BENSON Ross SUNDHEIM AND JORDAND. KELLNER
1204
Thermoelectric Properties of the Molten Silver Nitrate-Sodium Nitrate System'
by Benson Ross Sundheim and Jordan D. Kellner Department of Chemistry, -Vew York C'niversity, Washington Square, New York, -\'eta York (Received October 6 , 1964)
10003
Xeasurements of the final thermoelectric potential (in the Soret steady state) are reported for the system AgX03-SaN03 over the range of composition between mole fraction of silver nitrate = 0.90 and 0.05. By combination of these results with the previously reported initial thermoelectric powers for this system, the various quantities of transports are calculated. The single ion entropy of the silver ion in pure silver nitrate is estimated a t 19.0 e.u., leading to a value for the entropy of transport of the silver ion which is quite small. The reduced heat transported across a number fixed reference plane in an isothermal diffusion experiment (Dufour effect) is found to be everywhere positive, ranging from 1.1 to 3.6 kcal./mole and displaying a pronounced minimum near 2 = 0.7. S o adequate basis for the theoretical interpretation of results of this kind seems to be extant.
Thermoelectric measurenients in electrolytic solutions are of considerable interest because of the light they may be expected to shed on transport mechanisms.2 Studies in fused salts are of particular interest because the absence of the usual dipolar solvent simplifies the system so as to present the ion motion in an especially direct way. Measurements of the initial thermoelectric potentials (uniform composition) have been previously reported3 for the molten AgN03N a X 0 3 system with silver electrodes. We report here on the determination of the final (Soret steady state) thermoelectric potentials for this system. By combining the two sets of data, the various quantities of transport of the system are obtained.
Experimental Procedure The design of the therniocell establishes a temperature gradient across a fine porosity sintered Pyrex disk whose upper surface is in contact with a large thermostated reservoir of the fused salt solution containing one electrode. The lower surface of the sintered disk is closed by a layer of silver which serves as the second elect rode and which is therniostated independently. Considerable care is required to make a lower electrode which is adherent and leak-free and to which a good electrical connection can be made. First a coating of silver was laid down on the disk by vacuum evaporation. This was then reinforced by electroplating more silver on it. Finally, the sealing of The Journal of Physical Chemistry
pinholes was accomplished by painting with a suspension of silver in amyl acetate, evaporating, and baking. A silver lead and a thermocouple junction were then mounted firmly against the surface and held in place by porcelain cenien t . The temperature of the lower electrode was established by an air thermostat and that of the upper one maintained a t a constant preselected difference from it by a differential thermocouple controller driving a saturable reactor which in turn operated the heater which was iniinersed in the reservoir. A thermocouple, corkscrew stirrer, and silver electrode completed the assembly in the upper compartment The temperature of the lower electrode was only approximately controlled since the steady-state therniopotentials are independent of the mean temperature over a range of a t least 20". The differential temperature controller maintained the temperature difference across the diaphragm to ~vithinseveral hundredths of a degree. The dried salts were weighed, niixed mechanically, and introduced into the cell. The assembly was in(1) Abstracted in part from a dissertatzon submitted bx J D Kellner to the Graduate School of Arts and Science of New York 1-niversitx in partial fulfillment of t h e requirements for the degree of Doctor of Philosophx (2) B R Sundheim, "Fused Salts." McGraw-Hi11 Book C o , Inc , New York. N Y 1964 Chapter 3 (3) R Schneebaum and B R Sundheim D z s c z ~ s s ~ o nFaraday s Sac 32, 197 (1961)
THERMOELECTRIC PROPERTIES OF MOLTENAgN03-NaN03
1205
Table I : Quantities of Transport for AgNOa-NaNOa a t 310'
-
Q1*.
-ic--
in-
.---------€'
cal./deg.
kcal.
...
..
156 205 239 248 242 215 149
3.60 4.74 5.50 5.72 5.5s 4.96 3.43
.. 2.09 1.56 1.18 1.12 1.21 1.75 3.61
21
rv./deg.
cal./deg.
pv./deg.
