A Thermodynamic Theory of Electric Potential Generated on Crystallization of Electrolytes

Vol. 130, No. 6 DROP-FALL DETECTOR 1327

f RECTIFIED SIGNAL (A)

4-15

-150"l

I F I

INTEGRATED SIGNAL(B)

DIFFERENTIATED SIGNAL(C)

INTERRUPT PULSE(D)

Fig. 2. Waveforms in the trigger circuit. (Integrated circuits: National Semiconductor or equivalents.)

work ing e lec t rode is measured ( inver ted) at the ground re tu rn of the floating d -c power supply.

The ampl i tude of the signal of the 91 kHz signal genera to r is control led th rough a negat ive feedback loop, to avoid the p rob lems connected wi th very di- lu ted < 10-aM) or ve ry concent ra ted ( > 10M) solutions, i.e., noise, instabil i t ies , or amplif ier saturat ion. With this a r rangement , the (averaged) a -c cur ren t

th rough the cell, r a the r than the vol tage across it, is kep t constant .

Applications and Results This device has been used for over 2 years in our

computer contro l led appara tus for d rop - t ime measures . The accuracy has been verified to be be t t e r than 500 ~sec; the e r ror ra te due to false t r igger ing is be t t e r than one e r ror over a run of about 104 drops, despi te the e lec t romagnet ic noise genera ted by o ther l abo ra - tory devices or atmospherics .

Manuscr ip t received May 3, 1982; revised manuscript received ca. Oct. 1, 1982.

A n y discussion of this pape r wi l l appea r in a Dis- cussion Section to be publ i shed in the December 1983 JOURNAL. Al l discussions for the December 1983 Dis- cussion Sect ion should be submi t ted b y Aug. 1, 1983.

The Universite di Firenze assisted in meeting the publication costs of this article.

REFERENCES 1 P. D. Tyma, M. J. Weaver , and C. G. Enke, Anal.

Chem., 51, 2300 (1979). 2. B. K. Hahn and C. G. Enke, ibid., 46, 802 (1974). 3. M. De Blasi, G. Giannell i , P. Papoff, and T. Rotunno,

Ann. Chim., 65, 315 (1975). 4. J. Lawrence and D. M. Mohilner , This Journal, 118,

259 (1971). 5. G. Papeschi, M. Costa, and S. Bordi, Electrochim.

Acta, 15, 2015 (1970).

A Thermodynamic Theory of Electric Potential Generated on Crystallization of Electrolytes

R. P. Rastogi and S. A. Khan ~

Department of Chemistry, University of Gorakhpur, Gorakhpur 273001, India

ABSTRACT

A theory of precipi tat ion potential based on thermodynamics of irreversible processes has been developed. In connec- t ion with the verification of theory, thermochemical exper iments have been performed which suggest that during the deve lopment of precipi tat ion potential associated with the crystallization from supersaturated solution, lattice energy is converted into electrical energy. The theory is found to explain quali tat ively all the observed facts.

I t has been fa i r ly wel l es tabl i shed tha t potent ia l differences are developed dur ing (i) c rys ta l l iza t ion of e lec t ro ly tes f rom the i r supe r sa tu ra t ed solutions (1-4), (ii) dissolut ion of e lec t ro ly tes in a solvent (1-4) , (iii) f reezing of w a t e r containing ionic impur i t ies (5-9), and (iv) crys ta l l iza t ion of e lec t ro ly tes f rom the i r mel ts (10, 11).

The phenomenon of prec ip i ta t ion and dissolut ion po ten t ia l was discovered 2 by Rastogi, Dass, and Bat ra (1). Comprehens ive s tudies on these potent ia ls have been r epor t ed by Rastogi et al. (1-4). I t has been suggested by these worke r s tha t the s ign and magni tude of these potent ia ls depend on the unequal mobil i t ies of cations and anions. They have pos tu la ted tha t the p rec ip i ta t ion poten t ia l develops due to a difference in the ra te of a t t achmen t of cations and anions dur ing

1 P r e s e n t address : C h e m i s t r y D e p a r t m e n t , St. A n d r e w ' s Col lege , G o r a k h p u r , India .

K e y w o r d s : c rys t a l l i z a t i on , e l ec t ro ly t e s , t h e r m o d y n a m i c s , elec- t r i c po t en t i a l .

O u r a t t e n t i o n has b e e n d r a w n to t h e w o r k of J a e n i c k e a n d co- w o r k e r s b y one of t h e r e f e r e e s . T h e s e w o r k e r s [Chem. Abst., 47, 10973d (1953); 48, 5612h (1954)] w e r e c o n c e r n e d w i th t h e s t u d y of t h e d e v e l o p m e n t o f p o t e n t i a l c h a n g e s d u r i n g d i s s o l u t i o n of spa r - i n g l y so lub le sa l t s l ike A g B r u s e d in r e v e r s i b l e e l e c t r o d e s , and t h e y i n t e r p r e t e d t h e resu l t s on the basis of d i s t u r b e d i n t e r f a c e e q u i l i b r i u m on a c c o u n t of d i sso lut ion ,

crys ta l l iza t ion and tha t the dissolut ion po ten t ia l de - velops due to a difference in the rate of de tachment of cations and anions dur ing dissolution.

