Physical Chemistry. Electrochemistry I. Dr. Rajeev Jain School of Studies in Chemistry Jiwaji University, Gwalior 11

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1 Physial Cheistry Eletroheistry I Dr. Rajeev Jain Shool of Studies in Cheistry Jiwaji University, Gwalior 11 CONTENTS Condution in Metals and in Eletrolyte Solutions Metalli Condutors Eletrolyti Condutors Condution in Eletrolyte Solutions Strong and Weak Eletrolytes Strong Eletrolytes Weak Eletrolytes Speifi Condutane and Molar Condutane Measureent of Molar Condutane Deterination of Cell Constant Variation of Molar and Speifi Condutane with Dilution Kohlraush s Law of Independent Migration of Ions Migration of Ions Arrhenius Theory of Eletrolyti Dissoiation Ostwald s Dilution Law Appliations of Ostwald s Dilution Law Debye-Hukel-Onsagar Equation Transport Nubers Appliations of Condutivity Measureents 1. Condution in Metals and in Eletrolyte Solutions Condutors an be divided broadly into two ategories: (i) Metalli or eletroni ondutors (ii) Eletrolyti ondutors (i) Metalli Condutors Metals are the best ondutor and it reains unhanged with the passage of urrent. A etalli ondutor behaves as if it ontains eletrons whih are relatively free to ove. So eletrons are onsidered as harge arrier in etals. Therefore, these ondutors are also alled eletroni ondutors. Metalli ondution or eletroni ondution is the property possessed by pure etals, ost alloys, arbon and ertain solid salts and oxides. (ii) Eletrolyti Condutors 1

2 Condutors, through whih passage of an eletri urrent through the results in atual transfer of atter or brings about a heial hange in the, are alled eletrolyti ondutors or eletrolytes. Eletrolyti ondutors are of two types: - (a) In the first ategory are eletrolyti ondutors, whih ondut eletrolytially in the pure state, suh as aids, bases and salt in water. e.g. NaCl, NaNO 3, K 2 SO 4 et. (b) In seond ategory are generally put eletrolyti ondutors whih onsists of solutions of one or ore substanes. Eletroheistry is ainly onerned with this type of eletrolyti ondutor. Generally eletrolyti solutions are prepared by dissolving a salt, aid or base in water or other solvents. There is a speial lass of ondutors, whih ondut partly eletronially and partly eletrolytially, are known as ixed ondutors. For exaple, solution of the alkali and alkaline earth etals in liquid aonia are ixed ondutors. Fused uprous sulphide onduts eletronially, but a ixture with sodiu or ferrous sulphide also shows eletrolyti ondution. 1.1 Condution in Eletrolyte Solutions The passage of urrent through solutions of salts of etals suh as zin, iron, nikel, adiu, lead, opper, silver and erury results in the liberation of these etals at the athode and fro solutions of salts of the etals. If the anode onsists of an attakable etal, the flow of the urrent is aopanied by the passage of the etal into solution. When the anode is ade of an inert etal, e.g., platinu, an eleent is generally set free at this eletrode; fro solutions of nitrates, sulphates, phosphates, et., oxygen gas is liberated, whereas fro halide solutions, other than fluorides, the free halogen is produed. The deoposition of solutions by the eletri urrent, resulting in the liberation of gases or etals, is known as eletrolysis. 1.2 Strong and Weak Eletrolytes Solutes giving onduting solution in a suitable solvent are alled eletrolytes. On the basis of degree of ionization, these eletrolytes have been divided into two ategories. (i) Strong eletrolytes (ii) Weak eletrolytes Strong Eletrolytes Substanes, whih are highly dissoiated and give solutions with high ondutane in water, are alled strong eletrolytes. Due to the high degree of dissoiation of strong eletrolytes these substanes are good ondutor of eletriity i.e., aqueous solutions of these substanes have high value of olar ondutane and on dilution the inrease in their olar ondutane is very sall. This is due to the fat that suh eletrolytes are opletely ionized at all dilutions therefore on further dilution the nuber of urrent arrying partiles does not inrease in the solution. Thus, solutions of eletrolytes that have high olar ondutane, and inreases very slowly on dilution has a high degree of dissoiation is alled strong eletrolyte. During the passage of an eletri urrent through solutions, flow of eletriity is assoiated with the oveent of partiles, whih are alled ions. The ions arrying positive harges and oving in the diretion of the urrent, i.e., towards the athode, are referred to as ations and those arrying a negative harge and oving in the opposite diretion, i.e., towards the anode, are alled anions. 2

3 1.2.2 Weak Eletrolytes Weak aids and weak bases, e.g., aines, phenols, ost arboxyli aids and soe inorgani aids and bases, suh as hydroyani aid and aonia, and a few salts, e.g., eruri hloride and yanide, are dissoiated only to a sall extent at reasonable onentration; this group of opounds in general are alled as weak eletrolytes. The olar ondutane of the solutions of these eletrolytes inreases rapidly on dilution. The reason of this is that ore oleules ionize on dilution inspite of this they are never opletely ionized. For these eletrolytes, the nature of the solvent is also iportant; a partiular opound ay be strong eletrolyte, being dissoiated to large extent, in one solvent, but ay behave as weak eletrolyte in other solvent due to low degree of dissoiation. Q. Whih of the following are strong eletrolytes? (a) CH 3 COOH (b) H 2 SO 4 () NH 4 Cl (d) NaCl (e) HNO 3 (f) NH 3 () & (d) are soluble hydroxides; (b) and (e) are first proton. 2. Speifi Condutane and Molar Condutane Condutane Consider a unifor bar of ondutor of length l and ross-setional area a sq. and the ross setion is retangular and that the whole body is plaed into ubes of one. side, as shown in Figure 1. Fig. 1 3

