T. Fprland, 1. U. Thulin,
and T. fisfvold Institute of Physical Chemistry 'University of Trondheim
I
Concentration Cells with Liquid Junction
The concept of L1singleelectrode potential" is commonly used in physical chemical calculations where the emf of a galvanic cell is calculated as a difference between the separately calculated potentials of the two electrodes. Eventually a "liquid junction potential" is calculated when a concentration cell is considered and added to give the total emf of the galvanic cell. If the two compartments of the cell are separated by a semipermeable membrane, altogether five separate potentials are added to give the emf of the cell. None of these can he measured separately. Only the outer emf is obtainable by measurements. This use of unmeasurable quantities in the derivations causes unnecessary problems for the understanding of emf calculations. Another unmeasureable quantity which is connected to the single electrode potential, is the single ion activity or the chemical potential of a single ion. Only the chemical potential of a neutral salt is obtainable by ordinary thermodynamic measurements, as the process which takes place by the measurements will imply the transfer of a neutral component from one state to another. The transfer of charged particles, if not compensated for, would create electrical charges in the container of the electrolyte solution. The free energy of transfer urould then depend, among other factors, on the geometry of the container and the dielectric constant of the container material. It is generally not practical to operate with thermodynamic properties of a solution that depend on the properties of the container. (However, the concept may be of some value in connection with electrode kinetic processes.) Further, it is well known that the energy due to interaction between positive and negative particles is expressed as a total energy for both kinds of particles together and can only formally be divided between the two kinds of particles. The concept of difference in electric potential between two points in an isothermal system implies some of the same problems. One may define this difference as the work required to transport a unit positive charge under isothermal reversible conditions from the one point to the other. When this definition is to he connected to a measurement, the type of particle has to be specified. It could be, e.g., a proton (or an electron transported the opposite way). If however, the local surroundings of the particle at the two points are different, that is, the transport is along an inhomogeneous substance, the work will depend on the local binding forces at the two points and therefore on the kind of particle transferred. The difference in electrical potential from one point (1) to another point (2) in a homogeneous isothermal conductor (Fig. 1) may be connected to a well defined measurement.
v
Direction of c h a r g e t r a n s f e r Figure 1
Hornogsneour isothermal conductor.
Suppose electrons are transported from left to right in the conductor in Figure 1. The charge transported per unit time, I, can be measured using a coulombmeter. The electric energy per unit time, dE/dt, transformed to heat over the regionfrom (1) to (2) of the conductor can be measured using a calorimeter. The difference in potential between (1) and (2) is defined as
This reasoning would also apply to an inhomogeneous conductor where the charge transfer does not cause any change in composition. If the conductor is inhomogeneous and contains different carriers of charge and their contribution to charge transport vary along the conductor, the charge transfer will cause changes in composition or changes in quantities of the different phases present. Thus the Gibbs free energy of the conductor may increase or decrease by the charge transport. In this case the electric energy lost in a section of the conductor is not equal to the heat evolved. This loss of energy is partly converted to chemical energy, which can he determined knowing the content of each neutral component and their chemical potentials, p . The electric energy lost per unit time may also, as before, he expressed by the product A* X I, which defines, A*, the difference in electric potential in the inhomogeneous and isothermal conductor. EMF Calculation Using Measurable Quantities Only
We will now consider a simple isothermal galvanic cell A d s ) I AgCKs) I HCI(C,) I I HCI(C2) I AgCKs) I A d s )
as shown in Figure 2. (Compare derivations given in ref. ( 1 ) and (s).) The cell is a concentration cell and the concentration change takes place within the region from (1) to (2). The emf of the cell can be measured. using a potentiometer or, what amounts to the same, using a very high resistance, R in the circuit and measure the heat produced per unit time, RIZ,where I is Volume 48, Number 1 1 , November 1971
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741
where q is the area of the crossection. In a similar way one can express the change in content of the second component, A d n ~ , ~by , the deriv~r tive of the flux J q o .
The fluxes of components and flux of electric charge, I, can according to the postulate of irreversible thermodynamics be expressed as homogeneous linear functions of the correspondingforces: the gradients of the chemical potentials VPEOLand Vp,o and the gradient of electric potential V*.
Figure 2.
A sehern~tkdiagram of the Isothermal gdvenlc cell Aglsll.