Prec ip i ta t ion poten t ia l has been defined conceptu- a l ly by I~astogi, Shukla, and Bhagat (3) as follows: "The prec ip i ta t ion potent ia l is the poten t ia l difference ar is ing f rom an e lec t r ica l double l aye r deve loped at l i qu id / c rys t a l in terface by the ion migra t ion f rom the l iquid phase. The a lgebra ic s ign of the prec ip i ta t ion potent ia l is the same as the e lect ros ta t ic po la r i ty ( + ) or ( - - ) of the end of the d ipolar double l aye r on the crys ta l side of the interface." However , the opera t iona l definition is as follows: "The prec ip i ta t ion poten t ia l is the potent ia l of a p l a t i num elec t rode coated wi th a l ayer of an e lec t ro ly te deposi ted by crys ta l l iza t ion f rom a supersa tu ra ted solut ion at a t e m p e r a t u r e T less than the sa tu ra ted t empe ra tu r e Ts. (Ts -- T) is called the undercooling. The reference poten t ia l is a s imi la r bare p l a t inum elec t rode which is d ipped in the above supe r sa tu ra t ed solut ion once the crys ta l l iza t ion has s tar ted. The observed prec ip i ta t ion potent ia l is corrected for any a s y m m e t r y potent ia l exis t ing between the two p l a t inum elect rodes before the p rec ip i - ta t ion run. The poten t ia l difference is measured elec- t ros ta t ica l ly by means of a L indemann Elec t rometer ."

) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 134.129.182.74Downloaded on 2014-11-19 to IP

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1 3 2 8 J. EIec trochem. Soc.: ELECTROCHEMICAL SCIENCE AND TECHNOLOGY J u n e 1983

Recently Girdhar and Matta (12, 13) have studied the precipitation and dissolution potentials of a number of electrolytes in aqueous medium. Their results are in general agreement With those reported by Rastogi and co-workers. Ibl et al. (14, 15) have also recently reported careful experiments on crystallization and dissolution potentials using single crystals. In their experiments the crystal is always kept in contact with near-saturated solution so that the effect of diffusion is minimized. These workers have used a "corrosion" model based on the difference in the ki- netics of dissolution of cations and anions to explain their results.

The experimental studies confirm that the precipitation and dissolution potentials depend on (i) temperature, (ii) concentration of the medium, (iii) nature of the electrolyte, (iv) time , and (v) ( U + - U - ) / ( U +

U-) . Typical data obtained by different workers are summarized in Table I for the sake of comparison.

Quite recently, a thermodynamic theory of dissolution potential based on the thermodynamics of irreversible processes has been developed by Rastogi and Khan (16). In the present communication, an attempt has been made to develop a corresponding thermodynamic theory of precipitation potential. It is found that t h e theory gives a satisfactory explanation of the observed facts. In Order to have a quantitative check, an inequality based on the theory has been tested for which calorimetric experiments had to be performed. These are described in the present communication. The thermochemical experiments suggest that it is the lattice energy that is converted into electrical energy during the development of precipitation potential.

Crystallization Process The process of crystallization may be visualized as

being made Up of the following steps: 1. Formation of nuclei. 2. Transport of the cations and anions toward the

nuclei. 3. Diffusion through a locally solute depleted zone at

the growing interface. 4. Partial or complete desolvation of the moving

ions. Com~)lete desolvation would be required for the crystallization of NaC1 and partial desolvation would be required for the crystallization of BaC12 �9 2H20.

5. Adsorption or chemisorption of ions on the surface of crystal nuclei.

6. Two-dimensional diffusion of the adsorbed or the chemisorbed species to a step.

7. Partial or complete desolvation of the moving ions toward the step.

8. Attachment to the step. 9. One-dimensional diffusion along the step to a

kink.

10. Attachment to a kink. 11. Dissipation of the heat of crystallization which

may occur stepwise after each of the attachments. It is obvious that the precipitation potential would

be generated at step 5. Further, on account of step 4, a higher temperature would develop in the solution layer near the crystal interface as compared t o t h e

bulk, so that a thermo-emf would atso be generated. In addition, the diffusion potential would also develop o n

account of process 3. The equilibrium phase potential at the crystal/solution interface and the Nernst potential would also be included in the observed potential. Most of the experimental studies reported so far in- volve the observation of the net potential from which the value of the true precipitation potential can be easily estimated when the contributions due to other potentials are experimentally known. Rastogi and Shukla (2) have demonstrated that the values of thermo-emf and Nernst potential are negligible in most cases. Thus, under appropriate experimental conditions, the crystallization potential would largely con- sist of (i) precipitation potential and (ii) phase potential. The phase potential would include the unknown contact potential between metal electrodes and salts.

Irreversible Thermodynamics of Precipitation Potential

Theoretical We may represent schematically the cell used for

measuring the precipitation potential as follows

Supersaturated Pt Solid electrolyte solution of

(a-phase) electrolyte Pt (/~-phase)

We represent the electrochemical system as made up of two subsystems a and ~. We may consider, for ex- ample, the experimental conditions maintained by Ibl et al. (14) such that a uniform supersaturated solution is in contact with the a-phase. Further we may suppose that the temperatures of the two phase remain the same. This is a little simplification since phase trans- formation is always accompanied by enthalpy changes. We also assume that the crystal nuclei have similar surface energy and that they have average surface energy per unit area of the crystal face.

The total entropy change of the system dS would be given by

dS = dS~ Jr dSS [1]

where dS~ and dS~ are the entropy changes for t h e

two subsystems. The Gibbs equation for the two subsystems is given by

Table I. Dissolution and precipitation potential of some electrolytes

Observed Observed True True dissolution precipitation Diffusion Phase dissolution precipitation potental potential potential potential potential potential

Electrolyte Medium (mV) (mV) (mY) (mV) (mV) (mV) Reference

N a C I W a t e r - 1 7 0 - 78 - 73 - 2 4 - 50 - 5 4 ( 3 , 2 ) W a t e r - 1 2 0 - - 90 - 40 - 8 - 72 - 8 2 (4 , 12, 13) D~O - 97 - - 73 - - 31 - - 2 4 - - 42 - 4 9 ( 3 , 2 ) A c e t o n e + 2 1 1 + 69 - 5 2 +,194 ( 3 , 2 )