4 Oh s law states that The agnitude of urrent (I) passing through a ondutor is diretly proportional to the potential differene (E) applied aross it and inversely proportional to the resistane (R) of the ondutor. Q. The olar ondutivity of a solution ontaining 2.54 g of CuSO 4 /L is /Ω.ole. What is the resistane of a 3 of this solution when plaed between two eletrodes 1.00 apart, eah having an area of κ N, where C ole/ 3 C g of CuSO4 L 1ol 159.5g 2ol ol M CuSO κ Ωol ol x 10-3 S -1 We know that, 1 1 R. κ A 2.89 x 1 10 S Ω i R E (1) Thus, etalli ondutor and eletrolytes obey Oh s law. Condutane is the reiproal of resistane and is expressed in Sieen (S). 1 Condutane (2) R 4

5 2.1 Speifi Condutane or Condutivity The resistane of the bar (Fig.1) to the passage of eletriity through it is proportional to its length l () and inversely proportional to the area of ross setion a, therefore, the resistane R is given by the relationship. 1 1 R or R ρ ohs (3) a a Where, ρ is a onstant known as speifi resistane or resistivity. If, l 1., a 1 2 Then, ρ R oh Speifi ondutane of any onduting aterial is defined as the reiproal of its speifi resistane. It is given the sybol κ and is stated in reiproal oh -1, nowadays alled S -1. Condutane G is then defined as a G κ S (4) l G R 1 The ondutane (G) is the reiproal of resistane, i.e. G R 1 (5) 2.2 Molar Condutane Molar ondutivity of a solution is the ondutivity of that volue ontaining 1 ole of an eletrolyte when plaed between two suffiiently large eletrodes, whih are 1. apart. It is represented by Λ. Suppose 1 ole of an eletrolyte is dissolved in Λ 3 of a solution. Suh a solution will over an area of ν 2 of the eletrodes kept 1 apart. The ondutane of this syste, whih is the l olar ondutane Λ, ay be derived fro equation R κ a, where a is equal to ν 2 and l is 1 ; thus (6) Λ κ ν Where ν is the dilution of the solution in. per ole. If is the onentration of the solution, in ol /l, then ν is equal to 1000/, then it beoes Λ 1000 κ (7) 5

6 However if is expressed the equivalent ondutane of any solution an thus be readily derived fro its speifi ondutane and onentration. Sine the units of κ are S -1, those of Λ oes out fro eq. (6) or (7) to be S 2 ol -1. If one uses SI units then the units for ρ is Ω -1 for κ is S and for Λ is S 2 ol -1. In earlier the ter equivalent ondutane Λ was used. It is defined as the ondutane of 1gra equivalent of eletrolyte in solution with a given onentration. 3. Measureent of Molar Condutane Condutane is the reiproal of resistane and the resistane an be deterined by a Wheatstone bridge iruit in whih the ondutivity ell fors one ar of the bridge, this ethod is known as Null Method (Kohlraush ondutane bridge). (A) (B) Fig. 2: (A) Wheatstone bridge iruit for easureent of ondutivity, (B) Condutivity ell with one ar of a resistane bridge for easureent of ondutivity of an eletrolyte. The ars AB and BC represented by resistane R 1 and R 2 are usually in the for of a single alibrated slide wire resistor with a sliding ontat onneted to the null detetor. The solution whose ondutane is to be deterined is plaed in ondutivity ell. When the bridge is balaned, assuing that the ondutivity ell behaves as a pure resistane, then the voltage between B and D is equal to zero. R1 R x R3 R 2 (8) 6

7 By adjustent of the ratio R 1 /R 2, a wide range of resistanes an be easured. However, whenever possible, this ratio should not deviate too far away fro unity. The ell apaitane is balaned out by providing a variable apaitor in parallel with resistane R 3. It is so adjusted that the detetor gives a sharply defined balane point. The null detetor is not an ordinary galvanoeter as it is not sensitive to alternating urrent at the frequeny, whih is eployed to exite the bridge. The ost popular detetor in use is the agi eye, or the athode ray osillosope. The proble in deterining the resistane of the solution of an eletrolyte by the above ethod is that eletrolysis of the solution also ours siultaneously with the ondution of the urrent and due to this: - a) Polarisation sets in and auses the resistane to vary. b) The onentration of the solution hanges. In order to avoid these opliations an alternating soure of power with frequeny ~ 1000 Hz is used. 3.1 Deterination of Cell Constant The eletrodes in the ell are not exatly 1 apart and ay not have surfae area of 1 sq. (1 2 ). Thus the value of observed ondutivity is not equal to speifi ondutane but is proportional to it. 1 R R ρ or R ρ x or x (9) a ρ Where, x l/a ell onstant Cell onstant, x κ/g or Speifi ondutane, k ell onstant a X observed ondutane G. Taking an exaple of N/50 KCl solution, the speifi ondutane at 25 o C is S -1. We know that, Cell onstant, x /observed ondutane (G) By putting the value of observed ondutane in the above expression, one an alulate ell onstant. One the value of ell onstant has been deterined, great are ust be taken not to hange the distane between the eletrodes during further easureents in any way. 4. Variation of Molar and Speifi Condutane with Dilution 4.1 Effet of Dilution on Molar Condutane With dilution following variations on olar ondutane are observed: (i) The value of olar ondutane inreases on dilution. The inrease is due to the fat that olar ondutane is the produt of speifi ondutane (k) and the volue (V) of the solution ontaining 1 ole of the eletrolyte. As the dereasing value of speifi ondutane is ore than opensated by the inreasing value of V, thus the value of olar ondutane (Λ ) will inrease 7