AgClld~HCIlC~~~~HC1lGl~AgCIlsl~Agl1) with a plot of tho HCI consentrotion renw position, y, through the cell.
the electric current which is also measured separately. Thus the emf, RI, is obtained. The current drawn from the cell has in either case to be so small that no significant polarization occurs at the electrodes. By the transfer of charge through the cell, changes will occur in the content of HC1 in the different parts of the cell. These changes can be determined, e.g., analytically in a transfer experiment of the Hittorf type, and the result is generally expressed in terms of ionic transport numbers. In principle, if we treat the processes takng place by using the laws of thermodynamics and of irreversible thermodynamics, we do not have to know any details about the strnctural units of the system. We could express concentration changes using ionic transport numbers, or just as well express the changes in composition found from a Hittorf experiment by the use of transport coefficients for the neutral component HCl. (In the Hittorf experiment all movements of components refer to the water as a frame of reference, thus the transport number of water is zero.) For a certain amount of electric charge being transferred certain changes in HCI content are observed. To correlate these changes with the electric work in the outer circuit (or the emf), no knowledge is needed about ionic or other structural species and their mobilities in the electrolyte. It seems therefore also logical not to introduce such concepts in the first part of the emf calculations. The changes taking place by the transfer of a small charge, A& Faradays, during a short time At can be expressed by changes in fluxes of the components contained in the cell. We may first consider the changes taking place in a narrow section of the electrolyte between the plane of the crossection a t y and the plane of the crossection at y dy in Figure 2. The frame of reference for all movements is the wall of the container. If we denote the flux of the component HC1 by I,,, (moles/cm2sec), the change in HCl content between the two planes of crossection during the time At will be
+
where I = (AQ/At) is expressed in Faradays per cm' per sec and where L with subscripts is a phenomenological coefficient which is independent of the forces. (See, e.g., ref. (S).) One may right away identify some of the coefficients. Since the phenomenological coefficients are independent of the forces we may consider the case when V,.,HO, = Vpalo = 0 to identify the ratios Lx8/Ls3 and Lz3/Lsa. These ratios are obtained by dividing eqns. (3) and (4) by (5). We thus have
and
where we also have defined the new transport coefficients taoland taZ0. Further -LS3 can be identified as the specific electric conductivity K . By introducing the Onsager reciprocal relations (3) - L..I. L.. .J eqn. (5) may be written as
The right hand side of eqn. (8) contains well defined measureable quantities and can be used to define the quantity V*. The expression for V* may now be introduced in eqns. (3)and (4) giving and
+
These equations can be differentiated with respect to y and the result introduced in eqns. (1) and (2) to obtain the changes in composition in the small volume between Y andy dy.
+
and 742
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Journal of Chemical Education
+
JEZO = l(tao~Ln- LLI)V!JECI (ta,obs - L~dVfia~oI ta,d
E may here be given in units corresponding to the units = ji(a,Va)q
d y At
- dta,a
AQ
(10)
where jl(a,Va) and ji(a,Va) are functions of the activity of the electrolyte and of the gradients in this activity. The first term of eqns. (9) and (10) containing the time, At, (and not AQ) is due to diffusion and the second term containing AQ (and not A t ) is due to the charge transfer. The change in Gibbs free energy in the small volume between y and y dy is
+
AdG
=
+ rra,oAdna,o
pmAdnxm
(11)
and the change in Gibbs free energy over the whole volume containing the concentration gradient is
+
AG = ~ b E c ~ A d n x c l m 9 o A d m z o )
(12)
(1)
As the eqns. (9) and (10) consisted of two terms, one containing At and one containing A&, AG will contain two terms AG
=
AGct,
+ AG'CQI
(13)
containing respectively At and A&. By combining eqns. (9) and (10) with eqn. (12) we obtain
of chemical potential, that means in joule per Faraday. To calculate AGntQ)one has to know the changes in composition by the charge transfer, which is generally found by passing a charge through a cell having no concentration gradient and then determining the change in HC1 content by chemical analysis. By this type of experiment (the Hittorf experiment) one finds how HCI (or H + and C1-) have been moved with respect to water. So our frame of reference is no longer the walls of the container, but the HiO molecules. Using this frame of refei.ence ta,o = 0 and dt,o = 0. The result of this experiment is generally expressed in terms of ionic transport numbers. Thus the increase in HC1 content in the right hand side and decrease in the left hand side of the cell (Fig. 2) per Faraday passed is denoted by t ~ + . In the present case this will he the same as tacl previously defined (eqn. (6)). tacl is the number of moles of the component HC1 which from the chemical analysis can be considered transferred from left to right per Faraday passed. Details about how the transport is carried out do not have to be known in order to calculate the emf. Equation (16) may then be changed to E
=
=
-
(r
AGt.t.1
=
AGw
+ AQAG"m - AQ
=
AGO,
+ AG~QI
s"
+
( p ~ o ~ d t a c l m,odtx,a)
(1)
(14)
