TEXTBOOK ERRORS:'
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VZZZ: The DeJinition of Transference Numbers in Solution M. SPIROZ University of Toronto, Toronto, Canada
S o m considerable time after the discovery that electrolyte solutions conduct electricity, it was found that during electrolysis the concentrations of the solutes changed near the electrodes. Hittorf in 1853 first attributed this phenomenon to the different velocities of the ions in solution, and to measure these concentration changes quantitatively the term transference or transport number was introduced. With the development of theory various definitions of this term appeared, hut the quantities given by some of these definitions, though useful theoretically, correspond to the measured concentration changes only in certain cases. A definition fitting into the latter category is the only one given in practically all textbook^,^ without any added comments as t o its restricted applicability. This causes much confusion when attempts are made to interpret experimental data. This definition will be compared below with two others in common use, and their relatiouships both to the experimental concentration changes and to the concentrations, mobilities, and charges of the species in the solution will be discussed. The general textbook definition reads: The transference or transport number of an ion in a given electrolyte solution is the fraction of the total electric current carried in the solution by that ion ( I ) . To distinguish this quantity from that given in other definitions, it will be called the electrical transport number of ion i, ti. The nomenclature and symbols adopted here are modifications of those suggested by M ~ B a i nAlberty? ,~ Scatchard: and others, and are summarized in the accompanying list. I t follows that ti must always be positive and that:
Cti = 1
(1)
Suggestions and material suitable for this column are eagerly sought and will be acknowledged. They should be sent with as many details as possible to Karol J. Mysels, University of Southern California, Los Angela, California. Contributors of discussions in a form suitable for publication directly will be aeknowledged as guest authors. Present address: University of Melbourne, Melbourne, Australia. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the source of the errors discussed will not be cited. The error must occur in at least two indeoendent standard books to be oresented ' MCBAIN,J. W., "Colloid Science," D. C. Heath and Co., Boston, 1950, pp. 253-5. 6 ALBERTY, R. A,, J. Am. Chem. Soc., 72,2361 (1950). 6 SCATCHARD, G., J. Am. Chem. Soc., 75,2883 (1953).
As an example of the application of this definition, let us consider a Hittorf electrolysis experiment with aqueous zinc chloride solution. I n the cell: @ Zn, A I M I C, AgC1-Pt @
p Q A , M, and C represent the anode, middle, and cathode compartments, respectively. If A and Care sufficiently
large, observation shows that after one faraday of electricity (96,500coulombs) has passed through the cell the concentration of ZnClr in M is unaltered while in A and C i t is changed in such a way that the total ZnClz content of both compartments taken together has increased by one (electrochemical) gram-equivalent (provided a small correction is applied for the current carried by the solvent itself'). The experiment thus yields only one new quantity, namely the increase in the ZnClz content in either A or C. The experiment has not told us bow much of the current has been carried by the various ions (Zn++, ZnCl+, ZnCls-, ZnC1,--, C1-) known to exist in the solution: and therefore the electrical transport numbers of the various ions cannot. be measured directly. Let us see, however, whether they can a t least be calculated from various types of experimental data. By consideration of the flux of ions passing across a plane in the solution, it can be shown that:
The individual ionic concentrations can be found from spectrophotometric measurements, for example, and the problem becomes one of determining the individual ionic mobilities. I n the above example we have five unknown mobilities, corresponding to the five ionic species, and only two equations connecting them, namely the conductance and the quantity found in the above Hittorf experiment. Calculation of ti for any of the ions is therefore impossible, no matter how much experimental information on the solution is at hand. However, it follows from the above analysis that it is possible to calculate the electrical transport numbers and the mobilities from experimental data if only two ionic species are present in the solution, but i t will be LON~~SWORTH, L. G., J. Am. Chem. Sm., 54, 2741 (1932). ROBINSON, R. A., AND R. H. STOKES,"Electrolyte Solutions," Butterworths Scientific Publications, London, 1955, pp. 415-19.
