = Nl/8,? (4) 812 = NIDI NzOi ( 5 ) Bidni 8zd7tz = 0 ( G ) dN1 = - dN, 1 5, (7) 812 = - = 121 nz 1 (52 -
+
+
+
+
(8) dN1
SYSTEMS
=
+
Zll)nl
$dn1 02
+
na Zll
(10) N z = -___ 1 (81 - 5z)nz
111. A GENERALIZ 4TION O F THE SUTHERL4XD-ICINSTEIiY
DIFFUSION EQUATION
The theory of diffusion has been dealt with in some detail in earlier publications (5-101, in which references also to the work of others are to be found. In these treatments of the subject, the diffusion coefficient is referred to coefficients of friction and to thermodynamic factors, the latter for a two-component system being the n-ell-known expression :
The identity follows from the Gibbs-Duhem equation (reference equation la). is a symmetrical property with respect to the two components, thanks to the
B12
ON -4 GENERALIZATION IN THE DIFFUSION THEORY
1065
use of mol fractions. The symmetry is the essential point of the present theory. The treatment might be of interest for the kinetic theory of gas diffusion. Although the latter has always been treated in a symmetrical manner, it does not seem to have led to the separation of thermodynamical and frictional factors. The main purpose of the theory is, however, to ensure a rational interpretation of diffusion measurements. For more advanced approaches to diffusion problems, the reader is referred to the works of Onsager (13 and later), and of Meixner (12) on irreversible processes in general, and to the newer “theory of rate processes’’ (2). Confining ourselves here to a two-component system (except in Section IX), we begin with a condensed proof of the generalized Sutherland-Einstein relation. The latter expresses a limiting law for ideal, dilute solutions. The diffusion process being a stationary movement, the driving force per cubic centimeter, K12, is the product of the relative velocity, C1 - C2, between the components and the mutual friction, viz, per cubic centimeter, so that:
Kiz = (Ci
- C2)~iz
(2)
I n order to determine C1 and Cz, the “bulk velocity’’ d (compare Onsager and Fuoss (13)) is introduced 2’
=
Clnlcl
+ C2n2Gz
(3)
giving
by elimination of CZand considering reference equation 2. The diffusion force per mol of component 1 (of activity a1 = Nlfl) is:
Now, K I 2 = nlkl and, by reference equation 3, K12 = Nlkl/G12,so we obtain from equations 4 and 5 :
The theory of flow provides the equation
the time derivative of the concentration equalling the negative derivative of the flow Clnl along the diffusion direction z. Thus
1
(8‘)
1060
OLE LAMM
and analogously for the second component
Adding left and right members respectively, after an extension of equation 8’ by 81 and of equation 8‘’ by D2 (a procedure due to AI. Thiesen (14)), and considering reference equations 2, 5 , and G \ye obtain
a
0 = --
ax
(-2’)
(9)
I n this operation (the partial molar volumes being inserted \Tithin the derivatives) we introduce the restriction that the mixing process is not accompanied by any noticeable volume contraction or expansion. This condition is often fulfilled even in cases of considerable concentration difference between the tliffusing liquids, and the volume effect may always be reduced t o negligible magnitude by choosing a small concentration difference. Equation 9 means that the velocity in bulk according t o equation 3 is independent of x ; as a constant velocity of the diffusion column is of no interest in pure diffusion theory, we may put x’ = 0. It is concluded that the diffusion process a t constant partial molar volumes is characterized by 2 Cnir
=
0
(10)
I n reference G the diffusion of the three-component system was treated according t o this principle. The physical meaning of this is that the resulting volume transport through an r-level is zero, a rather self-evident statement but here following from a general treatment and the conception of Onsager and Fuoss. Returning t o equations 8 we obtain:
and by reference equations 7 and 8:
This is a Wiener-Boltzmann (1) differential equation:
According to equation 12, D1is symmetrical with respect to the two components, so that we may write D1 D2 = D 1 2 . Considering the reference equations 7 and 12 the following identical formulations of the mutual diffusion coefficient are obtained:
OK A GESER.ALIZ.4TIOX I N THE DIFFL'SION THEORY
1067
and
from which the generalization as compared to the Sutherland-Einstein equation is evident: lim D12= X,-l
RT
@2
The partial generalization