KC1 W a t e r - 1 5 2 - " 7 6 - - 63 - 2 1 - 97 - ~ (3 , 2) W a t e r - 1 8 0 - 66 - 42 - 2 0 - 15 - 6 8 ( 4 , 12, 13) D~O - - 1 0 5 - 66 - - 33 - - 1 7 - - 55 - - 4 9 ( 3 , 2 ) , Acetone + 2 6 8 ~ + 83 - - 7 3 + 2 5 8 - - (3 , 2) F o r m a m i d e + 2 8 9 - - + 83 + 1 9 + 1 8 7 - - (3 , 2 ) D M F + 89 + 58 - - 4 3 + 76 - - ( 2 1 , 2 8 )

KDr D~O --131 ---~0 -- 42 --21 -- 68 --49 (3,2) D M F + 1 2 4 - - + 48 - - 2 6 + 1 0 2 (21 , 28)

BaCI~ W a t e r - 153 - 123 + 140 - - 10 - 280 - - ~ ( 3 , 2) - 1 6 8 - - -- 46 --12 - l l O - - (4, 12, 13)

* Results for acetone reported by Rastogi and co-workers and by Girdhar and co-workers differ in sign as well as in magnitude due to different procedures adopted for purification in the two cases. Girdhar et al. report that when the procedure of purification adopted by Rastogi and co-workers is used, the observed value of the dissolution potential agrees in sign with that reported by Rastogi and co-workers,



V o L 130, No . 6 THERMODYNAMIC THEORY 1329

TdS~ = dUa Jr P a d V a

- - (#ladMi ~ + #~.~dM~. ~)

-- (eir a jr e~r ~) -- v d A [2]

TdS~ : dUB + P~dV~

-- (#i~dMiB jr ~ B d M ~ )

- - (eir jr e~r [3]

If we assume that the nuclei are spherical having radius r i t can be easily shown that

dA : 8~ rdr [4]

and if ~ is the density, change in mass dM ~ is related to dA by the equat ion

d A : 2dM~/r~ [5]

Hence, Eq. [2] can be rewr i t t en as

T d S (~ : dU '~ jr P d V (~ -- (#i ~ jr eir ~)

dM1 ~ _ ( # ~ jr e~r ~) dM~ ~ _ ~27 dM ~ [6] ~r

Using Eq. [3] and [6] in Eq. [1] we obtain

TdS : dU ~ + dUB Jr P adVa jr P~dVB

- ~ (~ + e~r - ~ (,i~ + e~)di~i~ ~----1,2 ~:I,2

- 2~ dM~ [7] pr

If dQ is the heat supplied to the system as a whole, using first law of thermodynamics, Eq. [7] can be wr i t ten as

T d S : dQ -- ~ (#i ~ jr eic~a)dMi a i=I$ ,

- ~ (~i~ + e~)dM~B -- 2v dM= [8] t= 1.2 pT

Now dS can be wr i t ten as the sum of deS, the ent ropy exchange with the surroundings and diS, the change in ent ropy due to i rreversible processes inside the system, so that

dS = des jr d is where

Hence

TdiS = -- ~r

des -- ( d Q / T )

(#i ~ Jr- eir a) dMi a

[ 9 ]

[ i 0 ]

i = 1 ~ 2

_ ~ (#t B + ei~B)dMi B _ 2v dM~ [ i i ] t= 1,2 pr

or dis dMi a

T : T ~r (# l ~ j r e i r a) a t ~=1,~ a t

~ ( ~ i B dMiB 2-/ dM a - + e~B) - - [12 ]

i= 1,2 a t pr dt

On account of the law of conservation of mass

Ji = dMl~/dt -- -- dMiB/d t

S o that Eq. [12] can be rewr i t ten as

Tr = -- ~ (AFi a'B jr eiAqba'B)Ji jr k J c [13] i = 1 , 2 pr

where we have replaced -- dM~/d t by the crystall iza- t ion rate Jc. Thus, Eq. [13] can be wr i t ten as

: ~ J~Xi j r J~Xc [14] Tr t= 1 , 2

where Xi = -- (A#p,s + eiAr [15]

and X~ = (2v) / (pr ) [16]

Ji is defined with respect to the center of mass move- ment. Both, Ji and Xi are vectorial in character but Jr and Xc are scalar in character. Hence, according to Curie's principle, we would have coupling between J1 and J~ only~ so that we can write

J1 "-- L n X i jr Li2X2 [17]

Js -- L21X1 jr L22X2 [ is]

Je = LccXe [19]

where Ln , Lm L~I, and Lcc are phenomenological coefficients and L12 = L21 on account of the Onsager re- ciprocity relation.

I t should be noted that at equi l ibr ium, Jc = 0. F u r - ther, Xc is also zero since r -- ~o. In some cases, r may not be equal to oo at equil ibrium, if for such cases r -- re at equil ibrium, then Eq. [16] would be wr i t ten as (18)

Xc = (2-/) I p ( 1 / r -- 1/re) [20]

according to which Xc : 0 when r -- re at equi l ibr ium. For planar surfaces re = oo.

The force Xi takes into account the whole t ransi t ion region from the solid surface to the bulk of the solu- t ion and is associated with the change of electrochemical potential.

The cur ren t I flowing through the system would be given by

I = eiJi jr eaJs [21]

Since we measure the potent ia l when I = 0, we have

ei ( L n X i jr Li~X~) Jr e2(L21X1 Jr/-,s2,X2) : 0 [22]

Pu t t ing

and

We have

-- Aea.B =

e lLn jr e2L2l -- I1 [23]

elLis jr esL22 = ls [24]

h~i a,~ jr a#~ a.B eili jr e212 ell1 Jr e212

[25]

If ti~ denotes the t ransport n u m b e r of unhydra ted i th ion, its relat ion to the mass t ransfer f rom phase #s to a would be given by

and

elll ( t l O ) A R I = 0 , a ~ $ = 0 - - [26]

e l l l Jr e~12

e21~

( t2~ A~2=0 - - eIl l j r esZs [27]

Hence, from Eq. [25], [26], and [27] we obtain

1;1 o t2 o _ ( ~ ) ~m : _ _ ~ i ~ , B + _ _ ~ 2 ~ , B [28]

e l e2

We now consider ~ to be the solid phase and ~ to be the l iquid phase. Since ei : ZiF, where Zi is the valency of the i th ion which can be negative or posi- tive, we obtain for a un iva len t electrolyte

-- F(A@)a,B : tlo[mo(s) -- #1o -- R T In al (ss)]

- - t~o[~,~ o(s) -- #2 o -- RT In as (ss)] [29]

where the superscript (ss) denotes the quanti t ies in the supersaturated solution, ~i o(s) is the s tandard chemical potential in solid phase and ~i o is that in the l iquid phase and

8 It should be noted however, that the net change in mass frac- tion of the component (i = 1, 2) at a point is related to Jo and the divergence of Ji (17).



