P. R ~ A Z U R
Vol. 58
THERMOPOTENTIALS I N THERMOCELLS BYP. MAZUR' University of Maryland, College Park, A4d. Received J u l y 28, 1064
The thermodynamics of irreversible processes is applied to thermocells. The results of the theory are discussed in connect>ionwith the experimental data. Some general relations between thermopotentials of thermocells and th'e temperature coeffictient for kothermal cells are given.
1. Introduction.-Eastmanza and Wagner2b have discussed the thermopotential of thermocells in a way analogous to Thompson's treatment of the metallic thermocouple, that is, by a so-called '' quasi" thermostatic method. Recently Holtan, de Groot and the author3,* have given a treatment for the thermopotentisls of thermocells along t'he lines of the thermodynamics of irreversible processes.5tG It is our purpose to summarize this treaSment here and discuss the equations obtained in connection with experimental results. A detailed account of the comparison between theory and experiment has been published by Holtan elsewhere.' We will consider systems consisting of two electrodes, connected to a precision compensator, and in contact wit,h an electrically conducting medium, For simplicity's sake we will consider only pure thermocells, i.e., thermocells with identical electrodes. During the experiment the electrodes are kept a t different temperature. The thermopotentials are measured at zero current passing through the cell, and before any appreciable concentration gradients have been, set up due to the Soret effect. 2. The Homogeneous Thermopotentia1.-The general expression for the entropy source strength Q in a system of n components, in mechanical equilibrium, (IC = 1, .. ., n), excluding viscous phenomena and chemical reactions can be shown to be5
Jq Fk
=
- (VPk)T
-1Jfqq
=
+
(VT)/T
-Jq(vT)/T
+ x2=1Jk[Fk -
(VPk
)T)
(1)
cr=l
(VT)/T f
-dfkq
Afki J i
(2)
(3)
with the following Onsager reciprocal relations amongst the coefficients =
M k i
=
jfqk
(4) (5)
M i k -Mkq
As can be shown, these coefficients also satisfy the relations
c$-l
0
Pk M k q
ci,l Pk d f k i
(6)
('7)
= 0
where Pk is the density of component k. It is clear from (2) that Mqk represents the heat transferred a t uniform temperature with the unit of mass of component k. Therefore the kfqk are identical with the so-called heats of transfer &k* and must, according to ( 5 ) and (6), satisfy the relation 0
P k &k*
(8)
For the'system under consideration F k is given by F k
=
(9)
- e k V q
where ek is the charge per unit mass of k, and (o the electrical potential. With ( 5 ) and (9), equation (3) can be rewritten as VP =
ek-'(vPk)T
- f'k-'
1 dfkiJi
,
TLT =
x$=ldfqkJk,
ek-'
&k*(VT)/T
(10)
Let us now introduce the partial electric current density due to the kth charged component
Here T is the absolute temperature, J , the soI k = CkJk (11) called reduced heat flowG,and Jk the flow density of component k with respect to an arbit,rary refer- and the total electric current density ence frame. The external force per unit mass actI =~ ~ , l z k (12) ing on component k is denoted by F k , whereas pk is the chemical potential of k taken per unit mass. Let us also introduce the transference numbers t h The subscript T in (1) indicates that the gradient defined as the ratios of the partial currents Ik of p k must. be taken at uniform temperature. and the total current I , at uniform temperature and The phenomenological equations describing the chemical potentials irreversible phenomena occurring in t,he syst,em tk = (Ik/I)VT=O;Vpk=O; c F = 1 tk = 1 (13) may then be written as (1) On leave of absence from t h e University of Utrecht, NetherFrom (10) one finds, for uniform temperature and lands. uniform chemical potentials E. D. Eastrnan, .I. A m . Chem. Soc., 48, 1482 (1926); 49, 794 (2) (a) (1927); (b) C. Wagner, dnn. phys., 151 3, 629 (1930); I51 6 . 370 (1930). (3) H. Holtan ,Jr., P. M a z u r and S. R . de Groot, Physica 19, 1109 (1953). (4) H. Holtan J r . , Thesis, Utreclit, 1953. ( 5 ) S. R . de Groot. "Therlllodynarllics of Irreversible Processes," North Holland Publ. Cornu., rlmsterdaln, 1951. (6) I. Prigogine, "Etude Tl~er~nodynal~liclr~e des PhPinornBnes Irr6versibles," Desoer, LiBge, 1917. (7) H. Holtan, Jr., Proc. K o n . N e d . Akad. 7 1 . W e t . Amslerdam, BS6, 498 (1953): BS6, 510 (1953).