8 with dilution. The variation of olar ondutane at different dilutions of soe oon eletrolytes are shown Table 1: Table 1: Molar Condutane of soe oon eletrolytes at 25 0 C in S 2. Conentration (ole/l) NaOH KCl HCl CH 3 COOH AgNO (ii) Effet of dilution on olar ondutane is shown in the Figure.4, where the olar ondutane at different dilutions is plotted against onentration (M). Molar Condutane Conentration in ole per litre Fig. 3: Variation of equivalent ondutane with onentration 8

9 Fro Figure 3 it is lear that strong eletrolytes suh as KCl, an have liiting value at zero onentration obtained by extrapolation (i.e. at infinite dilution) whereas for weak eletrolyte suh as aeti aid, there is no indiation that a liiting value an be obtained by the extrapolation of the graph to zero onentration. It eans we annot experientally deterine olar ondutivity of weak eletrolytes at infinite dilution. (iii) The axiu value of the olar ondutivity is tered as the olar ondutivity at zero onentration (or infinite dilution) and is tered. 4.2 Effet of Dilution on Speifi Condutane The speifi ondutane depends on the nuber of ions present per unit volue of the solution. Sine on dilution the degree of dissoiation inreases but the nuber of ions per unit volue dereases, therefore it is expeted that the speifi ondutane of a solution derease on dilution (Table.2). Table 2: Speifi Condutane of NaCl Solution at 25 0 C: Conentration Molar Condutane (S 2 ) Kohlraush s Law of Independent Migration of Ions It has been observed that the ondutivity of solution inreases with dilution until it reahes its liiting value at infinite dilution is represented as. Kohlraush ade a systeati study of for different eletrolytes and onluded that eah ion ontributes a harateristi value of its own to olar ondutivity at infinite dilution irrespetive of the nature of the other ion present. Consider the values in Table.3 to appreiate the law: 9

10 Table.3: Values of Λ o for different eletrolytes Eletrolyte Λ o at 25 Differene (S 2 ) (S 2 ) I. KCl NaCl II. KNO NaNO III. KCl KNO IV. NaCl NaNO Eletrolytes in set I and II have a oon anion so that the differene an only be due to the differene in ontribution to Λ by K and Na ions. In the sae way in sets III and IV the onstant differene ay be attributed to the differene in ontribution to ade by the Cl - and NO - 3 ions. These observations an be explained by Kohlraush s law of independent igration of ions whih states that: At infinite dilution, where the eletrolytes are fully dissoiated and free fro interioni effets, eah ion igrates independently of its o-ion. As suh eah ion ontributes its definite share to the total equivalent ondutivity of the eletrolyte, whih depends only on the nature of the ontributing ions and not at all on the ion with whih it is assoiated as a part of the eletrolyte. In other words, the olar ondutivity at infinite dilution of an eletrolyte is equal to the su of the ioni ondutanes of the ions oposing it, provided the solvent and teperature are the sae. ν ν (10) Where, Λ a and Λ are the ioni ondutanes of the anion and ation respetively at infinite dilution and ν and ν is the nuber of ations and anions in whih one oleule of the eletrolyte. For anion and ation this value is onstant at a fixed teperature and in a given solution. It is expressed in S 2 ol -1 or S 2 ole

11 Q. The liiting olar ondutivities of KCl, KNO 3, and AgNO 3 are S 2 ol -1, S 2 ol -1 and S 2 ol -1 respetively at 25 C. Calulated the liiting olar ondutivity of AgCl at this teperature? The basis for the solution is Kohlraush s law of independent of ions. Swithing ounterions does not affet the obility of the reaining other ion at infinite dilution. ν λ ν λ (KCl) ( K ) λ( Cl ) λ S 2 ol -1 (KNO 3 ) ( K ) λ( NO3 ) λ S 2 ol -1 (AgNO 3 ) ( Ag ) λ( NO3 ) λ S 2 ol -1 Hene, (AgCl) (AgNO 3 ) (KCl) - (KNO 3 ) S 2 ol S 2 ol Appliations of Kohlraush Law Calulation of olar ondutivity of a weak eletrolyte at infinite dilution It is not possible to deterine the value of for weak eletrolytes sine we annot obtain the liiting value of the olar ondutivity for a weak eletrolyte. This is done indiretly by the olar ioni ondutane for the individual ions of the weak eletrolyte as follows: For e.g. olar ondutane of aeti aid at infinite dilution an be alulated fro the olar ondutane at infinite dilution of hydrohlori aid, sodiu aetate and sodiu hloride as follows: HCl H CH 3 COONa NaCl Add eq. 11 and 12 and subtrat 13 we get: HCl CH 3 COONa - NaCl Cl - x y z Na H H Cl (11) Na CH 3 COO (12) Cl (13) Cl - CH 3 COOH x y z Na CH 3 COO - CH 3 COO - - CH 3 COOH Na - 11