1.0 0.90 0.75 0.60 0.50 0.35 0.20 0.05
3 19 312 322 327 33 1 338 345 419
7.36 7.19 7.42 i.53 7.64 7.80 7.96 9.63
&I*
troduted into the furnace and held a t 310" to age the electrodes, since, a t this point, there was always a large e.ni.f., even a t zero temperature gradient. After about 18 hr., this base e.1n.f. was largely gone and the remaining asymmetry potential did not change much with time. After inserting the stirrer, thermocouple, and heater in the cell, the controller was energized and set for zero temperature difference. The cell was allowed to equilibrate for 1 hr. The silvercopper junctions from the electrodes were placed in an ice bath and the cell e.ni.f. recorded to the nearest microvolt. Successive temperature differences were applied to the cell and, after each, the cell was returned to zero teniperature difference to check the base e m f . The cell e.ni.f. us. the temperature difference was plotted and the best straight line drawn through the points by the method of least squares. ( S o systeniatic curvature was found.) The temperature difference ranged from 1 to 7". Seven different compositions were employed between 0.90 and 0.05 mole fraction of AgS03. The nieasurenients were repeated several times to establish an index of precision. The initial and final thermoelectric powers are given in Table I. The experimental method is capable, in principle, of following the approach to the steady state as a function of time. The curves of e.m.f. us. time were not used for the determination of quantities of transport because the initial temperature jump was not made sufficiently rapidly. However, their data were used as a rough check on the method by use of the characteristic time for the attainment of the Soret equilibrium. This quantity is the time required for the e.m.f. change to reach the fraction l / e of the steady-state value. It is possible to calculate the characteristic time from a knowledge of the ordinary diffusion coefficient and this value may be compared to the experimental one as a check on the internal consistency of the data. The approach to the steady state is exponential and is given by4
- Q2*, kcal.
... 18.8 4.69 1.78 1.12 0.65 0.44 0.14
-r*,
SAPi,
e.u.
SAa+. e.u.
-s*AgcI
kcal.
.. 17.9 19.0 19.8 20.0 19.9 19.3 17.7
19.0 19.2 19.6 20.0 20.4 21.1 22.2 25.0
3.1 1.3 0.5 0.1 0.3 1.2 2.9 7.2
..
2.10 2.76 3.21 3.34 3.23 2.89 2.0
( D ' / D ) ( l - exp[-t/B])
e.u.
=
s(t)
a, mole/deg.
...
0.0310 0.00926 0.0043 0,00332 0,00276 0.00324 0,00555
(1)
where D is the ordinary diffusion coefficient, 6 the characteristic time, and the measurement of the Soret coefficient, s, is made a t the time t. The characteristic time is4 6 = hz/n2D
(2)
where h is the cell height, in this case the length of the path of the diffusing species in the cell. The time a t which l / e of the temperature change is reached may be compared to the calculated 6. The actual path length presumably is somewhat longer than the thickness of the disk due to the tortuosity of the path. I t was concluded that these internal checks were satisfactory within the estimated experimental uncertainty, since the calculated value was 160 sec. and the experimental one was 180 sec.
Ciscussion The phenomenological equations describing this system may be presented in many ways.4 Since there is only one independent composition variable, only one matter flux is required (and is associated with one independent heat of transport). We select the flux
because it represents the relative motion of silver nitrate and sodiuiii nitrate, because it is independent of the reference point chosen for the velocities, and because the'heat of transport associated with it is a particularly useful quantity. [Here j1(22) is the flux of the neutral component AgN03 (KaKO,).] The force conjugate to is - V T p , . The other fluxes used here are 5, the electrical current density, which is conjugate to -V$ (external electrical potential gradient) and p', the reduced heat flux (ref. 4, p. 26), which is
5
(4) H. J. V. Tyrrell, "Diffusion and Heat Flow in Liquids," Butterworth and Co.. Ltd., London, 1961. A systematic survey of thermal diffusion with many references may be found here.