is also of interest, as Biz # 1 is quite possible even in very dilute solution, as for electrolytes (equation 43), or in the case of inappropriate choice of component molecular weight (see belon-). Equations like equation l G hare been proposed by many investigators (for references see 10). The factor B12may be determined by independent measurements, although this magnitude is undetermined unless we have made suppositions regarding the component molecular weights, Le., that they be chosen as the smallest possible values consistent n ith the chemical formulas, or multiples of these. It is, therefore, more satisfactory to introduce , 4 1 2
which is unambiguously delined without suppositions, as this is the case with D12, q12,and R T . Accordingly, the diffusion measurement gives p12. The distinct,ion is still more enforced in the theory of combined diffusion and sedimentation in a centrifugal field, which the author hopes to publish later, along the same lines as in this work. For the sake of clearness we may discuss a dilute solution, B12n2,which is independent of the choice of the n2i 0 . We have, then, A , ? molecular weight multiple. I t might seem self-evident that B 1 2 = 1, in the limit of dilute solution. But this is true only if \\-e calculate with the smallest actual molecular weight J12 and not with a multiple nor an integer fraction of this. BI?i 1, only if n2 represents the actual number of molecules or ions.' In the next section the thermodynamic factors are discussed for dissociated or associated solutes. It is, in this connection, necessary to recognize the difference between the standard state of the component and that of the molecular species in which the component may be present in the solution. The latter are always so chosen that activity can be replaced by concentration in the limit of zero concentration. This standard state, say for the double molecules of a dissociable compound, e.g., acetic acid in benzene, is not a realizable one, owing ---f
2The appropriate choice of Mzis a matter of experience. Thus, for the diffusion of oxygen in water a t ordinary temperatures and pressures we cannot choose M Z = 16 and B I Z= 1 , as the dissociation of oxygen will be unobservable. .lfz = 16 requires B I Z= $.
1068
OLE LAMM
to dissociation into simple molecules upon dilution. I t is an imaginary state. The standard state of the component may be represented by any one of the actual or possible species in the solution. So, for instance, acetic acid vapor consists mainly of double molecules, and the partial pressure of these may very well represent the activity of the acid as a component (it is supposed that the vapor is sufficiently ideal in other respects except for dissociation), Equally well may we calculate using the partial pressure of the simple molecules and let this represent the activity of the solute. Only the state of simple molecules is, honever, realizable as the limit condition of the substance which is reached by increasing dilution. Wow, by definitionf2 and B12 approach unity when the standard state is approached. Thus, in an actual diffusion experiment, Blz +. 1 can be true only if the chosen standard state is approximately realizable and actually is nearly reached in the experiment in question. The question whether the activities of the diJeerent species in an equilibrium may be represented by their conceptrations is not densive of the value of BIZ. IV. THE THERMODYNAMIC FACTOR .4ND .4SSOCIATIOK
In case the solute is associated, B12 is directly calculable if the equi!ibrium according to the mass action law is known ( 7 , 8). Assuming the activity coefficients of the solute molecular species t o be unity, we have
where 61, p2 . . . p b . . . are the fractions of the solute which are present as simple, double, . . . . and b-fold molecules, respectively. As discussed in the previous section, B12 depends on the choice of component molecular weight. In equations 18 and 19 the lowest value was chosen, corresponding to the fraction p1 of the component. Let the activity coefficients of the solute molecular species be y1, y2 . . . Yb . . . ; then the more general formula for a dilute solution
is valid too, considered in connection with equation 24. Finally it is, at least in principle, of interest t o give up the restriotion of dilute solution too. Assume the two-component system to contain, of component 1, among others a molecular species A,, with the concentration no mols per cubic centimeter. The molecular weight of the component is counted as corresponding to the formula A,. We use the notation f for the activity coefficient on the basis of mol fraction, and y on the basis of mols per cubic centimeter. In the expression for the differential of chemical potential, dpl = R T d In al, the activity may be written N1 fl. The activity is also ndy., so that f i = naya/Nl. Suppose a. to be the fraction of component 1 present as molecules A,; then n. = aonl, where nl is the total concentration in dols per cubic centimeter of this component. Thus f1 = cuanly,/Nl and, by reference equation 3,
