PAULMILIOSAND JOHN NEWMAN
298 view of correlation between B C H ~and the corresponding unsubstituted BH noted earlier by RIacLean and I14ackorsgfor seven compounds and extended in Figure 1 to include additional methyl polynuclears; analogous plots can be made for CH? and CH3 shifts in ethyl polynuclears. The regression data in Table IV have been calculated for all positions given in Table I, apart from the special cases in which the methyl group suffers shielding from its location above the ring plane.6gJ0 All three groups of shifts are highly significantly related to BH; for CH3 and CH&H3, the gradients (a) are
similar, while, for the CH2CH3 protons more remote from the ring centers, the gradient is lower. Acknowledgments. We are grateful to Mr. R. L. Cureton, who provided a computer program for calculating regression parameters. This work was carried out during the tenure of an I.C.I. Fellowship by K. D. B. (69) M. 5. Newman and M . Wolf, J . Amer. Chem. SOC.,74, 3225 (1952). (70) A. W. Johnson, J. Org. Chem., 24, 833 (1959).
Moving Boundary Measurement of Transference Numbers by Paul Milios and John Newman Inorganic Materials Research Division, Lawrence Radiation Laboratory, and Department of Chemical Engineering, University of CalQornia, Berkeley, California (Received March 8, 1068)
A moving-boundary system is analyzed, and an equation, valid for concentrated as well as dilute solutions, is obtained for the transference number. When the partial molal volume of the solvent is constant through the boundary, the equation reduces to the approximate equation now in common use. The cation transference number in 0.213 M NH4NOa was experimentally determined at 25’ and found to be 0.5140 k 0.0024.
Introduction Rloving-boundary measurements date back to the nineteenth century.‘ The two-salt boundary mas analyzed in 1900 by the mathematician Weber.% I n 1910, the chemist Lewis3 presented an analysis which corrected for the boundary movement caused by the electrode reaction. His equation was thought to be restricted to dilute solutions’ until B e ~ m a n in , ~ 1962, showed that it actually applies to systems in which the partial molal volumes are constant through the boundary-a slight,ly less stringent condition. KO other theoretical progress has yet been made, and Lewis’s equation is still used today.5 I n the present work a more detailed analysis of the moving-boundary system yields an expression of more general validity for the transference number. Analysis of the Two-Salt Boundary Figure 1 shows the two-salt boundary a t steady state. Solution A is composed of ions 1 and 3 and, being the lighter of the two solutions, is above solution B, which is composed of ions 2 and 3. The solvent is referred to by “0.” The equations necessary to describe the movingboundary system are the continuity equation or material balance (1) The Journal of Physical Chemistry
an equation relating the current density to the fluxes
i
=
FCX,C~V~ i
a statement of electroneutrality
cx,ct
=
0
z
and a flux or transport equation ape
ce-- = bX
where
RTC--(vjClCl j
C T Q ~ ~
- v,)
(4)
is the electrochemical potential and where Dtj = D j l . These equations are discussed in detail by Sewman.6 When the motion of the ions is referred to the solvent motion, the transference number (relative to the solvent velocity) is defined as the fraction of the current carried by an ion in a solution of uniform composipe
L. G. Longsworth, Chem. Rev., 11, 171 (1932). (2) (a) H. Weber, Sitzber. Deut. Akad. Wiss. Berlin, 936 (1897); (b) H. Weber, “Die partiellen Differential-gleichungen der mathematischen Physik,” Friedrich Vieweg und Sohn, Braunschweig, 1900, pp 481-506. (3) G. N. Lewis, J. Amer. Chem. Soc., 32, 862 (1910). (4) R. J. Bearman, J. Chem. Phys., 36, 2432 (1962). (6) R. Haase, G. Lehnert, and H.-J. Jansen, 2.Phys. Chem. (Frankfurt am Main), 42, 32 (1964). (6) J. Newman, Advan. Electrochem. Electrochem. Eng., 5, 87 (1967). (1) D . A. MacInnes and
MOVINGBOUNDARY MEASUREMENT OF TRANSFERENCE NUMBERS
299
tion, that is t+ =
g+c+(v+
z+c+(v+ - vo)
- vo)
+ z-c-(v-
- vo)
x = -a, Salt solution A composed of ions I & 3
(5)
for a solution of two ions. I n terms of the transport properties this becomes
By an analysis similar to that of Weber, but with a transport eq 4 applicable in concentrated solutions, these equations can be manipulated to yield the exact relationships
boundary
Salt solution 8 composed of ions 2 & 3
(7)
x = + a ,
where
is the velocity of the boundary, the subscript indicates that c3/co must be evaluated in solution A above the boundary, and the subscript x = m indicates that C3/cO must be evaluated in solution B below the boundary. This derivation is effected by observing that the system is in a steady state in a coordinate frame which moves with the velocity Vb of the boundary. Equation 1then becomes
x
vb
= -.