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VOLUME 33, NO. 9, SEPTEMBER, 1956
shown later that t,hese quantities can be rigorously derived from experiment only if it is known that the electrolyte is completely dissociated. I n consequence, an approximate computation of ti is possible even in other cases if mobility values derived from work on other solutions are available as well as good theories of the effects of ionic strength, the specific interaction between ions, etc. on the mobilities. I n our example, t;:?': can be approximately calculated if Xz,++ is found from measurements on Zn(CIOl)z solutions, where there is no reason t o suspect incomplete dissociation or complex-ionformation,0and tZs1'if Xcl- is obtained from measurements on solutions of the alkali chlorides which are known t o be strong electrolytes. The electrical transport numbers of the other ions in the ziuc chloride solution cannot be calculated in this way. In equation (2) the mobilities of both cations and anions were taken as positive, in accord with the general practice. However, several workers"lO." have found it convenient to define the mobility in such a way that the direction of motion is taken into account. For this purpose they introduce the concept of positive current or electricity which is that part of the current flowing through the solution which is carried by the positively charged ions moving from anode to c a t h ~ d e . ' ~The "signed" mobility, .u, is then the velocity, under unit potential gradient, in the direction of positive current. By definition (I) the electrical transport number must be positive, so that:
465
the ionic species (e. g., C2H60Hmolecules in an aqueous alcoholic zinc chloride solution). I n the latter case, the transport number of the second solute may be directly determined by analysis. Such polar molecules appear to be transported by being imbedded in, or carried along with, the layer of solvent surrounding the ions.la I n terms of the properties of the species: .ui mi
'Ti
=
C .ui mi zi
=
.Xi
mi
C ah; mi ei
.hi- mi =-=
C .hi .ei
ti
(39)
Zi
The method of calculating these values is analogous to that for calculating ti. Both these definitions suffer from the defect that they define transfer properties of the ions and molecules in the solution, whereas experimentally me can determine only the over-all material transfer of electrolyte (or over-all current), and only indirectly can we discover how much of this electrolyte transfer (or current) should be attributed to each of the various species in the solution. If, then, we want to define a directly observable quantity we must realize that we are measuring the concentration change, not of a free ion, but of an ion constituent5. lo (or radical1'. 14). An ion constituent is the ion-forming portion of an electrolyte without reference to the extent to which it may actually exist in the dissociated state.1° Thus in aqueous ziuc chloride solution, the zinc ion constituent exists only partly in the form of free Zn++ ions and exists also in the form of species such as ZnCl+, ZnClz,etc. It is the concentration of zinc ion constituent, not that of free zinc ions, that is found by chemical analysis (e. g., by precipitation as ZnS), and the concentration of zinc ion constituent in gram-ions is equal to the total number of gramions of zinc contained in all the individual svecies in the
A second definition, which has been employed in "GORDON, A. R., Ann. nev. Phv8. C h a . , 1, 59 (1950). the "signed" form,&yields what will be called the List of Symbols Employed transport number of species i, and reads: The transport number of a given species is the net number of gram- c = Normality of solute (gram-equivalents of solute per liter of solution) jormula-weights of that species that crosses an imaginary = Normality of species i; ci = mi Izi I ; .ci = mi zi plane in the solution, in the direction of positibe current, = Fmaday constant (96,500 coulombs gram-equivalent-') when one faraday of electricity passes across that plane Z = Electric current in amperes carried by solute (11s). The word "species" refers to the ion or molecule I : = Electric current carried by species i = M o l & y of species i (gram-formula-weights per liter of considered, and the plane is generally regarded as mi solution) fixed with respect to the solvent. This transport num- nn.i = Number of pram-equivalents of ion ron~tituentR in one
molecules. whether or not they are in equilibrium with ' S r n u ~ s ,R. H., AND B. J. LEVIEN,J . Am. Chem. Soc., 68, 333 (1946). " NOYES,A. A., A N D K. G. FALK,J. Am. Cheni. Soc., 33, 1436 110111
IIIIRTLEY,G. S., AND 3. L. MOILLIET,Pme. Roy. Soe. (London), A140, 141 (1933). It is probably better, from the educational point of view, not to stress this aspect, since enough confusion arises from the terms current flow and electron flow through a. wire without adding ta it by dividing the current through the solution into two parts. This method of treatment is given here as an alternative because it is becoming very popular, and it is valua.ble in cases such as electrophoresis where the direction of motion is not always known a priori.