LO69
O N A GESERALIZATION IN T H E DIFFUSION THEORY
For the second component we assume the notation B and consider specially a complexity Bb, analogously characterized by bb and 76. I n this way one obtains from equation 1 the general expressions:
These two expressions for B12 have a definite meaning only if the molecular weights A , and Bb are established for both components. Otherwise the mol fractions are undefined. Returning t o the dilute solution, + 0 gives (compare reference equations 10 and 11): (22) Here the molecular weight of the solute component corresponds to the chemical formula Bg, so that the right-hand member has to be divided by b t o accord with equation 19. The transformation of B1, from one molecular weight definition to another (a change of a or b or both) is thus very simple in the limit of dilute solution. I n the general case (equation 21) the dependence is most easily seen from equation l i . As -412is independent, BIZ varies as (nl n2)/nm. Stages of approximation lying between equations 21 and 22 might be of interest too.
+
V. T H E N C M B E R O F COMPOSE\-TS
The number of components is essential for the diffusion differential equation; the case of three components (6) is already much more complicated (compare Section IX). I n judging the number of components it must be borne in mind that the observation of the diffusion supposes a concentration gradient. If in the two-component mixture of hydrogen iodide and nitrogen the hydrogen iodide dissociates into hydrogen and iodine, there is a separation of the latter gases through the diffusion process, so that the composition along the diffusion column can no longer be measured by the number of mols of hydrogen iodide per cubic centimeter. Thus, this is a three-component system when diffusion is considered. I n a case like sodium acetate in water, in which the acetate ions are protolyzed by the medium, a partial separation of acetic acid and sodium hydroxide is brought about, the system being a three-component one. On the contrary, a mixture of sulfur trioxide and water remains a tn-o-component system also in relation t o diffusion. A further example is a molten mixture of two electrolytically dissociated binary salts, which can, in this connection, be a two-component system only if the salts have one ion in common. VI. T H E FRICTIONAL COEFFICIENTS
( I ) As self-diflusion is denoted the diffusion of a mixture of components, the latter of which are practically identical in all respects, which influence the diffusion process. The diffusion coefficient of the two-component system is, in this case, independent of the composition of the mixture (cf. 9). Also, the three-
1070
OLE W M M
component system has in this simple case real diffusion coefficients which are identical (9; equations 23 and 25), whereas in the general case it is characterized by three frictional coefficients in a more complicated way (compare Section IX). Leaving the higher systems aside we return to equation 13, putting BIZ= l.a Thus we must have @I @Z = constant, representing the simplest possible @2 = F(N2) is a deviation. case, from which the experimental function @I Owing to the relation (13)
+
+
@l
+ @z
=
(PlZB12/X1N2
we also have: (PI2
N z (a1+ a*) + -(BzN-2 )8JNz
(l
= VI
As it is in some respects most obvious to regard the friction per cubic centimeter, it is of value to make clear according to equation 23 (with @z = constant and 81 = 8,) how this depends on the composition even in the simplest case, that of self-diffusion. Incidentally, we also remark that @I is the mutual friction in the volume ( l / m 1 / n ~ )cc., which may perhaps be called a characteristic volume for the diffusion process. ( 2 ) Returning to the discussion on association or the forming of compounds between the components, the question arises as t o the possibility of determining single frictional coefficients of the different molecular species in the mixture. Katurally, the diffusion coefficient depends in a characteristic way on the special frictions between the different species in question, and it should be possible to relate the friction @I @Z or (DIZ to the single frictional coefficients mentioned. To determine the latter individually our knowledge will not generally be sufficient, apart from the well-known difficulty on the whole of getting a detailed insight into the