1330 J. E ~ e c t r o c h e m . S o c . : ELECTROCHEMICAL SCIENCE AND TECHNOLOGY J u n e 1 9 8 3

in al (s) = In a2 (s) = 0 [30]

where al (s) and a2 (~) denote the activities of ions 1 and 2 in the solid state.

Equat ion [29] can be rewri t ten as

- - F ( ~ ) ~ , ~ = t l o [ # l o~s) - - #1 o - - R T In a l c*~]

_ t2o[#2o(s) _ ~ o _ R T In a2 (*)]

+ [ti~ R T In a i ( * ) / a l (ss) - - t2 ~ R T In a2(*)/a~ (ss)] [31]

where a~(*) is the activity of the ions at the crysta l / solution interface.

There are three terms on r igh t -hand side of Eq. [31]. Let us designate the first two terms as (~r and the third te rm as (a r respectively, for con- venience. Thus we shall have

F ( ~ ) I = ti~ ~ -- ~l ~ -- RT in ai (*)]

_ t~o[~2ocs) _ ~o _ RT in a2 (*)] [32]

F ( ~ r = tl ~ RT In alC*)/a~ (ss) -- t~ o RT In a2(*)/a~ (ss)

[33]

If (~#)Phase denotes the potential difference at the crystal / l iquid interface, it is easy to show (19) that

~lO(S) _ ~o _ R T In ai (*) = -- F (~)Phase [34]

~o(s) _ ~2 o _ R T In a2 (*) -- + F(A~b)Phase [35]

where 1 denotes cation and 2 denotes anion. Thus from Eq. [32], [34], and [35] we obtain

F ( A ~ b ) I - - t l ~ - F ( A ~ ) P h a s e ] - - ~ 2 ~ F(A~b)Phase] o r

F(A~)~ = -- F(hr [36]

since tl ~ + t2 ~ = 1. From Eq. [33] we have

F (~O)n = (t~ ~ - - t2 ~ R T In a(*)/a(ss) [37]

where we have supposed that

ai(*) /a l (ss ) = a~(*)/a2(ss) -- a( . ) /a (ss )

Thus from Eq. [31], [36], and [37] we find

-- F(A~) a'~ = -- F(AO)Phase -~- (tl ~ -- t2 ~ R T I n

[38]

a(*)

aiss) [39]

The second te rm on the r igh t -hand side of Eq. [39] denotes the precipitat ion potent ial and, therefore, if (A~b)Pption iS the potential difference arising from an

electrical double layer developed at the l iquid/crys ta l interface by the process of the migrat ion of ions from solution phase to the crystal phase, we shall obtain

RT U+ ~ -- U- ~ a (*) (A~b)Ppti~ =- F U+ ~ + U - ~ In a(ss--- 7 [40]

where U+ o and U - ~ are the mobilit ies of bare un - hydrated cation and anions, respectively.

Making use of usual thermodynamic arguments, it can be easily shown that

aC*) (~H) ~ (Ts -- T) in . . . . [41]

a (ss) R TsT

where (~H) ~ is the entha lpy change associated with the development of the precipitat ion potential. Using Eq. [41], Eq. [40] can be rewr i t ten as

1 < U+~176 > (~H)~ (~T)

(~r Ppt~on = -- ~- U + o ~ U_o Ts [42]

U+o -- U_o < I [43]

U+o + U - ~ it is obvious tha t

1 (hH) ~ (AT) (A~b) Pption < [44]

F T s

Test ot the Theory Inequal i ty [44] is in a form which can be used for

exper imental verification. However, the question arises whether (~H) ~ is equal to the heat of precipitation, which is equal in magni tude but opposite in sign to the heat of solution (AH)s, i .e. , whether

(a l l ) ~ = -- (~H)s [45] or different?

The basic reaction that occurs dur ing the a t tachment of ions in the crystal lattice is

A + + B - = A B [46]

where A + and B - are bare unhydra ted ions. If t h i s is so (a l l ) ~ should be equal to the lattice energy (hH)L, i.e.

(hH) ~ = (hH)L [47]

For a quantitative check of the theory, the inequality [44] was tested. (~r was calculated both by sub- stituting (AH) ~ equal to --(aH)s and (AH)L in [44]. The results are given in Table II. The calculated values recorded in column 5 do not support the inequality [44]. However, the data presented in column 4 support the inequality. Hence, it follows that (AH) ~ would be equal to (AH)L ra ther than -- (AH)s.

It is, therefore, obvious that Eq. [47] becomes crucial for the exper imenta l test of inequal i ty [44]. If [46] is the main controll ing step for the development of the precipitat ion potential, the heat produced on crystal l i- zation of AB would be converted into electrical energy dur ing crystall ization and cooling instead of heat ing should be observed around the electrode at which crystall ization is occurring. In order to test this point, thermochemical exper iments were performed. These are described next.

Experiments on the Relationship of Lattice Energy and Precipitation Potential

In order to show that dur ing the development of precipitat ion potential the lattice energy is converted into electrical energy, the following three types of experiments were performed.