12 5.1.2 Calulation of Degree of Dissoiation of Weak Eletrolytes The degree of dissoiation of weak eletrolyte suh as NH 4 OH, aeti aid an be deterined by easuring the olar ondutivity Λ, of the solution of the eletrolyte at any given dilution. For e.g. the degree of dissoiation (α ) of a weak eletrolyte at the onentration C ole per liter ay be given by the following relation: α (14) Where, Λ is the equivalent ondutane of eletrolyte at onentration and Λ is the equivalent ondutane of the sae eletrolyte at infinite dilution. Hene, easureent of Λ perits evaluation of α if Λ is known Deterination of Solubility of Sparingly Soluble Salts Salts like AgCl, BaSO 4, CaCO 3, Ag 2 CrO 4, PbSO 4, PbS, Fe(OH) 3 et. are ordinarily regarded as sparingly soluble and have a very sall but definite solubility in water. The solubility of suh sparingly soluble salts is obtained by deterining the speifi ondutivity (κ) of a saturated salt solution. The olar ondutivity at suh high dilution an pratially be taken as, i.e. for sparingly soluble salts, κ.v (15) Where, V is the volue in 3 ontaining 1 ole of salt while, an be alulated using Kohlraush s law. Substituting the values of κ and in eq. 15; below the value of V an be alulated: V / κ ( ν ν But V 3 of saturated solution ontains 1 ole of salt of saturated solution ontains 1000 / V ole of salt Hene, solubility of salt, S 1000/V M 1000/V x M. Wt. g/l ) / κ (16) Deterination of ioni produt of water With the help of speifi ondutivity of water, the ioni produt of water an be deterined. The ionization of water ay be represented as, H 2 O H OH - The produt of the onentrations of H and OH ions expressed in ol/l is alled ioni produt of water and is represented by K w. 12

13 i.e. [H ] [OH - ] K w The easured speifi ondutivity of the purest for of water is x 10-6 S -1. The olar ondutane is given by Λ K v x 18 (17) Thus, x 10-6 x x 10-6 S 2 ol -1 The olar ondutane of water at infinite dilution an be obtained by (H 2 O) Λ (H ) Λ (OH - ) Therefore, Λ (H 2 O) S 2 ol -1 and for water, α Λ / Λ x 10-3 x 18/ C H C - OH x 10-3 x 18/ (Conentration of water 1000/ M C H C - OH α x x x x 10-7 Kw (1.003 x 10-7 ) (1.006 x ) 13

14 Q. Calulate the value of K w for water in M NaCl solution. The experiental value is 1.65 x The therodynai value of the equilibriu onstant for the dissoiation of water 2H H - 2 O 3 O OH K w a H 1.01 x O ( ah 2 Water is a liquid ating as a solvent, & its ativity is its ole fration, here 55.3/ Then 0 K w [ a ) O OH H 3O ] [ OH ]. 3O. f H f OH K. 0 w K w f H3 O f OH Where K w is the onstant in ters of onentration ativity oeffiients fro the extended Debye- Hukel eqn., f H3 O Using these values, we obtain and f OH K w x x K w 1.61 x Deterination of Ioni Mobility of an Ion Ioni obility of an ion is defined as the speed of the ion in entieters per seond, when a potential of one volt is applied between two eletrodes kept 1 apart. The usefulness of ioni obilities is that they provide a link between easureable and theoretial quantities. Following equation shows the relationship between an ion s obility and its olar ondutivity. λ zuf (18) where F is the Faraday onstant (F N A e) Equation (18 ) applies to the ations and to the anions and an be dedued to equation (18a ) for solution in liit of zero onentration when there are no interioni interations. For a syetrial ( z u ν z u ν ) F z : z eletrolyte this equation siplifies to (18a) 14

15 6. Migration of Ions z ( u u ) F (19) As in a solution of an eletrolyte the eletriity is onduted by igration of ions and the ions ove in solution independently towards the oppositely harged eletrodes. This fat an be illustrated by following siple experients: (i) Lodge s oving boundary evidene A glass tube of the shape as shown in the Figure 4 is taken and its iddle portion between two ars is filled with a jelly of agar-agar. A trae of sodiu hydroxide (or any alkali) and phenolphthalein are added during the preparation of the jelly. It beoes red due to phenolphthalein in alkaline ediu. The jelly is allowed to set. After that dilute sulphuri aid is added to the left ar ontaining the anode and sodiu sulphate solution to the right ar in whih the athode is plaed. On passing the urrent, hydrogen ions igrate along the solution towards the athode and their oveent an be onitored by observing the gradual fading of red olour in the jelly due to the neutralization of the alkali by the hydrogen ions and the oveent of the original boundary. Experient learly shows that positively harged hydrogen ions (H ) are oving towards negatively harged athode. Fig. 4: Lodge s oving boundary experient (ii) Moveent of oloured ions A U shaped glass tube is taken and its iddle portion is filled with an aqueous solution of 5% agar-agar and a ixture of opper sulphate and potassiu dihroate in distilled water. This dark green-oloured solution after ooling fors a jelly and sets. The position of the surfae of green solution in both the ars of the U-tube is arked by plaing sall aount of haroal on it as shown in the Figure 5(A). 15

16 (A) (B) Fig. 5: Migration of ions In both the ars then a solution of potassiu nitrate and agar-agar is filled. This on ooling also set as jelly. Over this jelly, solution of potassiu nitrate in distilled water is added and the eletrodes are iersed in it. With the appliation of potential differene aross the eletrode the blue olour of opper ions rises in to the jelly towards the athode. The reddish yellow dihroate ions ove up in the other ar of the tube towards the anode. In this experient two types of ions an be learly seen oving with well-defined boundaries {Figure 5(B)}. 6.1 Speeds of igration of ions during eletrolysis During eletrolysis ions are liberated aording to Faraday s law at athode and anode but their relative rate of oveent towards the eletrodes ay be different. 16