molecular constitution of the non-dilute mixture in the case of liquids. It will sometimes be possible, however, to take the indirect way of making the simplest possible assumption regarding the constitution,-sufficient to explain the experimental result; the latter being the curve which connects or 6 1 2 with the composition of the rhixture. If component 2 is in dilute solution, and if electrolytic dissociation is not taken into account, a simple relation connects the component coefficient @z with the coefficients of the different molecular species of the solute (cf. 7 , 8). We make the same assumption regarding the chemical equilibrium as in Section I V ; p b is the fraction of the solute present as Bb molecules, @ b the coefficient of one mol Ba. We have
+
+
+
+
+
a How are the component molecular weights to be chosen for a n "identical" mixture, in order that B12 = l ? As an example, ordinary and heavy nitrogen may serve. If the equilibria between the isotopic molecules of nitrogen are very rapidly adjusted, the molecular weights 14 and 15 give the correct answer. If the processes are extremely slow we have three components: Xi', E:', and N14N16. Between these extremes, the components are not defined in a n ordinary sense, the process requiring a special theory, which involves the chemical velocity constants.
O N -4 GESERALIZ.LTIOS
1071
IN THE DIFFUSIOS THEORY
This relation is valid also if the activity coefficients in the mass action equilibrium are different from unity. 4 in equation 24 refers to the molecular weight of simple B molecules. (3) It is of interest for the general diffusion theory to prove that the component frictions, etc., are functions of the composition and other properties of state, but independent of the gradient property characteristic of diffusion measurements. I n the concentration gradient the chemical equilibrium position depends on t'he linear coijrdinate x in the direction of the gradient, and it is hardly self-evident t.hat does not change with the magnitude of the gradient. This molar friction equals the molar diffusion force divided by the linear component velocity in centimeters per second. We claim that not only is the frictional coefficient between two definite molecular species a characteristic property of the solution as such, but also the coefficient of one component, against a second one, and of one molecular species against the rest of t'he solution. The condition for the frictional coefficients to be properties of state in this sense is that the diffusion forces of all molecular species change by the same factor when the magnitude of the gradient changes. If the difusion forces retain their relative values, the mechanical picture is maintained, the time scale changing solely. We shall proceed t o prove that this is the case. Suppose a, t o be the activity of a molecular species A, of component 1, the
a 111 a
activity of which is al. The diffusion force on -1,is - RT a--
ax
per mol.
We
shall prove that the relative change LL of the force ivith the component activity aa
gradient -! is the same for all kinds of molecules. K e have
ax
1
L:
PI
a
(a+)
=
is the chemical potential of component 1, and Ire w i t e dp, = RT d In a1 = RT d In a,
(26)
OJYing to this relation we are hereafter bound to regard the molecular weight of component 1 as corresponding to A,. Equation 26 gives
thus
a al ax
( 2) a
(2)
1072
Now we assume the activity
OLE LAMM a1
to be independent of the gradient, so that
and finally
For another species A. with the activity a., the mass action law gives (a,,)” = Ic(a,)”, that is,
a_ In a. - a Ina, (Y
ax
a
ax
The factor a l a vanishes in the expression for L‘, and we obtain L: = L:, so that equation 28 is valid for all species of component 1 . Analogously, for all species of component 2 one obtains: II
1
L* = aaz -
ax This corresponds, however, to the relative change of the diffusion force with the activity gradient aa,/ax. It is transformed to the derivative with respect to aal/ax by multiplying by
Reference equation l b gives: aa, - _ - nl az aa1 ax n2al ax The concentrations and activities being independent of the gradient property, we have
Multiplying equation 29 by this, and again using the reference equation mentioned :
ON .4 GENERALIZATION I N T H E DIFFUSION THEORY
1073
ivhich just conforms to equation 28. Hitherto, we have not considered compound formation between the components. Supposing an equilibrium
+
pbS, qaBb F? abA,B, (33) \\e do not restrict ourselves to one single compound, a, b, p, and q being arbitrary also in this respect. The mass action law gives ab In uPq = pb In a.