Mobility of species i (velocity in em. see.-' under z potential gradient of one volt em.-'); s, = h , / F Mohility of ion constituent R ; an = h n / F Potential gradient in volt cm.-' Algebraic electrovalence per formula of species i (gramequivalents por gram-formula-weight of species i); lz, / =
+,42
Algebraic electrovalence por formula. of ion constituent 42 R ; Iznl =
+
Equivalent conductance of species i in gram-equivalent-' ohm-' cm.l Equivalent conductance of ion constituent R Eauivalent eonduotance of the w.h-hole solution: Ac =
2 hi% C .hi =
2 i
unless otherwise indicated
JOURNAL OF CHEMICAL EDUCATION
solution. Svensson" calls species like Zn++, ZnCl+, ZnCls etc. "subspecies" of the zinc ion constituent. In symbols the latter is written either as Zn ion constituent, or as Zn++ constituent, and since all equations refer to solutions, the subscripts Zn and Zn++ distinguish hetween zinc ion constituent and zinc ion respectively. This brings us to the definition of what will be called the transference number of the ion constituent R, TR: The transference number of a cation or anion constituent is the net number of (electrochemical) gram-equivalents of that ion constituent that crosses an imaginary plane in the solution, in the direction of the cathode or anode respectively, when one faraday of electricity passes across that plane10, ll. ls (111). The plane is generally taken as fixed with respect to the solvent. I t follows that:
because the total number of gram-equivalents of all the ion constituents so transferred in both directions is equal to the number of faradays passed through the solution.'0 As an illustration, let us consider compartment A in the above Hittorf experiment after the passage of one faraday through the cell, the current having been corrected for the conducting matter present in the solvent itself.' One gram-equivalent of solid zinc has dissolved to give zinc ion constituent in solution, T"':; gram-equivalents of zinc ion constituent have migrated across plane P toward the cathode, and TEC" gram-equivalents of chloride ion constituent have migrated across P toward the anode. Thus A has = TE'" gram-equivalents of zinc ion gained 1 - T";:' constituent (equation (4)) and TEC1'gram-equivalents of chloride ion constituent, i. e., TEC" gram-equivalents of zinc chloride. Similarly C has gained T";:' gramequivalents of zinc chloride. The transference numbers are therefore directly obtained by chemical analysis of A and/or C, and no assumptions regarding the detailed composition of the solution are needed. Definition (111) is also applicable to a second, undissociated solute such as CzHsOHif the term gram-equivalent is replaced by the word gram-formula-weight as in definition (11). A detailed analysis of the flux of ion constituent carried by the various species across a plane in the solution gives:
Care should be taken to distinguish between rR and
a. The term (zn/lznl) in front, which equals +I if R
.. ... . -.
t i , 171 (1932).
'
is a cation constituent and - 1 if R is an anion constituent, allows for the directional factor mentioned in definition (111), while the analogous terms (zi/lzil) account for the fact that cations and anions carry a given ion constituent in opposite directions. As an example see the equation a t the bottom of this page. The uncertainty of sign preceding Xz,cl, signifies our uncertainty as to whether there is a net transfer of undissociated molecules in the direction of the anode or the cathode. I t seems likely, however, that the numerical value of the equivalent conductance of a molecular species like ZnC12 is much smaller than that of an ionic species. It has been found experimentally that in concentrated zinc and cadmium halide solutions the transference number of the metal ion constituent is negative, so that there is a net transfer of the metal ion constituent from cathode to anode. Inspection of the last equation shows that this is not surprising in view of the fact that in concentrated solutions of these salts a large proportion of the metal is in the form of the negatively charged and electrically mobile halide complexe~.~ However, there are some peculiar explanations of this phenomenon by those textbook writers who employ only definition (I) for theoretical discussions and who do not realize that the number obtained experimentally is that given in definition (111). McBain4 gives a detailed numerical treatment of the differences between definitions (I) and (111) by using as his example a simple polyelectrolyte solution in which micelle formation also leads t o negative cation constituent transference numbers. It is important to note that the transference numbers given in definition (111) depend on the choice of ion constituents. For example, in an aqueous solution of H3P04the only ions existing in appreciable numbers are H + and HzP04-, since the second dissociation constant of the acid is much smaller than the first. It is then possible to choose as the constituents either H + and H2P04-, or H + and HP04--, or H + and PO4F3, and applying equation (5) :