17 Fig. 6: Anodi and Cathodi opartents showing speeds of igration of ions during eletrolysis In the Figure. 6 A and C are two porous diaphrags, whih prevent onvetion urrents but allow the passage of ions. The ell is divided into an anodi and athodi opartent. Suppose initially 13 oleules were present. The nuber of oleules in eah anode and athode opartent is equal i.e. 4 and 5 oleules are present between the two segent in a {Fig.6(I)}. Consider the following possibilities in referene to the above experient. (i) {Fig.6(II)} shows the oveent of two anions alone, here only anions are apable of oveent. (ii) Both anions and ations ove at the sae rate towards the opposite harged eletrodes, the ondition as shown in {Fig.6 (III)}. (iii) In another situation ations ove at twie the rate of the anions (Fig.6 (IV)}. In all the above onditions ions are always liberated in equivalent aounts; the effet of differene in their rate only ause a hange of onentration around eletrodes. Further, fro the above experient the following expression an be dedued. Redution in nuber of anions around anode Redution in nuber of ations around athode Speed of ation towards athode Speed of anion towards anode 17

18 Q. At 25 C the olar ioni ondutivities of Li, Na, and K are 3.87 S 2 ol -1, 5.01 S 2 ol -1, and 7.35 S 2 ol -1 respetively. Calulate their obilities. λ We have, u zf S ol (i) u ( Li ) Col 4.01 x 10-5 SC x V -1 s -1 (1CΩ 1AsΩ 1Vs) S ol (ii) u ( Na ) Col 5.19 x V -1 s S ol (iii) u ( K ) Col 7.62 x V -1 s Arrhenius Theory of Eletrolyti Dissoiation Arrhenius (1887) put forward the theory of eletrolyti dissoiation, as a ore expliit for of one he had proposed in 1883, whih fors the basis of the odern treatent of eletrolytes. The assuption ade was that when an aid, base or salt is dissolved in water, a onsiderable portion beoes spontaneously dissoiated into positive and negative ions. Considering an eletrolyte Av B v, whih ight undergo oplete dissoiation to for ν positive ions and ν negative ions aording to the equation Av B v ν z A ν z B (20) We ust alulate the net nuber of partiles that result fro a degree of dissoiation α. If is the olality of the eletrolyte, α is the degree of dissoiation, the onentration of undissoiated z z eletrolyte will be -α (1 - α ). In addition, the onentration of A and B will be ν α and ν α, respetively. Here the onentration of partiles is (1 - α ) ( ν )α ( ν )α and let ν be the total nuber of ions yielded by oplete dissoiation of the eletrolyte, i.e, ν ν ν. With this notation the olality of partiles for the partially dissoiated eletrolyte is (1-α ) α ν rather than the value of expeted for no dissoiation. The van t Hoff I fator an be written as (1 α ) αν 1 α αν (21) 18

19 Fro this interpretation of i, one obtains i 1 α ν 1 This relation an also be used inspite of the relation α dissoiation of an eletrolyte. (22), for alulating the degree of 8. Ostwald s Dilution Law The weak eletrolyte ionizes to a very sall extent and their olar ondutivity doesn t attain a liiting value at high dilution. In suh solutions there is equilibriu between free ions and undissoiated oleules. The equilibriu an be written in the for: MA M A - (23) (1-α) α α Where M A - free ions MA undissoiated portion of the eletrolyte inluding both nonionized oleules and ion pairs. By law of equilibriu, K a M x a A - am A (24) Where, a s ativities of indiated speies K the equilibriu onstant alled dissoiation onstant of the eletrolyte Equation (24) an be written as the produt of onentration C, in g ions or oles per litre, and the ativity oeffiient f, above equation beoes; K M x MA A - f M f x MA f A (25) Here, α is the degree of dissoiation of eletrolyte, i.e., the fration of the eletrolyte in the for of free ions and is its total onentration, (M). Both M and A - are equal to αc while MA is equal to (1- α) then eq. (25) an be written as, 19

20 K 2 α - (1 α) f M f x MA In suffiiently dilute solution the ativity oeffiient is approxiaately unity, then eq. (26) takes the for: K 2 α (1 α) f A (27) (26) This is the expression of the dilution law, first derived by W. Ostwald in For weak eletrolytes like aeti aid (CH 3 COOH) or aoniu hydroxide (NH 4 OH) the value of degree of dissoiation is very sall, i.e., (1 α) 1 eq. (27) an be written as; 2 K α (28) 2 K α (29) α K (30) 8.1 Verifiation of the Dilution Law The degree of dissoiation α, is deterined by ondutivity easureents and is obtained by dividing the olar ondutane at a ertain dilution, Λv by the equivalent ondutivity at infinite dilution Λ, and is given by eq. (31). α (31) Deterination of Dissoiation Constant of Aeti Aid (CH 3 COOH) Aeti aid is a weak eletrolyte and when it dissolves in water it dissoiates as CH 3 COOH CH 3 COO - H 20

21 If g oles of aeti aid are dissolved per liter of the solution and α is the degree of dissoiation, then the dissoiation onstant K is given by K 2 α (1 α) (32) If is the equivalent ondutane of the CH 3 COOH solution at the given dilution and at infinite dilution, then α (33) The value of is deterined by ondutivity easureent at onentration C. The value of for aeti aid an be alulated with Kohlraush s law i.e., Λ for (CH 3 COOH) Λ (CH 3 COO - ) Λ for (H ) (34) hos The value of K an be alulated by substituting the values of, and. Q. At 25 C, the olar ondutane of propanoi aid at infinite dilution is S 2 ol -1. If its ionization onstant is 1.4 x Calulate olar ondutane of 0.05 M propanoi aid solution at 25 C? K a 2 x (1 x) If x << 1 than K a x 2 K a K a 1.4 x x x x x S 2 ol -1 21