m
+
ct(vt - et,)
=
constant
(9)
for all species, and v1 = 212
=
ut,
(10)
since the concentrations c1 and c2 are zero either above or below the boundary. Equation 2 then gives 1)s
vb
+
(11)
i/2$3F
The transport relation 4 for ions 1 and 3 now takes the forms -
ax
+
- __ E T [______ C O ( ~ ub) O ~ ~ c3(v3 - Ubi.'] Dl3
cT
(12)
(13) The chemical potential PA of the neutral salt A is defined in terms of the ion electrochemical potentials as /*.A
= W.JI
+
~ 3 . = ~ ~YI(M 3
- ZIPB/XB)
where vaA = - - x ~ v ~ / z Q .Hence eq 11and 12 yield
?EA - _&T[ _ (210 bX
CT
vb)
( co - zico ) + -
Dl0
__ 23%0
(14)
Figure 1. The 2-salt boundary.
As x --t - m , one approaches the region of uniform concentration of salt A above the boundary. Here c2 -+ 0 and bp&x -+ 0, and eq 15 becomes v0 v3
-
-
vb/ Ublr=
-m
2lDlO 21%
- z3a30
=
El
(16)
Equation 7 is then obtained by the elimination O f v3 - vb through eq 11. Equation 8 for tz in solution B can be derived in a similar manner by noting that c1 + 0 and apB/ax+oasx-+ ~ 0 . It should be noted that eq 16 can be obtained directly from 5 since both equations apply to a medium containing two ion types and of uniform composition. Thus the steps from eq 11 on can be bypassed. These steps merely show that the modern theory of multicomponent diffusion is self-consistent with respect to the basic definition 5 of the transference number. Such a theory would be necessary for calculating the detailed concentration profiles in the two-salt boundary and the thickness of this boundary and for showing that the steady boundary is stable if the current is passed in one direction, but not if it is passed in the opposite direction. Incidentally, the thickness of the boundary can be shown to be inversely proportional to the current density, and, roughly speaking, the boundary is stable if the direction of current is such that
+
Xlitl
< XlitZ
i.e., the leading solution should have the higher conductivity. The detailed transport theory is not necessary if one is interested only in the conditions on the two sides of the sharp boundary. Similar situations are encountered in the treatments of a hydraulic jump and of a planar shock wave in a compressible gas. Equations 7 and 8 are the principal result of the analVolume 7.9,Number 8 Februarzi 1069
PAULMILIOSAND JOHNNEWMAN
300
ysis of the two-salt boundary, and t’hey immediately yield the Kohlrausch regulating function’
This is identical with the relation derived by Smits and Duyvis’ in their criticism of work done separately by B e t ~ m a nHaase,8 ,~ and Spir0.O If the concentration of the solution following the boundary does not initially satisfy the Kohlrausch regulating function, it will adjust itself so that it does. I n this case a concentration boundary between the initial solution and that following the boundary will be left behind as the two-salt boundary progresses up or down the channel. This boundary between solutions of two different concentrations of the following ion has been studied optically by Longsworth. lo The “Volume Correction.” In eq 7 and 8, the term co(vo- vb) is the solvent flux relative to the boundary and is constant through the boundary. The application of these equations requires the determination of vo either above or below the two-salt boundary. Usually one electrode is selected with a. well-defined reaction, and this electrode chamber is tightly closed so that the volume changes a t the electrode can be assessed. Thus the system shown in Figure 1 must be expanded to include the closed electrode chamber. Figure 2 shows this arrangement with the two-salt boundary and a concentration boundary, formed as described in the preceding paragraph. To illustrate the method of assessing volume effects,
let the electrode reaction be that of metal dissolution, with the two-salt boundary progressing up the channel, so that solution A is the leading solution. This is the usual arrangement for determining tl. On account of the dissolution process, the concentration of solution B near the electrode will be higher than its initial value. The solvent velocity vo in the solution below the twosalt boundary can be calculated by means of material balances over the region below a plane x fixed in the solvent (see Figure 2), allowance being made for the dissolution of the anode, the concentration dependence of the density of solution B, and transference of ions across the solvent plane. Consider a volume Vo bounded by the electrode and by the plane x moving with the solvent velocity vo. This volume changes with time as dVo
dt = d
z+F
(18)
Pe
d svdV =0 while a material ba’lanceon the electrolyte gives
Solution A
s,.
