SIXGLE ELECTRODE ~'OTENTIALS
2959
Single Electrode Potentials
by Irwin Oppenheim Department of Chemistry, Massachusetts Institute of Technology Cambridge, Massachusetta (Received A p r i l SO, 1964)
A nonthermodynaniic technique for measuring absolute values of single electrode potentials is proposed. The technique involves the measurement of the quadrupole radiation emitted by an electrode which is made to execute hariiionic motion by niechanical or ultrasonic means. Thus, the concept of a single electrode potential becomes operationally meaningful.
Introduction Therc has been for. many years a controversy concerning the operational significance of single electrode potentials. Indeed, single electrode potentials cannot be determined by thermodynaniic type measurements. Thus, in a sense, the nieasurenient of a single electrode potential does not yield theriiiodynaniically useful information. Since charge cannot be transported independently of iiiass (ie.,it is always associated with electrons, ions, etc.), the work done in transporting a charge across a phase boundary consists of two parts: electrical work against the electric field and nonelectrical work against the electromotive force due to the difference in composition of the two phases. Therefore, the evaluation of the difference of electrostatic potential between two phases, diff ering in composition, cannot be determined by a charge-transfer experiment. This is in distinction to the nieasurenient of the electrostatic potential at any point in a honiogeneous systcni by determining the electrostatic work done in transporting a unit test charge from infinity t o the point in question. It is, of course, a siniple matter to determine the difference in electrostatic potential between the electrodes of a galvanic cell, and the single electrode potentials which are often listed in the literature are actually potentials of the given electrode relative to the nornial hydrogen elcctrode as a standard. We take the point of view that the electrostatic potential a t any point in space in any medium is determined by the charge distribution over all space. Thus, if there exists a n experimental technique which ~iicasures the charge distribution, the elcctrostatic potential is deterniined and beconies operationally significant. In this paper, we propose an experimental
method for measuring the relevant charge distribution which determines a single elcctrode potential. The experiment is described and the analysis of thc experinient is sketched. The analysis is not carried through in complete detail because of coinplicated geonietric effects. However, it is carried through far enough to demonstrate that single electrode potentials can be measured at least in principle. We note that it is, of course, sufficient to iiieasurc thk absolute potential of only one electrode since the potentials of all other electrodes can be deterniined by usual techniques using that electrode as a standard.
Model of a Single Electrode WC consider, for definiteness, a half-cell which consists of a nietallic electrode A 1 partially ininiersed in an electrolyte solution JIX which contains the metallic 1.. We denotc the nietallic electrode as phase ion 1 (1) and the clectrolyte solution as phase ( 2 ) . The phase boundary is inipernieable to sonie charge carriers. The constaril electrostatic potential in the interior of the nietal is +' and the constant electrostatic potential in the interior of the electrolyte is @. The experiment that we shall propose is capable of nicasuring + I and/or $1
- $2.
We assume that the chargc distribution which gives rise to the electrostatic potential difference + I - +z is a double layer a t the surface of the electrode ininiersed in the electrolyte. The electrostatic potential + I can be deteriiiiiied by a nieasurenient of the double layer a t the surface of the elcctrode which projects above the surface of the clectrolyte solution. Thus, by a nieasurenient of the strengths of the doublr layers andlor +' - +2 can be deterniined. Either or d1 - +2 niay be considered to be the singlr
V o l u m e 68, .Yumber 10 Oclober, 1,984
IRWIN OPPEXIIEIM
2960
electrode potential depending upon which convention one wishes to use.
Description of Proposed Experiment We proposc to ineasurc the strengths of the double layers described above by observation of the quadrupole radiation emitted when the double layers are accelerated. The half-cell is made to execute siiiiple harmonic motion perpendicular t o one of the faces of the electrode. The oscillation is produced either niechanically or by using ultrasonic tcchniques. The oscillating double layers enlit yuadrupole radiation which is characterized by its angular dependence. The intensity of the radiation can be measured a t least in principle by properly tuned receivers. The frequency of the radiation is the same as the frequency of the oscillation. The quantities which inust be iiieasurcd in order to determine the strength of the double layers are the intensity and frequency of the radiation, the amplitude of the oscillation, and the pertinent geometric factors. A complete analysis of the expeiiiiient would also require the description of the hydrodynamic effects set up by the oscillation of the double layer. These effects can be predicted, a t least in principle, for given experimental conditions and for a given model of the double layer.
where i is a unit vector along the z-axis. of eq. 3 into eq. 2 yields
Substitution
where the syinbol (a, b , c) stands for a vector with .ccomponent equal to a, y-component equal to b , and z-component equal to c. The electric field vector &(ii,t) a t a t time t due to a dipole with dipole nioinent 1728 which executes simple harinonic niotion along the z-axis centered a t the origin with aniplitude A and angular frequency w is easily obtained from ey. 4 in the forin
-
PEED
=
-m
bE - --
dx
Analysis of the Experiment An accelerating point charge e gives rise to a n electric field vector E’ of the form
-
4TE - E(G,t) = e
-
I ~ - - - - I
aa
-
dz
0 3
=
x
-A - c r
COS
~ ( -t
T/C)
(6)
where E is the dielectric constant of the medium, -+ r (t’) is the time dependent distance between the point of obspivation and the accelerating charge Fa, u(t’) is the time dependent velocity of the charge, G(t’) is the acceleration of the charge, t’ = t - r / c is the retarded time, t is the time of observation, and c is the vclocity of light. If u/c < < 1,me can neglect, rclativistic effects and eq. 1 siniplifies to
We next consider a dipole layer with dipole inoinent per unit area ~i which executes simple harnionic niotion in the z-direction centered a t the origin with aniplitude il and angular frequency w . The z-axis passes through the cent cr of the dipole layer. The vector position of the point of observation
