J. J. Campion Stote University College at New Paltz New Poltz, New York 12561
An Alternate Approach to Liquid Junction Potentials
The concept of the liquid junction potential (diffusion potential) is one of the most difficult t o understand in physical chemistry. I believe the difficulty lies in the failure to realize the interdependence of diffusion and conductance (transference). Consider the following Galvanic cell M/MX (1)JMX (11)lM
an
* F.
(t+d In au+ - t-d In ax-)
The general equation3can be written as
> a1
where MX is a uniunivalent electrolyte and ~ I and I a1 are the respective activities. The potential difference across the boundary arises from the different rates of diffusion of the ;\I+ and X- ions. I n the above example MX will diffusefrom region I1 to I, that is from the region of greater activity to the one of lower activity. If it is assumed that the M+ ion initially diffuses more rapidly than X-, then the left hand side of the diffusion boundary will be positive with respect to the right (see figure). This separation of charge (the liquid junction potential) decelerates the M+ ions and accelerates the X- ions until their velocities are equal.' The general equation for the liquid junction potential can be derived from both a thermodynamic and a diffusion approach. That this is so should not be surprising. The Nernst equation can be derived from electrode kinetics; equilibrium constants can be obtained from the law of mass action as well as thermodynamic approaches. Thc thermodynamic approach2 considers the boundary region to consist of an indefinite number of layers so thin that the variation of the solution's composition from one layer t o the next is infinitesimal. If in the above cell a faraday of current is reversibly passed from region I1 to I, t+ molcs of M+ will pass from left to right, and t- moles of X- will pass right t o left in a given layer. The change in potential (Gibbs frec energy) for this process will be dEj =
between the two solutions is obtained by integrating eqn. (I), therefore
(1)
where E j is the junction potential; R, the gas constant; T, the absolute temperature; F,, thevalueofthefaraday; and 1, the transference number. The junction potential
' Though diffusion is an irreversible process, it is assumed that it is such a slow process that departures from equilibrium are negligible. 2 M n c I ~ ~ r , :D. s , A,, "The Principles of Electrochemistry," Dover Publications, Inc., New York, 1961, Chap. 13. Implicit in eqn. (3) is the fact that solvent transport has been neglected. A,,, AND STOKES, R. H., "Electrolyte Solutions" I ~ O B I N ~R. ON (2nd ed.), Buttemorths Puhlieations, Ltd., London, 1959, Chap.
..
..
where 2, is the algebraic valence on the i t h ion and n is the number of differentions present. It is immediately evident that the thermodynamic derivation makes no use of the model in the figure, namely the separation of positive and negative charge. The failure t o consider the microscopic level is a limitation of all purely thermodynamic approaches. At the risk of overstating the case, the thermodynamic approach is an example of the logic of Plato, that is, an object isn't present unless I'm observing it; or junction potentials do not exist except when measurements are being made on certain types of Galvanic cells. If the model in the figure is indeed the easiest concention of the liouid "iunction notential then ean. (3) shbuld be derived by an approach consistent u&h t h i model. No recourse t o Galvanic cells should be necessary since the potential exists at the junction of two dissimilar folutions whether electrodes are present or not. The remainder of this paper is devoted to deriving eqn. (3) by an approach that has been applied t o diffusion coefficients. This approach, though consistent with the model, has the limitation of being mathematically more complex than the purely thermodynamic one. The equations used in the initial part of thc derivation were obtained from Robineon and stoke^.^ No attempt is made t o consider the relation of junction potentials t o single ion activities and the limitations these quantities present to measurements of standard potentials and an absolute pH scale since they are discussed el~ewhere.~The experimental difficulties in measuring liquid junctions6 and various attempts to evaluate eqn. (3) have been discussed by M a c I n n e ~ . ~ The simplest liquid junction involving the same electrolyte at different activities has already been mentioned. A more complicated junction involves different electrolytes at the same or different activities. For example
.
.11..