It+ z+F
I z+F
c+dV = - - -
The first term on the right represents the transport of cations across the solvent plane, and the second term represents the contribution of cations due to dissolution of the anode. An assumption is now made which allows for the explicit determination of the transference number. h s sume that the density of solution B below the solvent plane a t x is linearly related to its concentration
Solution 8
Solvent
-I IMO -
where A is the cross-sectional area of the channel, I is the total current (negative for an anode), and M e and pe are the atomic weight and density of the anode. The first term on the right represents the displacement of the solvent plane, and the second term represents the volume rate of dissolution of the anode. Since the amount of solvent in the volume Voremains constant, a material balance on the solvent yields
d 2 - salt boundary
svclV = -voA
X
PB
plane
=
LYB
+ PBCB
(21)
Since Concentrat ion bound a r y
/ I
this implies that
\1
E l e c t r o d e chamber Electrode
Figure 2. The closed electrode. The Journal of Physical Chemistry
(7) L. J. M. Smits and E. M. Duyvis, J. Phys. Chem., 70, 2747 (1966). (8) R. Haase, 2. Phys. Chem. (Frankfurt am Main), 39, 27 (1963). (9) iM.Spiro, J. Chem. Phys., 42, 4060 (1966). (10) L.G.Longsworth, J . Amer. Chem. Soc., 66, 449 (1944).
MOVINGBOUNDARY MEASUREMEXT OF TRANSFERENCE NUMBERS where Equation 19 then becomes
301
concentration equal to c and with E equal to either in the anode chamber or €2 across the concentration boundary, whichever is larger. If the initially prepared following solution is of higher concentration than that required by the Kohlrausch relation, one can, with little error, approximate e by el E~ and take c to be the initially prepared concentration (rather than the lowest concentration of solution B present). The value e2 due to the Kohlrausch adjustment must be experimentally minimized or measured. The part €1 due to the electrode reaction can be calculated from one of the following approximate equations.ll For a uniform concentration in the anode chamber (an unlikely situation)
+
Substitution of eq 18 and 20 then gives the desired relation for vo
or
€1
where V is the volume through which the boundary moves when current I is passed for time t. For eq 28 to be fruitfully applied to a system, eq 23 must be a good approximation. If the density of the following solution is given by p =
+ kzc +
kl
k3Cn/=
+ k4C2
(2%
then the solvent concentration is co = e
+ fc + gc*h + hC2
(30)
where e = ICl/Mo; f = (kz g =
k,/Mo; h
- M)/fwo} k4/Mo
=
v/u
(34) where U is the volume of the anode chamber. If the anode can be treated as a vertical electrode with free convection El
Finally, substitution of this equation into eq 7 with elimination of t+ (here ion 2 is the cation reacting at the electrode) by means of eq 8 yieldsll
=1.62(
c
)’.’(9 ) O o 2 -
Itz+v+FDA,
0.00932(I c
Z+v+Ae
(35)
where in the latter expression I is the current in mA, A , is the effective electrode area in cm2, c is in mol/l., L is the effective electrode height in cm, and p = (l/p)dp/dc is in l./mol. The latter expression is based on a diffusion coefficient D of loe5 cm2/sec, a transference number t- of 0.5, a kinematic viscosity v of loW2cm2/sec, and a gravitational acceleration g of lo3cm/sec2. For a horizontal electrode
(31)
rather than eq 23. A least-squares fit of eq 23 to eq 30 over the concentration range of interest, from c to (1 E)C, yields
+
where in the latter expression A , is in cm2,I is in mA, ‘V is in cm3, c is in mol/l., and the value t- = 0.5 has been assumed.
Discussion
1
+ +6 B
(32)
and b =f
+ 32 g P [ 1 + 1 + 0 ( 4 ] + 2hc(l + ;e) 4 E
(33)
where 0 indicates the order of neglected terms.