+ qa In + In K ab
(34)
thus
and, considering the expressions previously obtained for L: and L ; ,
We have :
LaT =
1
a In ap9
(37)
ax the first equality being the definition of Lk,, the second following from equations 35 and 36. The result, again, conforms to equation 28. I t is concluded t,hat the relative diffusion forces are independent of the magnitude of the gradient, if this is the case vith activities, a supposition the correctness of ivhich is not questioned for the slow gradients t o be met with in this connection. Thus the frictional coefficients are solely determined by the ordinary properties of state, provided the solution or gas mixture is in inner, chemical equilibrium during the diffusion process. The proof is, however, restricted t o the tn-o-component system. owing to the use of equation 30. Problems of this kind are likely to be most efficiently treated by aid of the methods of Onsager or Meixner; nevertheless, the more elementary procedure used here might be of interest. 1'11. ELECTROLYTES
I n order to calculate the thermodynamic factor for a binary, completely dissociated, and non-protolyzed electrolyte il.Ba n-ith the ions A", B", and the concentration n2 mol A,Bb per cubic centimeter, we have to transform Bl2 into concentration units (instead of mol fractions). This is reached by derivation of y2n2= j J V 2 completely Tvith respect to n2and the use of equation 1 and reference equations 10 and 11, an operation which leads to:
1071
OLE LAMM
The activity aABof the electrolyte is: f2N2 = y2n2 = a:bb = n:nby"+r! = aabbn;ibyz+6 (39) where y i is, by definition, the mean activity coefficient of the electrolyte. Thus, y2 = aabbn;f"l a+b Yi
(40)
and from this,
Equation 38 gives:
I n the limit of ideal, dilute solution, this reduces to
BIZ= a
+b
(43)
The molar frictional coefficient of the solute is, for an electrolyte, the sum of the frictional coefficients of the ions in 1 mol of the salt. This is quite different from the relation 24, oiving to the condition of electroneutrality in this case. We do not a t all treat the connection of with conductivity in this article. I t is, however, concluded that equations 16 and 43 lead to a simple proof of the nellknown diffusion equation of Nernst-Haskell (4) for the ideal, dilute electrolyte. On the other hand, this confirms the general applicability of the method here developed, especially regarding the treatment of the thermodynamic factor. I n concentrated or otherwise not sufficiently ideal solution, equation 43 must be replaced by equation 42 and equation 16 by equation 14, the last alternative. We obtain the simple expression:
The corresponding general equation is
The factor 1 - nzCa= nBI is a result of the symmetrical treatment of the two components. Formula 14 supposes ( I ) a real two-component system of and solvent (compare Section V) in inner equilibrium, and of mechanically normal fluidity4; (2) constant partial molar volumes and irl. The latter is The validity of formula 44 does not nsecesitate that the electrolyte be completely bB'-, owing dissociated, nor that it be dissociated according t o the scheme A,B& = aA'+ t o the formal possibility of defining activity a8 a"bbn;%~+,fbin any case. On the other hand, i t is necessary to choose the molecular weight according t o the formula A,Bb for the oalculation of n2 and a,.