Volume 69,Sumber 4
April 1965
BENSON Ross SUNDHEIM A N D JORDAN D. KELLNER
1206
conjugate to -0 In T . In terms of these variables, the equations are
,?
=
-L11(Crcll
+ Q*O In T ) - L~z(?++ a*?
r
+
= L 2 1 ( 0 ~ 1 Q*V In
T ) - Lzz(?+
+ r*c In
p’
= >Q*
t Jr*
z ~ ~ +
where transported of the silver ion, equal to isthethesum of S A g + andentropy S * A g + and where 8, -
In T )
is the “transported entropy” of the electron, equal to the sum of 3, and Se*.
T)
+ [(Lii + Lzi)&* + LIZ + LM* - La316 In T
r 0.1
1
J
The phenomenological differential coefficients are
-T&,
=
-(&c
In T);T,pi,?= t&*
+
H*
-(O+/V In T ) J ,=~ r* (J/T)or,r_l,oIn T = Washburn transference
- - T E ~=
where t = number; &* = ( ~ / J ) < ; ; I ~ T - = heat of transport = qQ1* Q Q ~ * ; H* = ( p / ~ ) ; ,In~ T = Peltier heat; E , = initial thermoelectric power; &f = final (Soret steady state) thermoelectric power. See the Glossary for fuller definitions and for important relations. In the AgNO3-XaNO3 system it has been that the potential of a concentration cell between silver electrodes
+
A+ = - (1/5) liltqFpl.dT
(9)
is represented adequately over the entire concentration range by
A 4 = ( 1 / ~ ) ( ~ ’’
PI’)
(10)
so that in this particular case t = -1 and is independent of composition. Consequently, here Ef =
-n*/T
(Et - E,) =
(11)
&*/T
The experimental values of these quantities are shown as functions of composition in Figure 1. The Peltier heat can be related to the transported entropy2 of the silver ion by constructing an entropy balance a t the electrode The entropy absorbed in the reaction at the electrode is SA^ - 3 ~ -~ S e-( * g ) where SAais the entropy of Silver metal, which is 14.3.5 cal. mole-‘ deg. - l a t this &(A~) is the partial molal entropy of the electron in silver metal and S A a - is the same quantity for the silver ion i n the fused salt. The entropy transported away from the electrode is - S * A , + - S*,~A,) where S * A , + is the entropy of transport of the silver ion and S * e ( A g ) is the entropy of transport of the electron in silver metal. Summing these terms stid equating to the Peltier heat The Journal of Physical Chemistry
0
0
0.1
0.2
0.3
0,4
0.5
0.6
o.,
0..
0.9
,
Figure 1. The initial and final thermoelectric powers as a function of composition.
The transported entropy of the electron in silver metal has been estimated’ to be virtually zero, i.e., smaller than the uncertainty in the thermocell e.m.f. Thus we have -&f
=
SAg
- SAg+ - S*A,?
(13)
By estimating the value of the partial ionic entropy of silver ion and combining it with the experimentally obtained steady-state thermocell e.m.f., a value for the entropy of transfer of the silver ion can be obtained. The entropy of a single ion is not experimentally accessible but if the ions are similar except for the sign of the charge, one could estimate it by taking one-half of the entropy of AgNOs obtained from heat capacity measurements and third-law calculations.2J In the case of a polyatomic ion such as X03-, a correction term first must be subtracted to allow for the entropy of rotation and vibration. This correction is reduced somewhat due to restriction of rotation of the anion in the fused salt.g These considerations lead to a value (5) (a) F. It. Duke, R. W. Laity, and D. Owens, J Electrochem. SOC., 104, 299 (1957); (b) R. W. Laity, J . A m . Chem. Soc., 79, 1849 (1957). (6) K . K. Kelly, U. S. Bureau of Mines Bulletin 584, U. S. Government Printing Office, Washington, D. C., 1960. (7) M . I. Temkin and A. Z . Khoroshin, Zh. Fiz. Khim., 26, 500 (1952). (8) C . Wagner, A n n . Physik., 3, 639 (1929). (9) K . S. Pitzer, J . Phys. Chem., 65, 147 (1961).