22 9.2 Appliations of Ostwald s Dilution Law Appliations of ostwald s dilution law are any. Soe of the are disussed below: Dissoiation onstant of onobasi aid Dissoiation onstant of weak eletrolytes suh as weak aids, weak bases an be deterined with the help of Ostwald s dilution law. Consider the solution of a weak aid HA with onentration. If α is the degree of dissoiation at equilibriu, then HA - H A 0 0 (initial on.) (35) (1- α) α α (equilibriu on.) By applying the law of ass ation, K a [ H ] [ A ] [ HA] (36) where K a is the dissoiation or ionization onstant of the aid. Substituting [H ], [A - ] and [HA] in eq. (36), we get K a 2 α (1 α) (37) Dissoiation onstant of a weak base Let us onsider a weak base BOH dissolved in water. With onentration. Let α be the degree of dissoiation at equilibriu, the following ioni equilibriu exist in solution, - BOH B OH (1-α) α α The dissoiation or ionization onstant is given by (38) 22

23 K b [ B ] [ OH ] [ BOH ] 2 α (1 α) (39) (i) Experiental deterination of dissoiation onstant We have already shown that the dissoiation onstant of weak aids and weak base an be represented by K a 2 α (1 α) (40) Here α an be deterined fro the expression, α K a 2 1 (41) 2 (42) ( ) The value of is alulated fro Kohlraush law i.e. of the solution is deterined fro its speifi ondutivity. a of this solution (43) Where, a and a are ioni ondutane of anion and ation of weak aids or bases. By knowing the value of and values of α an be alulated. The dissoiation onstant an also be alulated using equation (40). 23

24 Q. Molar ondutivities at infinite dilution at 25 C of NH 4 Cl, NaOH and NaCl are 129.8, & S 2 ol -1. For 0.01 M NaOH olar ondutane is 9.33 S 2 ol -1. Calulate ionization onstant of NH 4 OH. - NH 4 OH NH 4 OH By Ostwald s dilution law, ionization onstant K b of weak base is 2 x K b (1 x) X degree of ionization (0.01N) 9.33 S 2 ol -1 By Kohlraush s law (NH 4 OH) (NH 4 Cl) (NaOH) S 2 ol x x K b (1 x) (NaCl) 0.01 (1 x (0.0392) ) x Debye-Hukel-Onsagar Equation In order to explain the abnoral behaviour of strong eletrolytes nuber of sientists worked in this field viz. Noyes (1904), Sutherland (1906), Bjerru (1909) and Milner (1912), Debye and Hukel in 1923, and Onsagar in 1926 put forward the odern theory of strong eletrolytes known as Debye- Hukel- Onsagar theory of strong eletrolyte. Debye-Hukel treatent deals with the distribution of ions around a given ion and the net effets of these neighbouring ions have on the entral ion. Debye and Hukel derived an equation based on the quantitative treatent of inter ioni attration. This equation was later on odified by Onsagar and is known as Debye-Hukel-Onsagar (DHO) equation for strong eletrolyte. It shows how the potential energy of an ion in solution depends on the ioni strength of the solution. In the ase of strong 24

25 eletrolytes the value of olar ondutane at infinite dilution is uh less than unity due to following effets: (i) Relaxation effet Interioni fores are present and eah ion has a tendeny to be surrounded on the tie average by ions of opposite harge alled the ioni atosphere. A negative ion is surrounded by the ions of opposite harge alled the ioni atosphere. When an EMF is applied, the negative ions igrate towards the anode where the ioni atosphere of positive ions is left behind to disperse, at this tie a new ioni atosphere is under foration. The rate of foration of new ioni atosphere is not the sae at whih the previous ioni atosphere disperses and the later takes ore tie. This tie is alled the relaxation tie. In the ase of the oving ion there will always be an exess of ions of opposite harge. The ions will always be dragged bak. This effet will derease the obility of the ions and is known as relaxation effet or asyetri fator. (ii) Eletrophoreti effet The solvent oleules attah theselves to ioni atosphere and the ions ove in the diretion opposite to that of entral ion. It produes frition due to whih the obility of the entral ion is retarded. This effet is alled the eletrophoreti effet. The eletrop horeti effet redues the olbility of the ions and hene also redues their ondutivities. The quantitative forulation of these effets is far fro siple, but the Debye-Hukel-Onsager theory is an attept to obtain quantitative expression at about the sae level of sophistiation as the Debye-Hukel-theory itself. The theory leads to a Kohlraush like expression in whih κ A B (44) 2 ef 2 1/ 2 z 2 with A 3πη εrt Where, η flux of oentu ε eletri perittivity of the solvent and q for a 1,1-eletrolyte (Table 4) B 3 1/ 2 qz ef 24πεRT 2 εrt (45) 25

26 Fig. 7. The dependene of olar ondutivities on the square root of the ioni strength, and oparison (shown by dotted lines) with the dependene predited by the Debye-Hukel-Onsager theory. Figure 7 shows the dependene of olar ondutivities on the square root of the ioni strength and oparison with the dependene predited by the Debye Hukel theory. The agreeent is quite good at very low olar onentrations (less than about 10-3 M, depending on the harge type). Green Kubo relationship an be applied on eletri ondution, whih expresses a transport properly in ters of the flutuations in irosopi properties of a syste. The eletrial ondutivity is related to the flutuations in the saple that arises fro variations in the veloities of the ions. 1 κ KTV 0 ( ) j( t) j 0 dt (46) j N zieν i (47) i where ν i is the veloity of the ion i at a given instant and the angular brakets denote an average over the saple. If the ions are very obile, there will be large flutuations in the instantaneous urrents in the sae, and the ondutivity of the ediu will be high. If the ions are loked into position, as in an ioni solid, there will be no instantaneous urrents, and the ioni ondutivity will be zero. The veloities of the ions are alulated expliitly in a oleular dynais siulation, so j 0 j t, an be evaluated reasonably siply. the orrelation funtion, the quantity ( ) ( ) 26