+
OK
A GESERILIZ.4TION I N T H E D I F F L 3 I O S THEORY
1075
always true if D12is the real diflerential diffusion coefficient, as obtained with a sufficiently small concentration difference between the diffusing liquids. Thus, the molar volumes by no means need to have the same values at different concentrations n?. Diffusion co-fficients of this kind have recently been measured by the method of Onsager and Harned (see 3 and later papers). VIII. GENERAL CONSIDERATIOhX I K T H E FORMZCLATION O F T H E THERMODYNAMIC F4CTOR
According to equations 1 and 17 we have
By consequent transformation to units of mols per cubic centimeter one obtains:
This coefficient has the advantage of being a direct, unambiguous property of the mixtiire. Dividing this by another property, nln*fil&,and multiplying by R T , we get a neiv quantity called Q12 :
RTAn -__ nl n2 81 6%
The differential of the osmotic pressure 02
dpl
IZT
a In a2
012 = T v2 a In n2 pl
(47)
of component 1 is given by
+ RT d In a2 = 0
(48)
We obtain:
and equation 47 gives
I t is not surprising to find (212 closely related to a directly measurable property of the solution as is the osmotic pressure. The older treatments of liquid diffusion \\ere based on the van't Hoff osmotic-pressure expression. The generalized diffusion theory has a close connection ivith the more general osmotic pressure interpretation. It should be noticed that Q12 is n symmetrical property as indicated by this notation. Kriting equation 50:
1076
OLE LAMM
we sec that Q12 is a property suitable for the characterization of a binary mixture, although from practical reasons it must generally be measured indirectly. Finally, we write the values of the magnitudes in question in the limit of zero concentration. n2 + 0 gives:
BIZ+ 1 The last equation is the simplest, but supposes the correct molecular weight multiple of component 2 to be chosen. Measuring the other limiting values, Q 1 2 or AI,, determines this molecular weight. IX. KOTE O S THE DIFFUSIOX OF A TEltSARY SYSTEM
In previous papers ( G , 9) a differential equation for the diffusion of a ternary liquid mixture was written in the form
T w o other equations are obtained by cyclic permutation of indices, 1 + 2 3 --t 1. The notations are 6123
+ Nz6z + N &
= NIDI
+
%P = (~iz(p1s
kiz = e d Y 2 ki3
4
= Ma(Y2
+
(OSIP~Z
+ Ya) + + Ys)+ ( P I ~ Y ~ 981y?
+ +
Y1 = n161 etc. are the volume fractions, Y1 Y 2 Y 3 : 1. The possibility exists of using mol fractions consistently, but apparently this leads to very complicated expressions. The formulation 53 has the disadvantage of mixed variables, as it contains both mols per cubic centimeter and mol fractions. The best way to avoid this seems to be to introduce volume fractions throughout. For this purpose we write activity Y1 rl, rl being the activity coefficient on the basis of volume fraction. We obtain (cf. 6, equation 14)
where
a In rl o1 = 1 +--i) In Y,
a In a In nl a In G I + a x l
1 f-
alld
Equation 54 now gives, more clearly and with disappearing
i1?3,
The 0-values are regarded as empirical functions of the two independent volume concentrations. Besides these, the dihsion process is determined by the fric. the gradient in activity of a component tional coefficients pI2,( ~ 2 8 ,and ( P ~ ~ As 1 can have a finite value even if aYl/az = 0, owing to the gradient in C O I I C ~ I L tration of the other two components, the functions 0 incidentally may become infinite. (The connection betmeen the Ol-values and the B1-values cannot be written in concentration variables belonging to component 1 solely.) The use of concentrations n makes very little difference, related as they are 60 volume fractions through 11 ' = nl& etc., and 81 here can be regarded as constant, as has already been assumed in the deduction of equation 53. The equation IS
HC~IXIIRT
The scope and features of a theory of diffusion, detailed in earlier works, have been outlined. A simple proof of a generalized Sutherland-Einstein equation for the diffusion of a binary solution in general is given, and the interpretation of diffusion measurements discussed. The properties of 6he thermodynamic factor and the frictional coefficients involved are treated in some detail, considering the chemical equilibrium between the components in arbitrary or dilute solutions. An equation for the diffusion of electrolytes is.computed. A note is added on the diffusion of a ternary solution, this being most simply described in terms of volume fractions or mols per cubic centimeter. REFERENCES
(1) BOLTZMANS, L.: Wied. Ann. Physik 63,959 (1894). (2) GLASSTONE, S., LAIDLER, K. J., AND EYRING, H . : The Theory of Rate Processes. BfcCrswHill Book Company, Inc.. Kew York and London (1941). (3) H A ~ N E D H ,. S., AND FRENCH, D. 31.:Ann. N. Y. Acad. Sci. 48, No. 5,267 (1945). (4) HASKELL, R.: Phys. Rev. 27, 145 (1908). (5) LAMM, 0 . : ArkivXemi, Mineral. Geol. 17A, No. 9 (1943).