THERMOELECTRIC PROPERTIES OF MOLTEN AgNO3-KaNO3
of 19.7 e.u. for the internal entropy of the nitrate ion. The difference in mass between the two ions (appearing in the translational entropy) may be taken into account by adding 3/2 In [MAg+/Mx0,-]to the entropy of silver nitrate. A value of 19.0 e.u. for the entropy of Ag+ in pure AgNOs finally is obtained. The entropy of Ag+ in solution is estimated a t the various concentrations eniployed by assuming ideal entropy of mixing, i.e., by subtracting R In xl. Table I shows the values calculated in this way for the entropy of transfer of the silver ion a t each experimental composition. I t rnay be noted that the uncertainties in some of these corrections will affect the absolute magnitude of the calculated entropy of transfer, perhaps to the extent of k l e.u., but may be expected to have a minor effect on the relative values. The values are small over the greater part of the concentration range although they rise at the extremes. Pitzerg has suggested that SI*may be generally near zero in fused salts. We now turn to the quantity Q*. If we consider the isothermal, zero current interdiffusion of AgNOs and N a N 0 3 in a number fixed reference frame (51 = -j 2 ) we , find
p’
jQ*
=
jiCQi* - QL*)
=
51Q1*
+ JzQz*
=
=
-
~ 1 & 1 * / ~ 2 (14)
+
since xlQ1 x2Q2 = 0. In this special case since 51/x2 we see that Q*
=
&I* = (&I* - Qs*)/xz
5= (15)
The quantities Q* and Q1* - Q2*are shown as functions of composition in Figure 2. The latter quantity is the net (reduced) heat transported across the number fixed reference plane in an isothermal diffusion experiment (Dufour effect). Q* is everywhere positive, its magnitude ranging from about 1.1 to 3.6 kcal. for the concentrations studied. I t is very nearly synimetrical about x1 = 0.5. [The curve of Q* us. 51 is fit quite well by Q = In z1 In (1 - X I ) . ] Q*/xz = Q1* - Qz* has a more pronounced ininimuni and is skewed toward x1 --t 1. This quantity has been determined for a number of aqueous electrolytic solutions and the dependence on concentration generally observed resembles that found here. There does not appear to be any really satisfactory basis for interpreting the shape of this curve. The Soret effect reflects a difference in mobilities in a thermal field, the more rapidly moving species accumulating in the hotter zone.10 I t is interesting that the mobilities of the cations in this system are approxiniately equal to each other in an electric field.5
+
1207
9I
1
0,
0 1
0 1
0 1
0 6
0 5
0 .
1 0
0 ,
0 )
01
Figure 2. The quantities &* and Q1* - Q2* (reduced heat transported across a number fixed reference plane in an isothermal diffusion experiment).
Fundamental theories of transport properties in liquids, for example those expressed in terms of perturbations to the equilibrium distribution functions, have not reached the point of being applicable to specific systems. A less sophisticated but more pictorial model views transport as taking place by a series of activated transitions. Since the loci of the activation are distributed across the temperature field, differences ir the net entropy flux arise. In its original forni,llasb it utilized implicitly a quasi-lattice niodel of the liquid state, which perhaps is better suited to fused salt systems than to other liquids. In a simplified farmulation,llcjdit is found that &I* - Q2*
(16)
= PHI - q H 2
where q H ,is the energy required for the breaking free of the given species from its surroundings. By assuniing regular solutions with no volume effects it niay be shown that Qi* - Q2*
=
( ~ f / 2 (Eiil” ) -
G221/2)(x1G111’2
+ (17)
x2&2z1”)
where gl1 is the energy of activation of a molecule and = &11&22. In all simplified theories of this sort, the quantity Q*/x2 is found to be only weakly dependent on the concentration. ~~
~
~
~~
~
~~~
(10) H. S. Green, “Molecular Theory of Fluids,” Elsevier Publishing Co.. Amsterdam, 1952. ( 1 1 ) (a) K. Wirtz, A’aturwiss., 27, 369 (1950); (b) K. Wirtz, Physik.