The schematic diagram of the first type of experiments is shown in Fig. 1. The precipitat ion cell C was made of a double-wal led Pyrex glass vessel (outer diameter 6 cm, height 12 cm; inner diameter 3 cm, height 10 cm). The annu la r space A of the cell was evacuated in Order to minimize the heat losses. Two p la t inum electrodes E1 and E2, which were fused in Pyrex glass capil lary tubes P1 and P2 of 1 mm diam, were used in the cell. Electrode E1 was in the form of a coil through which a precision thermometer T could be inserted which could be read correctly to _ 0.2~ Both the electrodes were connected to a Cam- bridge L indemann Electrometer.

Table II. Comparison of experimental and the theoretically calculated values of (~r for some alkali halides using Eq. [44], [45], and [47] at Ts = 311 K and AT = 12~

(~r Experl-

(AH)L -- (AH)s Theoret ical Theoretical menta l Eleetro~ (kcal/ (kcal/ us ing Eq. us ing Eq. [Ref. (2) ] lyte mol) mol) [47] (mV) [45] (mV) (mV)

Since the mobilit ies of bare ions are not known, Eq. KCl 169.9 4.1 -284 -5.74 -54 K B r 162.1 4.8 -- 272 - - 6.72 -- 52

[42] could not be tested quanti tat ively. However, since NaCl 184.7 0.9 -310 -1.26 -56



Vol. 130, No. 6 THERMODYNAMIC THEORY 1331

"Vk-- -

E 1 -

L

T ~/p2 "/////////~]t~.

B

S ~ C

Fig. 1. Experimental setup for the measurement of cooling produced on the development of precipitation potential. (C - - double walled calorimeter; S ~ saturated solution of KCI; El, E2 = platinum electrodes; B = inner chamber of the cell; R ~- rubber stop- per; P1, P2 - - Coming glass capillary tubes of 1 mm diam; T --- thermometer; V = vacuum pump; and L = Lindemann electrometer.)

The following exper imenta l procedure was adopted. 10 ml of KC1 solution, saturated at 60~ was placed in the precipitat ion cell. The electrode El along wi th the thermometer T was inserted into the solution. Now the system was allowed to cool gradually. The second electrode E2 was inserted into the solution as soon as a t iny crystal appeared on the first electrode. The development of the precipitat ion potential was indicated by the electrometer while the tempera ture of the calo- r imeter was recorded with the help of the thermometer as a funct ion of time.

The second type of exper iment was similar in pr inci - ple except that a copper-constantan thermoco~uple was used for the measurement of temperature. The schematic diagram of the exper imenta l setup is shown in Fig. 2. The same double-wal led calorimeter C was used in this set of experiments also. P la t inum electrodes E1 and E2 were fused in Pyrex glass capillary tubes PI and P2 of 1 mm diam. The electrode E1 was in the form of a coil through which the capil lary tube P3 conta in- ing the copper-constantan thermocouple junct ion T1 (Omega In terna t ional Corporation, Stanford, Connec- ticut) could be inserted. The other junct ion T2, used as cold junct ion, was placed in mol ten ice kept in a Dewar flask. Thermocouple junct ions were used in conjunct ion with an OSAW direct reading precision potent iometer with which the emf could be read cor- rect ly to _ 10 #V. Electrodes E1 and E2 were connected to an Electrometer Amplifier (Annadigi, Hyder- abad, India) with which the voltage could be read correctly to __. 0.1 mV and which had a m a x i m u m input impedance of 1014~.

The exper imenta l procedure was similar to that adopted in the earl ier exper iment except in the following respects. The p la t inum electrodes E, and E~ were kept a l i t t le above the aqueous solution of KC1 saturated at 60~ When the crystal started appearing in the bulk, the electrode E~ was dipped into the solu- t ion and immediate ly raised up, so that on the evaporat ion of water, a few seed crystals were left on the

ELECTROMETER AMPLIFIER

:J-i PR~C~I Pl TAT I0 N

POTENTIAL P.t i

I ' O,SATI~ POTENTIOMETER

COLD JuNCT{O~ P2

Y

Fig. 2. Modified experimental setup for the measurement at cooling produced on the development of precipitation potential. (El ~- platinum electrode wrapped around the capillary P8 containing copper-constantan thermocouple junction T1; E2 ~ second platinum electrode; S ~ suppersaturated solution; T2 ~ the junction of thermocouple kept at ice point.)

electrode. Both electrodes E1 and E~ were now s imultaneously inserted into the solution. The bui ldup and decay of precipitat ion potential and thermo-emf were s imultaneously recorded.

In order to provide a more convincing proof, a third type of exper iments was planned. An improved exper imenta l setup was designed as shown schematically in Fig. 3. It consisted of twin cells R1 and R~ which were identical. These were in the form of two Coming glass tubes closed at one end. Both the tubes were 8 cm long and had a diameter of ! cm. Each tube had a system of electrodes and tempera ture probe as shown in Fig. 3. The copper-constantan thermocouple junctions T1 and T2, kept in the cells R1 and R2, respectively, were connected to a PYE precision potent iometer with which the potent ial could be read correctly to _ 1 ~V in conjunct ion with a sensit ive Cambridge Spot Galvanometer. Further , the electrodes of cell R1 were connected to an Electrometer Amplifier (Annadigi, Hyderabad) while this was not so for the electrodes of cell R2 which acted as a dummy cell.