cy=l ek-1
ei-'
dfki
ti = -vq/Z
=R
(14)
where R is the coefficient of electrical conductance. We can now multiply both members of (10) by tk and sum over k. We then obtain with (4), (11), (12) and (14) vq =
-
1 tkek-'
(VPk)T
-
tk ek-'
&k*(VT)/T
- RI
(15)
THERMOPOTENTIALS IN THERMOCELLS
Sept., 1954
With the conditions of vanishing total electric current ( I = 0) and vanishing concentration gradients ( v p k = O), we finally obtain the following expression for the homogeneous thermopotential. FVp
-
x;,llkZk-lQk’*
(v‘T)/T
(16)
where &k’* is the molar heat of trhsfer, 8k the molar charge in faraday’s and F the faraday number. Formula (16) has been derived with transport quantities tk and &’* with respect to an arbitrary reference frame. In the following, they will be defined with respect to a frame in which the electrodes of the thermocell are a t rest. The homogeneous thermopotential in the metallic wire connecting the electrodes through the compensator is a special case of (16) and is given by F
V P = Qei’*
(VT)/T
(17)
where Q d l * is the heat of transfer of the electrons in the wire. 3. The Heterogeneous Electrode Potential Difference.-The electrical potential difference a t the interface between the electrodes (phase 1) and the electrolytic medium (phase 2) in the state of vanishing electric current is given, as in ordinary thermostatics, by the condition that the electrochemical affinity of the electrode reaction equals zero. We can therefore write for the potential diff erelice a t a junction between electrode and medium F(ppl
- VPZ)= AG
(18)
where AG is the change in Gibbs free energy, due to the electrode reaction, when one faraday of electricity passcs from the electrode to the medium. For thermocells with identical electrodes, one a t temperature T, and the other atJ T AT, the resulting heterogeneous thermopotential is given by
+
FAphet = AGT+AT - A &
=
-ASAT
(19)
Here AS is the change of entropy due to the electrode reaction, when one faraday of electricity has passed through the cell. As in 5 2, it has been assumed in (19) that concentration gradients do not occur in the medium between the electrodes. 4. The Total Thermopotential of a Thermocell. -The resulting potential of a pure thermocell can be obtained by adding up the homogeneous and heterogeneous potential differences around the circuit Aqtot
=
A(Pmed
-I- A p m e t -I-
AVhet
(20)
ApLOt is the total potential measured in the ex-
periment, “med” refers to the medium, “met” to the metallic wire. From ( l G ) , (17), (19) and (20) one finds for the total thermopotential of the cell F A V ~ , ~ / A=T - x z , l t k
Zk-‘&kl*/T
-
Qei’*/T - A S ( 2 1 )
Again, of course, this formula is valid only when no concentration gradients occur in the medium. Introducing the so-called entropies of transfer S* defined by * S k * = (&kl*
f 2’sk)/2’
(22)
701
where Sk is the partial molar eiitropy of componeiit IC, equation 21 can be rewritten in the form FAp/AT =
-
lk Zk-’Sk*
- Asel* f
xi=ltkZk-lSk
- SeI
(23)
It can easily be verified that, when A S is evaluated for each special case, the last ‘three terms on the right-hand side of (23) will not contain ion entropies, which are iiot known, but only known salt entropies. As an example consider the cell with solid electrolyte Pb; PbCln; Pb 1‘ 2’ A2’
+
In this case - A S equals -AS = 4
SPb
- 4 Spbtf
-
se1
(24)
Since lead chloride is a pure anion coilductor, the transference number of the anion is equal to unity. Equation (23) then reads FAq/AY’ = S d - * - S d * -k S p b - 8 S P ~ C( 2I5~ ) The right-hand side of this expression contains, apart from the eittropies of transfer, only the partial molar entropies of lead and lead chloride. As another example and special case of (23) consider the metallic thermocouple Mea; Mea; Mea T T A2’
+
We have here simply -AS = AS^ - SS’J
(26)
aitd therefore from (23) FAQ/A’~’ = AS,,*
- bSel*
(27)
that is, the usual expression for the thermopotential of a metallic thermoco~ple.~ Turning back to formula (23), we thus see that the sum of the last three terms on the right-hand side contains only known quantities (transference iiumber and partial molar entropies). Let us for convenience denote this sum, which can be calculated, by F(A(p/OT),, so that F ( A ~ / A T )= F ( A ~ / A T ) , -
cF,l k z k - ’ S k * - S e i *
(28)
Comparison of (Ap/AT), with the experimental results Ap/AT therefore yields information concerning the entropies of transfer S k * and s d * . Holtan performed such a comparison for a number of cells. It turned out that for most thermocells with solid electrolyte, which were considered, the calculated values coincide with the experimental ones within the limits of experimental This seems to indicate that the last two terms of (28). are of negligible magnitude in these cases. For thermocells containing electrolytic solutions such a result has not been found. It should be noted however that the hydrated ions in the electrolytic solutions must be considered 8s components, as defined in the derivation of expression (23). For the calculation of ( ArplAT), in this case, one therefore needs data concerning the transport of solvent during electrolysis. Since the available data are riot accurate enough, no tlefinitc conclusions concerning the magnitude of
702
M. EIGENAND E. WICKE
the terms containing entropies of transfer can be drawn. 5. Relations between Thermocells and Isothermal Cells.-The formalism developed in the previous paragraphs enables us to derive some general relations between thermopotentials of thermocells, and the t