BATES,R. B., "Determination of pH Theory and Practice," John Wiley &Sons, Inc., New York, 1964, Chap. 3. 'For example potentials of cells containing junctions involving two or more different electrolytes depend on the manner in which the junctions are formed.
Diffusion Boundary Diagram of diffusion boundary.
Volume 49, Number 12, December 1972
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827
HBr (I)/KCl (11)
where KC1 will diffuse from right t o left and HBr from left t o right. The equation expressing the junction potential of this system will be derived from the diffusion approach. A particular system rather than a general one was chosen for reason of clarity. The chemical potential (partial molal free energy) of any electrolyte is given by Gxx =
+ Gx-
G,+
(4)
The force (F) on any ion (i) is equal t o the force on that ion due to a chemical potential gradient plus the effect of the electric potential (+) which retards the faster and accelerates the slower ions
where N is Avogadro's number and e is the protonic charge. I n the system under consideration the electroneutrality condition requires that OK+ m i = m - VCL(6) where v is the velocity of the ion. The velocity of any ion is equal to the product of the force on that ion times the ion's absolute mobility (u,') ". - F. ,Ui t (7)
+
+
1
Combination of eqns. (5), (6), and (7) with appropriate subscripts lead t o
The relationship between the absolute mobility (u,') (the velocity in cm/sec under a force of 1dyne) and the electrical mobility (ui) (the velocity under a force of 1 volt/cm) is u,' = u,/lZ,le. Substituting for the ahsolute mobility in eqn. (7) and making use of the relationship that the faraday (F,) = eN, we have
Since the transference number of any ion is (ti)
=
uiai
------
C uiai n
eqn. (9) may now be written in terms of transference numbers
The chemical potential gradient may be expressed in terms of single ion activities through the following relationship
Substituting eqn. (11) into eqn. (10) and integrating over the whole diffusion layer yields the expression for the liquid junction potential 828
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Journal o f Chemical Education
It can he seen that eqn. (12), which has been derived without any consideration of Galvanic cells, is a particular case of eqn. (3). Correlation of the Model with Results from Galvanic Cells
The potential developed at the junction of MX solutions at different activities may be calculated from the emf values of two concentration cells-one with transference (previously mentioned) and one without transference. The cell without transference might he
where E, is the potential of the cell with transference and E , is the potential of the cell without.? It is interesting t o consider eqn. (13) at fixed activities of the MX solutions for three different values of t+. I n the 1. For this first case assume t+ >> t - , that is t+ value of t+ eqn. (13) shows that E, is most negative, and E , is most positive compared t o E,. Stated somewhat differently, the maximum work available from the cell without transference is largest compared t o the cell with transference. With t+ essentially unity the left hand side of the diffusion boundary is positively charged while the right hand side is negatively charged (this will be the case for any value oft+ > 0.50, and the larger the value of t+ the greater the potential developed). The transfer of M + from region I to the left hand side of the diffusion boundary is opposed since that side of the boundary is positively charged. There is also opposition t o the transfer of ill+ from the negatively charged right hand side of the boundary t o region 11. I n transferring ill+ from region I to I1 appreciable amounts of work must be expended t o overcome the junction potential, and this expenditure of work lowers the maximum work available from the cell with transference. At the other extreme assume t- >> t+ or t+ N 0. I n this case assume E, is at its maximum positive value, and the cell with transference can perform more work than the cell without. Here the left hand side of the diffusion boundary is negatively charged while the right side is positively charged. The transfer of X- from region I1 to I is not opposed, but instead is favored by the junction potential. This situation can be considered as the junction potential performing work on the cell and thus increasing the maximum work available. The last case involves t+ = t where the junction potential is zero. Here no work is performed by thc cell or on the cell in transferring ions across the houndary, and the maximum work available from the two cells is identical.
-
7 To avoid misunderstanding it should be noted that Ers does not refer to the just mentioned concentration cell without transference, but rather to a hypothetical cell namely the original cell with transference if transference were not present. In actuality Ew equals the potential of the just mentioned concentration cell without transference divided by two.