All these equations are equally valid and there is no ambiguity as long as the ion constituents are stated and the experimental concentration changes interpreted on the same basis, for each mole of H3PO1conta.ins one, two and three gram-equivalents of H3P04, HP04, and
must be possible t o express the chemical formula of
VOLUME 33, NO. 9, SEPTEMBER, 1936
every species in the solution as an additive function of used in deriving equation (2), it is often overlooked bethe chemical formulas of the ion constituents. Thus in cause definition ( I ) does not refer to it. This plane is the zinc chloride solution Zu++ and C1-, but not Zn++ generally regarded as fixed with respect to the solvent, and ZnCl+, can be chosen as the constituents, for in the and the transference numbers are calculated from the latter case the formula of, for example, the chloride ion experimental data on this basis.15. l6 could be represented only as the difference ZnCl+The alternative "signed" form of definition (111) Zn++ instead of as the sum of some combination of the reads: The transference number of a n ion constituent is ion constituent formulas. It follows that if, in the the net number of (electrochemical) gram-equivalents of former example, the dissociation constants of phosphoric that ion constituent that crosses an imaginary plane in the acid had not been known, it would have been necessary solution, in t.4e direction of positive current, when one to pick H + and PO4iSas the constituents in case ions faraday of electricity passes across that plane (111s). is then usually positive for a cation constituent and such as HP04-- and P 0 4 i 3existed in the solution in negative for an anion constituent. The corresponding appreciable numbers. Since it is the ion constituent transference number equations are: that can be related directly to the experimental concentration changes, it is evident that the electrical transport number can be experimentally measured if the two numbers are identical. For this to be so, the electrolyte must be completely dissociated into its ion constituents and no complex ions or molecules must exist that contain more than one ion constituent, so that, practically speaking, the terms ion and ion constituent become synonymous. I n the special case of a I n the above discussion, the only experimental strong electrolyte that is also uni-univalent, such as method for measuring transference numbers that was NaCl in aqueous solution, all three definitions reduce cited was that devised by Hittorf, and this is largely reto the same form: placed today by the moving boundary and other methods. The same arguments, however, apply to all of these, and it is the transference number of the ion constituent that is experimentally observed. As an The transference number chapter in most textbooks in- example, the passage of current in the illustrated twocludes as examples mainly electrolytes of this kind, so salt moving boundary system results in the replacement that the limited practical utility of definition ( I ) is of the ZnClz solution by the "indicator" LiCl solution not apparent. as the boundary descends, provided the solutions are Two common misconceptions arise from the confusion not too concentrated. The same number of grambetween definitions (I) and (111). One is that the equivalents of zinc ion conproduct of the transference or transport number and stituent that is mept downa the equivalent conductance is proportional to the moward by the boundary bility of a single ion. This, however, is true only if LiCl passes also across plane MN the electrical transport number is employed, and since further down the tube, and ---1 it is the ion constituent transference number that is hence, by definition (111), measured the multiplication gives: ZnC4 the number of gram-equivalents of zinc ion constituent .I[in that volume of solution that the boundary sweeps 0 out when one faraday flows G o u g h the cell is T;:~". The velocity of the boundThe ion constituent R vill therefore migrate as a uniary is C ~ " , ~ T ~ " ~ " . The general proof, and details of form substance with a mobility E,, provided the time two small corrections that must be applied, are given by of existence of each specie^ in the sdution is small compared to the time of migrati~n.~. l4 If the electro- MacInnes and Longsworth.'s lyte is completely dissociated, ERis proportional to the ACKNOWLEDGMENT mobility of one ion if a proper choice of ion const~tuents I would like to thank the Natioual Research Council has been made. The second misconception is that the mobilities of the ions, obtained from work with com- of Canada for the award of a University Post-doctorate pletely dissociated electrolytes, are absolute velocities, Fellowship, Professor A. R. Gordon for his interest and whereas equation (7) shows that they are really discussion, and Professor I