27 11.Transport Nubers The transport nuber (t ) is defined as the fration of total urrent arried by the partiular ioni speies in the solution. In a siple ase of a single eletrolyte yielding the ions designated by the suffixes ( )and (-), the orresponding transferene nubers are given as follows: q t (48) Q Where q is the quantity of eletriity arried by the ation and Q is the total quantity of eletriity arried by all the ions through the solution. Siilarly, the anion transport nuber t is defined as: q t (49) Q Where q is the quantity of eletriity arried by the anion, equations (48) and (49) an be expressed as t u u u (50) t u u u (51) The quantities and whih represent the onentrations of the ions, are equal and therefore for this type of the eletrolyte. t u u u (52) t u u u (53) and t t 1 27

28 u and u are the obilities of the ions in the sae solution and we know that the speed of an ion in a solution at any onentration is proportional to the ondutane of the ion at that onentration and therefore the transferene nuber ay be represented in the for; t and t (54) Where the values of and (ion ondutane) and Λ (olar ondutane) of the solution, are at that partiular onentration at whih the transferene nubers are appliable Deterination of Transport Nuber Three ethod have been generally eployed for the experiental deterination of transferene nubers: the first, based on the proedure originally proposed by Hittorf, involves easureent of hanges of onentration in the viinity of the eletrodes; in the seond, known as the oving boundary ethod, the rate of otion of the boundary between two solutions under the influene of urrent is studied; the third ethod, is based on eletrootive fore easureents of suitable ells Hittorf s Method Apparatus This ethod of deterining transport nubers was desribed as long ago as To understand the priniple involved onsider the overall desription of the ethod given below. The apparatus onsists of two separated opartents joined by a substantial iddle opartent and ay be of any of the shape as shown in figure 8(A) and (B). Let us take silver nitrate solution in a ell having silver eletrodes. Before the experient begins, the onentration of AgNO 3 is the sae throughout the ell. The experient involves passage of a diret eletri urrent fro a power soure through the ell. 28

29 (a) (b) Fig.8: Hittorf s ell At the left-hand eletrode, Ag dissolves and inreases the AgNO 3 onentration in its opartent. In the right-hand opartent, Ag ions deposits so the AgNO 3 onentration dereases in the solution. Measureent of the hanges in onentration in eah opartent after a 2-3 h passage of urrent yields the transport nuber of the anion (sine t t - 1, it also gives that of the ation). The urrent is passed for a fixed tie. Thereafter, the anolyte (Figure 9) has an inreased onentration 1 and the atholyte a dereased onentration 3. The iddle opartent does not hange its onentration of silver nitrate, whih will be designated 2. Fig. 9: The Priniple of Hittorf s experient (Adapted fro J.O.M. Bokris and A.K.N. Reddy, Modern Eletroheistry, volue I, 2 nd edition, Plena Press, N.Y. (1998)). After t seonds (s) at urrent I, the nuber of oles of Ag introdued into the anolyte is 29

30 N F It (55) Where F is the faraday or eletrial harge on one 1 ole of Ag. In the entral opartent, in whih the onentration is shown by experient to reain onstant, urrent is given by the equation: I A 2 F (u 2 u -2 ) (56) Where A is the ross setional area of the entral opartent and u 2 and u -2 are the ioni obilities, respetively, of Ag and NO - 3. Therefore, fro eq. (55) and (56), N A 2 (u 2 u -2 )t (57) In the left- hand opartent, Ag ions are produed and also oved out. Hene, dn 1 dt 1 - A F 1 u (58) 1 Aording to priniple of eletroneutrality the onentration of both positive and negative ions in the left -hand opartent should be the sae. Therefore, dn 1 dt A 2u (59) 2 whih represents the rate at whih anions introdued by dissolution fro the silver eletrode ove into the left hand opartent to join the ation. Integrating eq. (59) gives 0 N1 N 1 2u2 A (60) 0 Where N 1 is the nuber of oles of AgNO 3 in the opartent before the urrent I was swithed on. In the atholyte opartent, Ag ions are reoved by deposition and transported fro the iddle opartent, dn 3 dt 1 A 2 u - 2 F (61) NO 3 - oves out to allow eltroneutrality to be aintained. It ust ove out at the sae rate as Ag disappears. Thus Integration of eq. (62) gives dn 3 - A 3u dt (62) 0 N 3 - N 3 A3u 3t (63) It has been assued that the entral opartent keeps a onstant onentration while the AgNO 3 is inreasing on the anolyte and dereasing in the atholyte. 30

31 Hene, N 1 - Now fro eq. (57) and (60), t u 2 Again fro eq. (57) and (63), N u2 u 0 N 3 N 3 2 N N N N 1 0 N 3 - N 3 (64) Gain in weight in anolyte Loss of weight in anode Loss of weight of silver in atholyte Gain in weight of athode Q. A solution ontaining g of AgNO 3 per gra of water was eletrolyzed between silver eletrodes. During the experient g of silver was deposited in a silver ouloeter plaed in series. At the end of experient, the anodi solution ontains 23.14g of water and 0.236g of AgNO 3. What are the transport nuber of Ag - and NO 3 ions? Before eletrolysis: Mass of water 1g Mass of AgNO g Nuber of gra equivalent of Ag After eletrolysis: Mass of water g Mass of AgNO g x 10-5 Nuber of gra equivalent of Ag in 1g of water x x 10-5 Inrease in onentration of Ag in the anodi opartent where no silver ions igrate x Inrease in onentration of Ag in the anodi opartent when Ag ions igrate. Conentration of Ag after eletrolysis Conentration of Ag before eletrolysis x x x 10-5 Fall in onentration of Ag in anodi opartent x x x 10-5 Transport nuber of Ag ions Fall in on. Silver deposited around on Ag anode eletrode (65) (66) x 10 x Also transport nuber of NO 3 ions an be alulated as 1 - t Ag