The following procedure was adopted. Both the cells were filled with the aqueous solution of KC1 saturated at 60~ which was heated a little above the saturat ion temperature. The calorimeter was also filled with the same solution. Care was taken so that cells R1 and R2 and the l iquid in the calorimeter were at the same temperature. Initially, the electrodes in both cells were kept a little above the surface of the solution. The as- sembly was allowed to cool gradually. When crystal l i- zation started, the electrodes E1 of cell R1 and E2 of cell R2 were lowered into the respective solutions and then raised up so that on evaporat ion of water seed crystals appeared on their surface. Now, electrode systems of both cells R1 and R2 were s imul taneously in - serted into the respective solutions. The bui ldup and decay of precipitat ion potential and thermo-emf were



1332 J. Electrochem. Sac.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y June 1983

J E | . E C T R . O H E T t : K A 14Pi,~FI~

IT P re~il:,L~a

pote.t

PTE pOTENTIOI~ETER

11

No electrical co.~eJ~~ i e. No precL~tatLo"

pot en~fAa~

Fig. 3. Experimental setup for the simultaneous measurement of precipitation potential and thermo-emf during crystallization. (R1, R~ = twin precipitation cell; El, E2 ~ platinum electrodes on which crystallization takes place; El', E2' ~ bare platinum electrodes; T1, T2 ~ copper constantan thermocouple junctions; S --- supersaturated solution of KCI; A ~ annular space of the vesssel C; and B ~ inner chamber of the vessel.)

s imul taneous ly recorded with the he lp of the Elec- t romete r Amplif ier and PYE potent iometer , respec- t ively.

Results and Discussion Equat ion [40] shows how the prec ip i ta t ion poten t ia l

depends on the mobi l i ty of the cations and anions. Fur ther , i t fol lows f rom Eq. [40] and [42J that, (i) the sign of ( A r would depend on the sign of (U+ o -- U-o), (ii) the magni tude of (Ar would depend on the magni tude of rat io (U+ o -- U - o ) / ( U + o -t- U_o), (iii) the magni tude of (A~b)Ppt ion would also depend on the magni tude of undercool ing (Ts -- T), and (iv) the decay of the observed prec ip i ta t ion potent ia l would be re la ted to the ra te wi th which the concentra t ion of the supersa tu ra ted solut ion approaches the sa tu ra - tion concentrat ion. Thus, in the course of t ime, when crys ta l l iza t ion stops and a(*)/a ~ss) -) 1, (5@)Pption would vanish and the observed prec ip i ta t ion potent ia l would become equal to (5@)Phase.

Expe r imen ta l da ta r epor ted by RaStogi and co- workers (2, 3, 20, 21) show (Table I I I ) tha t these con- clusions are justified. However , i t m a y be argued tha t the resul ts on the prec ip i ta t ion potent ia l of NaC1 and KC1 for wa te r (Table I I I ) and D20 [Ref. (3)] do not suppor t the theory, since one would expect the p r e - c ipi ta t ion potent ia l of NaC1 to be g rea te r than that of KC1 owing to the fact tha t the mobi l i ty of Na + ion (hydra ted ) is lower than that of K + ion (hydra t ed ) . This a rgument is not sound since as suggested in Eq. [40] one has to take into account the mobil i t ies of unhyd ra t ed ions ra the r than the inabi l i t ies of hydra t ed ions. Al though such values a re not avai lable , some guess can be made about the i r re la t ive magni tudes .

Table Ill. Sign and magnitude of (U+ - - U-)/(U+ ~ U-) and (Ar

Elec tro l y t e ( U+ -- U+) / (U+ + U - ) (Ar Pption ( m V )

K C I - 0.0187 - 54 K B r -- 0,0303 -- 52 NaC1 - 0.2072 -- 56

The t ranspor t number of hydra t ed and unhyd ra t ed ions wil l be re la ted (22) by the equat ion

t+ o = t+ -t- C ~ ~nn~o [48]

t_a = t - "t" Co ~-~jo [49] w h e r e

~nH2o = rich*) - - nch-) [50]

Taking into account the appropr i a t e values of h y d r a - t ion numbers (23, 24), AnH20 for KC1 would be a round un i ty and tha t for NaC1 would be a round 2. Hence, f rom Eq. [49] and [50] we observe tha t the t r anspor t numbers of unhyd ra t e d Na + and K + ions would be closer than tha t of the i r hyd ra t ed ions. I t may also be noted that the t r anspor t numbers of Na + and K + ions are equal in mol ten NaC1 and KC1 (25).4 Accordingly , i t is not surpr i s ing tha t the magni tude of the p rec ip i t a - t ion poten t ia l in the case of NaC1 and KC1 are s imi lar wi th in the e xpe r ime n t a l error .

I t is difficult at this s tage to have a quant i ta t ive check of the theory deve loped in the sect ion on Irre- vers ib le Thermodynamics and Prec ip i ta t ion Potent ials . However , the inequa l i ty [44] is satisfied when ( a l l ) ~ is pu t equal to (AH)L as Table II shows. The expe r i - ments descr ibed in the sect ion on Exper imen t s on the Rela t ionship of Lat t ice Energy and Prec ip i ta t ion Po- ten t ia l show tha t such a subs t i tu t ion is justified. We discuss the thermochemica l resul ts fur ther , in the following paragraphs .

In the first set of exper iments a cooling of the order of 1~176 was observed in the v ic in i ty of the e lect rode on which crys ta l l iza t ion occurred when the deve lop- men t of the prec ip i ta t ion potent ia l occurred, as was indica ted b y the e lec t rometer . On the o ther hand, a b l ank expe r imen t wi th the same cell wi thout any elec- t r ica l contact be tween the electrodes, gave evidence of heat ing of the o rde r of 0.5~176 Thus a cooling in the v ic in i ty of e lect rodes on which crys ta l l iza t ion occurs and the prec ip i ta t ion potent ia l is developed indicates that the la t t ice energy is conver ted into e lec t r ica l energy.