Application If the following solution is initially prepared so that its concentration is less than that required by the Kohlrausch relation, eq 17, then eq 32 and 33 are immediately applicable with the initially prepared value of the
Equations 7 and 8 are exact relations describing the two-salt boundary. Equation 28, relating the transference number to the experimentally measured quantities, depends additionally only on the very good approximation of a linear concentration dependence of the density of solution B over the range of concentrations encountered between the electrode and the two-salt boundary. Equation 28 is derived without ever referring to the details of the two-salt boundary itself. I n particular, no assumptions are made concerning the partial molal volume of the solvent through the boundary or of the density variation within the boundary. (11) P. Milios, “A Theoretical Analysis of the Moving Boundary Measurement of Transference Numbers,” Master’s Thesis, University of California, Berkeley, Sept 1967 (UCRL-17807). Volume 73,Number 8 FebruaTy 1060
302 For the system being analyzed, the equation of the type derived by Lewis1v3is
It can be shown” that
where a A is given by an equation analogous to eq 23. Thus eq 28 reduces to a Lewis-type equation in the special case where the partial molal volume of the solvent is constant through the boundary, precisely the assumed situation in the derivation of the Lewis-type equation, I n the present analysis a consequence of the assumed concentration-density relationship below the two-salt boundary is that the partial molal volume of the solvent below the boundary is constant. The distinction to be made is that this consequence refers to “below” the boundary while the assumption associated with the Lewis-type equation refers to “through” the boundary. I n the general case it is not necessary to consider the partial molal volume of the solvent in the analysis of the two-salt boundary. In applications, eq 28 is not very sensitive to the value of E, but it is of course better to keep e small. For most situations encountered, eq 35 and 36 indicate that the use of a vertical anode results in an appreciably smaller value of €1 than a horizontal electrode. Then the linear approximation, eq 23, is better, and U B is more accurately known. This should be expected since with a vertical electrode there is a tendency toward a uniform concentration in the anode chamber (the optimum case), while with the horizontal electrode stratified layers of solution form with very little mixing. Equation 36 suggests that an autogenic boundary (one formed with solution A initially adjacent to the electrode surface) should not be used since then €1 is very large-either A , is small or V is large-and the analysis is not applicable. I n the derivation of eq 28, it was assumed that ion 2 is the cation reacting at the electrode. If, on the other hand, the common ion 3 is positive, eq 28 should be replaced by
where V b and i now have opposite signs. If the boundary moves down the channel, eq 28 or 39 is still applicable, but now the concentration boundary is above the two-salt boundary, and ion 2 is the leading ion. If the electrode operates as a cathode, €1 should be taken to be negative. The analysis treats an electrode involving metal dissolution or deposition. The analysis can be modified The Journal of Physical Chemisirzl
PAULMILIOSAND JOHN NEWMAN to treat other cases with well-defined volume changes a t the electrode. The transference number of the leading ion can always be determined more accurately than that of the following ion, since the concentration is known. The transference number of the following ion can be calculated from eq 17 if the concentration difference across the concentration boundary is determined, say, in the manner of Longsworth.lo The uncertainty in the concentration so determined must be added to the uncertainty in the transference number of the leading ion in order to determine the uncertainty in the following-ion transference number.
Experimental Section The transference number of 0.213 M NH4N03a t 25” was determined” with AgN03 as the following solution. A standard Tiselius cell with a closed vertical silver anode was used. Equation 32 was used to calculate U B , with eq 35 being used to calculate €1 and e2 being measured by a Rayleigh interference optical method. All other procedures were standard methods for movingboundary measurements. Equation 28 yielded the result t+ = 0.5140 k 0.0024. Other than the value ti. = 0.5130 at 0.1 M determined by MacInnes and Cowperthwaite,12 this is the only value measured for XH4;“\T03. The value a t infinite dilution, calculated from the limiting ionic mobilities of the ions,13is 0.507. Equation 17 was used to estimate the cation transference number of the following AgN03 solution. The result t+ = 0.467 rt 0.009 at a concentration of 0.195 A4 can be compared with the work of Haase, et ~ l . , ~ 0.4708 and 0.4723 (two determinations) at 0.1995 M , calculated from a Lewis-type equation. If reasonable values are assumed for the concentration range of the following solution used by Haase, his data can be reevaluated with eq 28 to yield the results 0.4693 and 0.4708, respectively. Summary An equation for the transference number, applicable a t high as well as low concentrations, was presented. The difference between the present equation and the approximate equation now in common use lies in the calculation of the volume changes which accompany transference number measurements. The equation in common use was originally arrived a t by an intuitive approach and accounts for volume changes across the two-salt boundary. Approximations are thus introduced which limit the usefulness of the equation to dilute solutions. The present equation is derived from fundamental equations. It is seen that once a steady-state concen(12) D.A. MacInnes and I . A. Cowperthwaite, Trans. Faraday SOC., 23, 400 (1927). (13) G. Milazzo, “Electrochemistry,” Elsevier Publishing Co., Amsterdam, 1963.