1078
WILDER D. BAXCROFT
(6) LAMM,0 . :.4rkiv Kemi, Mineral. Geol. 18A, No. 2 (1944). (7) LAMM,0 . :Arkiv Kemi, Mineral. Geol. M A , Nd. 9 (1944). ( 8 ) LAMM,0.: Arkiv Kemi, Mineral. Geol. M A , No. 10 (1944). (9) LAMM,0.:Arkiv Kenii, Mineral. Geol. 18B,No. 5 (1941). (10) Lahihi, 0 . :Paper published in honor of T . Svedberg, Upsala, 1944, p. 182. (11) LEWIS,G . N., AND RASDALL,M.:Thermodynamics and the Free Energy of Chemical Substances. RlcGraw-Hill Book Company, Inc., New York (1923). (12) MEIXNER,J.: Ann. Physik [51 39, 333 (1941); 43, 244 (1943). (13) OXSAGER, L . : Phys. Rev. 37, 405 (1931). ONSAGER, L . , AND Fuoss, R . &I.: J. Phys. Chem. 36, 2689 (1932). 11.:Verhandl. deut. physik. Ges. 4 , 348 (1902). (14) THIESEK,
THE BIOCHEMISTRY O F PLAiCT PIGMENTS WILDER D. BANCROFT Cornell r n i u e r s i t y , Ithaca, .Vew Y o l k Received B p r i l 29, 194Y
A great deal of work has been done on the chemistry of plant pigments (l), much of it first class. When it comes to the biochemistry of plant pigments our scientific knon-ledge is still very unsatisfactory. This is apparently because the problem is one which should be attacked by chemists and botanists n-orking conjointly. This has not yet happened. I t is possible that the botanist could learn enough chemistry to enable him to dispense with the chemist; but no botanist has clone that. It is probable that the chemist could learn enough botany to enable him to dispense with the botanist; but no chemist has yet done that. Lycopene, Cd0HS6,is the red pigment of the tomato. "Lycopene has also been found in hips (Rosa c u n i n a ) , in thc ripe fruits of T a m u s com(C,'hristmns rose), i n deadly nightshade (Solanuw riulcomara), in the fruit of the watermelon (C'ucutnis cilrullus), in the berries of d l r i i m i n u c i i l a t u t i i , in the apricot (Prunus u m e n i a c a ) , in bryony fruit (Bryonia dioica), in the golden flowers of the marigold (Calendula o$kinaZis), in the fruit of lily-of-the-valley (Cowiilluria m a j a l i s ) , i n Kaki fruit (Dius,iUru,3 K u k i ) , in tropical fruits, in the dark orange blossom of Dimovlihoteca aurantia, i n Citrus yratidis, in Passiflorn c o e r u l e a , and in bacteria, ae in the thiocustio baclereuiii." (10) miitiis
Apart from lycopene the red and many of the blue regetable pigments are anthocyanins or anthocyans. They are all derivations of 2-phenylbenzopyrylium salts and all, with the exception of a fev amino derivatives, are hydroxy derivatives existing in the plant usually as glycosides. X11 blue flowers turn red when acidified. Some red flowers, but not all, pass through blue \\.hen treated Xvith ammonia. It is not known \That stabilihes the blue and ivhether it is always the same substance or mixture of substances. I t is now known that crude extracts of the anthocyans contain copigments such as tannin, gallic acid, etc., which possess the ability to intensify or modify the color. Thus the glucoside of 2-hydroxyxanthone is an active copigment for cyanin,