The da ta for a typical run in the second type of exper iments on the s imul taneous measu remen t of the prec ip i ta t ion potent ia l and t h e r m o - e m f are shown in Fig. 4. I t is c lear f rom the figure that as soon as the prec ip i ta t ion potent ia l is developed, as indica ted b y the sudden rise in curve a of Fig. 4, a sudden cooling occurs, as indica ted by a cor responding sharp decrease in thermo-emf . The m a x i m u m in curve a and min i - m u m in curve b, Fig. 4, coincide, confirming once again that cooling is p roduced in the v ic in i ty of the elec- t rode on which crys ta l l iza t ion occurs and the bu i ldup of the prec ip i ta t ion potent ia l takes place. On the o ther hand, b lank exper iments wi thout any e lect r ica l contact be tween the electrodes showed an increase of t empera tu re of the o rder of 0.5~176 due to the l ibe ra - tion of the hea t of precipi ta t ion.

The da ta for a typical run in the th i rd type of expe r iments on the s imul taneous measuremen t of the prec ip i ta t ion potent ia l and t h e r m o - e m f are shown in Fig. 5. I t is c lear f rom curve a of Fig. 5 that at the in- s tant of the deve lopment of the prec ip i ta t ion potent ial , the two cells show a t h e r m o - e m f equal to 48 • 1 /~V, which corresponds to a cooling of 1.24 • 0.02~ in the cell R1 which was the cold ]unction. I t has to be noted that the e lectrodes in the o ther cell, R2, were not in e lect r ica l contact and hence no prec ip i ta t ion potent ia l could develop and its t empe ra tu r e was s imply en- hanced on account of the release of the hea t of p r e - cipitation. In these exper imen t s both the prec ip i ta t ion cells R1 and R2 were at t he rma l equ i l ib r ium ini t ia l ly . S imul taneous crys ta l l iza t ion occurred in both of them.

4 I t i s d i f f i cu l t to d e t e r m i n e the transport n u m b e r o f u n h y d r a t e d ions . H o w e v e r , e f f o r t s h a v e b e e n m a d e t o d e t e r m i n e it by us ing a second so lute such as sucrose as a r e f e r e n c e for m e a s u r i n g the cha ng es of salt concentra t ion (26) . T h i s p r o c e d u r e has b e e n q u e s t i o n e d b y L o n g s w o r t h (27) s i n c e t h e added substance cannot be a s s u m e d to be mot ion le s s d u r i n g t h e passage of the current .



Vol. 130, No. 6 T H E R M O D Y N A M I C T H E O R Y 1333

-- 2.~2

2.io

2,oe

; /a .04 2,o2

i- 2.00

(6)

I I i t t

4. 6 a JO t2

"r~:ms (~.n.) .---"~

- 160

-120

-BO ~

-40 o

.,.,i

Fig. 4 Curve (a) buildup and decay of precipitation potential of KCI in water. Curve (b) thermo-emf corresponding to the cooling produced on the development of precipitation potential. The scale of the ordinate on the right-hand side refers to curve (a) while that on the left-hand side refers to curve (b).

-160

-80

- 4 0 . ~

0 o

o-

~ ~ ~ ~-- s

- Z0

~ . 6 0

I" 1 n I I

o 4 O 12 16 20

T.(me (H~n,) --'>

Fig. 5. Carve (o) buildup and decay of precipitation potential of KCI in water. Curve (b) thermo-emf corresponding to the cooling produced during the development of precipitation potential. The scale of the ordinate on the right-hand side refers to curve (a), while that on the left-hand side refers to curve (b).

One cell was involved in the deve lopment of the p re - c ipi ta t ion poten t ia l and the o ther was not. Therefore , a cooling in the fo rmer cell convincingly demons t r a t ed tha t la t t ice ene rgy is conver ted into e lec t r ica l energy on account of the deve lopment of the prec ip i ta t ion potential .

The above discussion leads to a re la ted quest ion as to wha t is the source of e lect r ica l ene rgy dur ing the deve lopment of dissolut ion potent ia l . Since

(AH)s ---- -- (AH)h -~- (AH)L [51]

i t was guessed tha t i t is the hydra t ion ene rgy which is conver ted into e lec t r ica l ene rgy dur ing the deve lop- men t of the dissolut ion po ten t ia l because the key r e -

action in this case would be

A + + B- + aq = A + (aq) + B- (aq) [52]

If this is so, much greater cooling than expected should occur on the electrode at which dissolution takes place. To test this point, experiments were performed with a twin calorimeter which confirmed this prediction (28). The pr inc ip le of these expe r imen t s can be unders tood f rom Fig. 6.

The above exper iments show tha t the energy re - leased in the la t t ice fo rmat ion is responsible for the genera t ion of the prec ip i ta t ion poten t ia l and the h y - d ra t ion energy is conver ted into e lec t r ica l ene rgy du r - ing the deve lopment of the dissolut ion potent ial . F u r - ther , these show tha t the inequa l i ty [44] is just if ied and the t he rmodynamic theory of the prec ip i ta t ion poten t ia l deve loped in this p a p e r is so far in qual i ta t ive agreement wi th the exis t ing data.

A c k n o w l e d g m e n t s The authore are extremely grateful to Professor

Graham Hills, of Strathclyde University for helpful comments. The authors are also grateful to UGC and CSIR for providing financial assistance. S. A. Khan is thankful to Dr. John Patterson, Principal, St. Andrew's College, Gorakhpur, for providing the necessary facili- ties.

Manuscript submitted Feb. 24, 1982; revised manuscript received ca. Dec. 15, 1982.

Any discussion of this paper will appear in a Dis- cussion Section to be published in the December 1983 JOURNAL. All discussions for the December 1983 Dis- cussion Section should be submitted by Aug. I, 1983.