SORPTIONOF WATERVAPORBY GENETICVARIANTSOF c ~ , ~ - C A S E ~ N tration profile is established in the two-salt boundary, no changes take place across the two-salt boundary which affect the determination of the transference number. Rather, the volume corrections are dependent only upon what takes place in the region from the electrode in the closed electrode assembly to, but not including, the two-salt boundary. The approximation is made that, in this region, the concentration of solvent is linear with respect to the concentration of solute, co = a bc. This approximation does not greatly limit the usefulness of the transference number equation. Equations are presented which allow the calculation of the quantities a and b if the density of the solution is known as a function of concentration and the concentrations on both sides of the Kohlrausch concentration boundary are known. The transference number of 0.213 M NH4NOawas measured a t 25".
Faraday's constant, 96,493 C/equiv Acceleration of gravity, cm/sec2 Current density, A/cm2 Current, A Defined by eq 29
+
Acknowledgment. This work was supported by the U. S. Atomic Energy Commission.
vi X
zi P E
Nomenclature Defined by eq 23 Area of channel, emz Surface area of electrode, em2 bB Defined by eq 23 Ci Concentration of species i, mol/cm8 CT Total solution concentration, mol/cma D Diffusion coefficient of salt, cme/sec asi Diffusion coefficient for binary interactions, cma/sec e, f, g, h Defined by eq 30 and 31 aB
A A,
303
Pi V
V+,
V-
P Pe
Electrode height, em Molecular weight of substance i, g/mol Universal gas constant, J/mol-deg Time, see Transference number of species i Transference number of species i as calculated using a Lewis-type volume correction, see eq 37 Temperature, OK Volume of anode chamber, cm3 Velocity of the two-salt boundary, cm/sec Velocity of species i, cm/sec Volume through which the two-salt boundary moves, em3 Partial molal volume of component i, crna/mol Distance variable, em Charge number of species i Coefficient of expansion, cma/mol Range of concentrations of solution B present below the two-salt boundary, relative to the lowest concentration in that range Relative concentration range in electrode chamber Relative concentration change across concentration boundary Electrochemical potential of species i, J/mol Kinematic viscosity, cmZ/sec Numbers of cations and anions produced by the dissociation of one molecule of electrolyte Density of solution, g/cm3 Density of electrode metal, g/cma
Sorption of Water Vapor and of Nitrogen by Genetic Variants of Ctst-Casein
by E. Berlin, B. A. Anderson, and M. J. Pallansch Dairy Products Laboratory, Eastern Utilization Research and Development Division, Agricultural Research Service, U . S . Department of Agriculture, Washington, D . C. 20260
(Received April 16, 1968)
Water vapor sorption b y dried preparations of a,l-caseins A and C was measured at 24.2'. A t lower relative pressures, both forms sorbed identical amounts of water, while at intermediate to higher relative pressures the A variant sorbed more water. A reproducible hysteresis loop was observed over the entire isotherm through two successive sorption-desorption cycles. Treatment of the Hi0 sorption data and nitrogen adsorption data according to the Frenkel-Halsey-Hill isotherin equation allowed for clear distinction between the natures of the sorption processes involved with HzO and Nz.
Introduction It is widely accepted that proteins sorb water vapor by binding water molecules to specific hydrophilic sites at lower relative humidities followed by condensation or multimolecular adsorption as the humidity increases.1a2 The nature Of these sites has, however, been the subject of much research, and there is no general
agreement a t present concerning the chemical groupings in proteins to which water is bound.3 The nature of these binding sites has been studied indirectly by com(1) L. Pauling, J . Amer. Chem. SOC., 67, 566 (1946). (2) A. D. MoLaren and J. W. Rowen, J . Polym. Sci., 7, 289 (1951). (3) F. Haurowita, $