Z.,44, 221 (1943); ( c ) I. Prigogine, L. de Broickere, and R. Amand, Physica, 16, 577, 851 (1950); (d) K. Denbigh, Trans. Faraday SOC., 48, 1 (1952); (e) K. F. Alexander, 2. physik. Chem., 203, 204 (1954).
Volume 69,iVumber
4
April 1966
BENSON Ross SUNDHEIM AND JORDAK D. KELLNER
1208
I n passing to electrolytic solutions, considerations of ionic interactions in dilute nonionic solvents suggest12 that the heat of transport should vary with m"'. The applicability of this approach to fused salts seems rather reiiiol e. The positive value of the Soret coefficient in the AgNOa-KaK03 system means13 that a silver ion is more difficult to remove from its environment than is a sodium iori. The greater ionic interactions in a silver nitrate melt as compared with those in a sodium nitrate melt are generally attributed14 to the higher polarizability of the silver ion. The enthalpy of mixing in this system is given by the expre~sion'~ AHm = z1z2(677 - 1 . 5 6 ~ cal./mole ~)
This is a curve falling to zero at the extremes with a maximuin near ~t:= 0.5, quite different in shape and magnitude From that of &* us. zl. Thus arguments directed along the lines of interaction energies alone are not helpful. It is remarkable that we know so little about the mechanisms of such elementary electrocheniical transport processes. Further speculation must wait upon the accumulation of data on other systems.
Acknowledgment. I t is a pleasure to acknowledge assistance to this work from the Office of Naval Research (Con tract Sonr-285-37).
The Journal of Physical Chemistry
Glossary &f Final thermoelectric power = [d+/dT]J,I &in Initial thermoelectric power = [d+/dT] T , l T Absolute temperature R Gas constant, 1.987 cal./deg. mole H Peltier heat ( q / Z ) y , v ln Q* Heat of transport = ( q / J ) T , I 21,zz Mole fraction of AgN08, NaNOI SA,+ Transported entropy of Ag S*A,+ Entropy of transport of Ag+ S,+ Partial molar entropy of Ag+ SA,+= S'A~+ - R In 51 [z; d FT5] u &ret coefficient in moles/deg. = +
~
f
So,+ Partial molar entropy of Ag+ when z1 = 1 (pure salt) S*A~N =O S*,+~ S*Noa-
+
S*NaNOa = S * N a +
+
H / T = -sf SAP Q* = T(E1. - cf) Qz*
S*NOa-
-
-
-
&(Ag)
=
SAg
-
SA, -
S*Ag+
(-~i/~i)Qi*
Flux of AgN03 (moles/cm.2 sec.) Flux of NaN03 (moles/cm.2 sec.) Electrical current flux (faradays cm.l/sec.) I Reduced heat flux = second law heat flux = convected q enthalpy J = J1 - (z,/zz)Jo= SlC(V1 - VZ) Washburn transport number = ( J / I ) T , ~ t Mass of i in atomic weight scale in a number fixed referMi ence: J = Jl/zz;Q* = &I*= (&,* - Qz*)xz J,
Jz
(12) E. Helfand, J . Chem. Phys., 3 2 , 857 (1960). (13) See ref. 4, pp. 272-290. (14) J. G . Jam, J . Chem. Educ., 3 9 , 59 (1962). (15) 0. J. Kleppa, R. B. Clark, and L. S. Hersh, J . Chem. Phys., 35, 175 (1961).