where AG"(Q) is the change in Gibbs free energy per Faraday transferred in the region close to the two electrodes, and A G i Q ) is the total change in Gibbs free energy due to charge transferred. The change in Gibbs energy AGtQ, occurs when a charge A Q is transferred and the outer electric work E A Q is produced, where E is the emf (when the potentiometer e'mf is slightly below the balance point). Conversely applying the electric energy EAQ from the potentiometer by transferring a charge AQ the opposite way (when the potentiometer emf is slightly above the balance point). This will cause the change in Gibbs energy -AG(Q) in the system. Thus A G i Q ) is the change in Gihbs energy which can be converted to electric work EAQ in the outer circuit. Or and
~ E C I ~ ~ H C I
(1)
i'
+
PEC~~HCI PEOI~~HO =I (1)
(1)
I n addition to the calculated change in Gibbs free energy in the concentration gradient there will be a change in Gibbs free energy due to changes taking place in the region close to the two electrodes, and this change is proportional to the charge transferred A&. One can write for the total change in Gihbs free energy in the cell
i' i'
- ( ~ a c ~ i q t x o-w ~p a c w , C c ~ o ~+) -
(17)
txcldp~c~
(1)
where the subscripts (1) and (2) denote the left and right hand side electrolyte respectively. If E is expressed in volt, the right hand side of eqn. (17) should be divided by Faradays number F. The result is the same as the well known expression for the emf of a concentration cell (see, e.g., the derivation in ref. (4) and (5)). The advantage of the present derivation is that it contains well defined measureable quantities only, and that it is simple to visualize that the emf originates from the transport of matter over a range where its chemical potential is changing. Comparison with A* from the Flux Equation
If we now go back to eqn. (8) for the electric potential gradient, we should also be able to obtain an expression for the emf by integration of this equation over the total length of the cell. By the potential measurements Iwill be extremely small, accordingly eqn. (8) becomes
as our transport number txCtagain refers to the water as frame of reference. We may choose the area of the crossection of the cell equal to 1 cm2, so dy can be replaced by dV, the volume between the crossection at y and y dy. We will now carry out the integration of eqn. (18) over the whole volume of the electrolyte
+
Am
=
S
d.P =
-
I
taoldpacl
(19)
over electrolyte
The integration will then include all changes taking Volume 48, Number 1 1 , November 1971
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743
place in the electrolyte of the cell. This also corresponds to the changes measured by the Hittorf experiment from which tm is determined. Equation (19) is identical to the expression for the total emf given by eqn. (17). This would imply that the contributions to the emf from the two electrodeelectrolyte interfaces must be equal and of opposite sign, and as the two electrodes are in contact with different electrolytes, it would he reasonable to assume that both interface potentials are zero, if one should deal with interface potentials at all. One has to be aware that the transport number taol in eqn. (19) depends on what reaction takes place a t the electrode. To see this better one may consider the cell (Pt)Hdg, 1 atm) / HCl(Cd I I HCl(Cn) I Hdg, 1 atm)(Pt)
By an analogous treatment one will obtain the following expression for the emf of the cell (2)
E, = -Jhc,d
!JHG,
(20)
(1)
The transport number of the component HCl is again obtained by passing positive current from left to right in a cell where c, = c,. One will from chemical analysis find that in this case the HC1 content of the left hand side of the cell is increasing, and that it is decreasing on the other side. This is opposite of what is found for the previous cell. The corresponding transfer of moles of HC1 per Faraday of charge transferred is generally denoted by tcl-, the transport number of C1-. I n our formalism the number of moles of HC1 transported per Faraday in the direction of the current is equal to t ' m = -to,-. Going back to eqn. (8), this equation can be written on the more general form for a multicomponent system
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for I = 0. In our formalizm we state according to this equation that the contribution to the emf from any region of the cell is given by the transport (coupled to charge transfer) of a component over a region with a gradient in chemical potential. When either Vrci or t , is zero, there is no contribution to the emf. There may be electrical double layers in a cell, e.g., at the electrode-electrolyte interfaces whose charge may depend on constituents or even minor impurities which do not contribute to the sum ZtiVpi. That means that such electric double layers do not contribute to the measured outer emf of the cell. They are connected to electrokinetic phenomena, and it creates confusion to consider them in connection with emf calculations. Conclusion
Instead of making the emphasis on undefined or irrelevant local electric potential differences in the cell by the calculation of an emf, ss is frequently done, we feel that the emphasis should be on the gradients of chemical potential of neutral components of the cell and on how the content of these components is changed by charge transfer, that means, on the coupling between transport of charge and transport of components. It is our belief that a derivation of the emf of galvanic cells based on measurable quantities only, will make it easier for the student to grasp this subject matter. Literature Cited (1) F ~ ~ G A T.. N DA, d o Cham. Soond.. 14, 1381 (1980). (2) F ~ B G A W T., Dw , o THOLIN, L. U.,A d a C h m . Scond., 22,3023 (1988). (3) Pnrooorae, I., "Introduotion to Thermodynamics of Irreversible Prcoesses," Interscience Publishers. New York, London. 1961. (4) PITSER, K. S.. m n BRewm, L., "Thermodynamics," McGrsw-HiU Book Company Inc., New York, 1961. (5) G ~ O O E N H E IE. M , A,. -Thelmodyn.mi08," North-Halland Publishing Co., Amsterdam, 1967.