LIST OF SYMBOLS a(*) ac t iv i ty of e lec t ro ly te at the c ry s t a l / so lu -

t ion in ter face a ( s s ~ ac t iv i ty of e lec t ro ly te in supersa tu ra ted so-

lu t ion ai (*) ac t iv i ty of i th species at the c ry s t a l / so lu -

t ion in ter face ai (ss) ac t iv i ty of i th species in supe r sa tu ra t ed so-

lut ion dA change in a rea of solid sur face

PYE pOTE H TIOt-IETER

NO ELECT~LCA L (ONNECT)ON

POTEHTIAL-

4 - Lmr~er ~lepos~.'~

E l - -

ELECTRO METER AMPLIFIER.

i l/m IWXI i ~'LEc.'r~CAt- coNNEc.'r~o~t

:::.;-:::::. . - B

- - C

Fig. 6. Experimental setup for the simultaneous measurement of dissolution potential and thermo-emf produced during dissolution. (El, E2 = platinum electrodes embedded with KCI crystals; Ez' E2" = bare platinum electrodes; Tz, T2 = copper-contantan thermocouple junction; R1, R~ = twin dissolution cells; and C = double walled vessels.)



1334 J. Electrochem. Soc.: ELECTROCHEMICAL SCIENCE AND TECHNOLOGY June 1983

C o ei F (hH) ~

(~,H) h (AH)L (a l l ) s I Jc Ji LiA Mi a, Mi~

dM~ h i pa, pB aQ r re dS des d~S

dS% dS~

t tio ti

T Ts dU% dU~

U+,U-

U+ ~ U_ ~

dV% dVB

x~ Xl

gi

#i s, /zi~

~i~

P {; r a, r

(4r ~,~

(4r Phase

(~b) PDtio. i, J 1,2

initial concentration of the solution charge per gram of ith species Faraday of electricity the enthalpy change associated with the development of precipitation potential hydration energy lattice energy heat of solution current crystallization flux mass flux of ith species phenomenological coefficients masses of ith species in phases a and ;~, respectively change in total mass of phase hydration number of ith species pressures of phases a and ~, respectively total heat supplied to the system radius of crystal nuclei radius of crystal nuclei at equilibrium total entropy change entropy exchanged with the surroundings entropy change due to irreversible processes ir~side the system entropy changes in phases ~ and ~, respectively time transport number of unhydrated ith species transport number of hydrated ith species which is the conventional transport number temperature of supersaturated solution temperature of saturated solution changes of internal energy in phases a and ~, respectively mobilities of hydrated cations and anions, respectively mobilities of unhydrated cations and anion, respectively changes in volume of phases a and ;~, respectively force conjugate to the crystallization flux force conjugate to the mass flux of ith species valency of ith species solid phase supersaturated solution of electrolyte surface tension at crystal/solution interface chemical potentials of ith species in phases

and ~, respectively standard chemical potential of ith species in liquid phase standard chemical potential of ith species in solid phase density rate of entropy production electric potentials of phases a and ~, respectively electric potential generated on the crystallization of electrolyte phase potential at the crystal/saturated solution interface precipitation potential components component 1 and component 2

REFERENCES 1. R. P. Rastogi, R. K. Dass, and B. P. Batra, Nature,

191, 765 (1961). 2. R. P. Rastogi and R. D. Shukla, J. Appl. Phys., 41,

2787 (1970). 3. R. P. Rastogi, R. D. Shukla, and S. Bhagat, This

Journal, 121, 1564 (1974). 4. H. L. Girdhar and R. P. Matta, Electrochim. Acta,

22, 173 (1976). 5. G. W. Gross, "Trace Inorganics in Water," Ad-

vanc.es in Chemistry Series, No. 73, American Chemical Society (1968).

6. E. J. Workman and B. E. Reynolds, Phys. Rev., 78, 254 (1950).

7. E. J. Workman and B. E. Reynolds, Nature, 169, 1108 (1952).

8. F. Heinmetes, Trans. Faraday Soc., 58, 788 (1962). 9. A. W. Cobb and G. W. Gross, This Journal, 116,

611 (1969). 10. R. P. Rastogi and R. D. Shukla, Indian J. Chem., 6,

611 (1968). 11. K. Azuma and H. Kametani, J. Phys. Soc. Jpn., 12,

313 (1957). 12. H. L. Gridhar and R. P. Matta, Indian J. Chem.,

14A~ 160 (1976). 13. H. L. Girdhar and R. P. Matta, J. Electroanal.

Chem., 89, 407 (1978). 14. N. Ibl, D. Richarz, and H. Federkehr, Z. Phys.

Chem. Neue Folge Bd., 988, 123 (1975). 15. N. Ibl, et al., "Topics in Pure and Applied Electro-

chemistry," Published by SAEST (1975). 16. R. P. Rastogi and S. A. Kahn, This Journal, 127, 1989

(1980). 17. S. R. de Groot, "Thermodynamics of Irreversible

Processes," p. 96, North-Holland Publishing Co., Amsterdam (1952).

18. R. A. Laudise, "The Growth of Single Crystals," p. 102, Prentice-Hall, Inc., Englewood Cliffs, NJ (1970).

19. J. V. Butler, "Electrical Phenomena at Interfaces," p. 14, Methuen & Co. Ltd., London (1951).

20. R. P. Rastogi, B. P. Batra, and R. K. Dass, Bull. Natl. Inst. Sci. India, 29, 231 (1961).

21. Vishnu, S. A. Khan, and E. Jahan, Electrochim. Acta, 24, 387 (1979).

22. L. I. Antropov, "Theoretical Electrochemistry," p. 74, (Translated from Russian by Artavaz Bek- nazarov), Mir Publishers, Moscow (1977).

23. J. Koryta, J. Dvorak, and V. Bohackova, "Electro- chemistry," p. 14, Methuen & Co. Ltd., London (1970).

24. J. O'M. Bockris and A. K. N. Reddy, "Modern Elec- trochemistry," p. 125, A Plenum/Rosetta Edition, New York (1976).

25. H. Bloom, "The Chemistry of Molten Salts," p. 97, Benjamin Inc., New York (1967).

26. E. W. Washburn and E. B. Millard, J. Am. Chem. Soc., 37, 694 (1915).

27. L. G. Longsworth, ibid., 69, 1288 (1947). 28. S. A. Khan, Ph.D. Thesis, Gorakhpur University,

Gorakhpur (1980).



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