32 Moving boundary ethod The oving boundary ethod is based on easuring the rate of igration of one or both of the ioni speies of the eletrolyte, away fro the siilarly harged eletrodes and by this ethod we an diretly observe the igration of ions under the influene of an applied potential unlike the Hittrof s ethod in whih onentration hanges at the eletrodes are observed. This ethod is very aurate and has been used in reent years for preision easureents. In the pratial appliation of the oving boundary ethod one boundary only is observed, and so the neessity of finding two indiator solutions is obviated; the ethod of alulation is as follows. If one faraday of eletriity passes through the syste, t equiv. of the ation ust pass at any given point in one diretion; if ole of the unit volue is the onentration of the solution in the viinity of the boundary fored by the M ions, this boundary ust pass through a volue t / while one faraday is passing. The volue ф taken out by the ations for the passage of Q oulobs is thus Ф Q t / F (67) Where F is one faraday, i.e., 96,500 oulobs. If the ross setion of the tube in whih the boundary oves is α sq.., and the distane through whih it oves during the passage of Q oulobs is l., then Ф is equal to l α, and therefore fro eq. (65) t l αf / Q (68) Sine the nuber of oulobs passing an be deterined, the transferene nuber of the ion ay be alulated fro the rate of oveent of one boundary. The apparatus used for the deterination of the transport nuber by this ethod, onsist a long vertial tube of unifor bore fitted with two eletrodes at the two ends as shown in figure

33 Fig.10: Cell for the deterination of transport nuber by oving boundary ethod Let us onsider the eletrolyte AgNO 3 the transport nuber of whose ation (Ag ) is to be deterined. A layer of a solution of AgNO 3 is introdued above the solution of another suitable eletrolyte KNO 3 having the oon anion NO - 3. The eletrolyte KNO 3 is seleted so that the veloity of K ions is less than those of Ag ions. Under the irustanes, there will be a lear-ut boundary between the two eletrolytes, even if the two solutions are olourless. On passing urrent between the two eletrodes, both Ag and K - ions ove upward towards the negative eletrode while NO 3 ions ove downward towards the positive eletrode as shown in Figure 10. Sine K ions have a lower veloity than Ag ions, a sharp boundary is always aintained between the two solutions. Also the K ions are not far behind for if it so happens the solution below the boundary would get diluted and its inreasing resistane auses an inreased potential drop thereby inreasing the veloity of K ions. Thus the boundary oves slowly in the upward diretion. On passing n faraday of urrent, n x t Ag will be arried by Ag ions and a orresponding aount (n x t Ag ) ole of it oves up. If represents the original onentration of Ag ions in ole l -1 and the boundary oves through a distane l in the tube of ross-setion S 2, the nuber of ole of Ag oving upward is given by S x l x C. Thus: n x t Ag S x l x C (69) 33

34 or t S x l x C Ag (70) n Knowing the experiental values of S, l, C and n, the value of t Ag the transport nuber of Ag ions an be alulated. Q. In a oving boundary experient 0.01 M HCl solution was treated on a lithiu hloride solution. The tube used had a diaeter of 1. when a urrent of 11.0 illiapere was passed for 20 in., the H ions Li ions boundary oved through Calulate the transport nuber of H, Cl - ions in HCl solution used? We have, t H Also s π r 2 22 x 0.5 x 0.5 s x l x Q F x C 7 l 13.9, F oulob, C 0.01N Q quality of urrent passed 11 x 10-3 x 20 x x 0.5 x 0.5 x 13.9 x x 0.01 t H 3 11 x 10 x 20 x and t 1 - t Cl H Appliations of Condutivity Measureents Soe of the iportant appliations of ondutivity easureents are as follows: 12.1 Deterination of degree of Dissoiation The degree of dissoiation of a weak eletrolyte is deterined by the appliation of expression: α Where is the olar ondutane of the solution and an be obtained experientally (71) is the olar ondutane at infinite dilution and an be found fro the literature. The ethod ay be illustrated with speial referene to the deterination of the dissoiation of phosphori aid. Consider phosphori aid, ionizing as a onobasi aid in to H and H 3 PO 4 - ions, at 25 º C. 34

35 (H 3 PO 4 ) (HCl) (NaH 2 PO 4 ) - (NaCl) The easured value of for 0.1M phosophori aid is 96.5 S 2 ol -1 and the approxiate degree of dissoiation is 96.5/339.3 i.e., and the onentration of the ions is approxiately M Deterination of Solubility Produt of a Sparingly Soluble Salt If a slightly soluble eletrolyte suh as AgCl, BaSO 4, PbSO 4 dissoiate in a siple anner, it is possible to deterine the solubility of suh salts by ondutane easureents. If s is the solubility in ole/liter, of a sparingly soluble salt and κ is the speifi ondutane of the saturated solution, then its olar ondutane Λ is given by the relation k 1000 s i.e. s 1000 k At infinite dilution, k s 1000 (73) The ethod for the deterination of solubility produts of a sparingly soluble salt ay be explained by taking the exaple of solubility of silver hloride in water at 25 0 C. First the salt is repeatedly washed with ondutivity water to reove any soluble ipurities. After that it is suspended in ondutivity water, war and ooled to 25 0 C. A very inute quantity of salt will pass in solution and the rest will settle down. The ondutane of the solution and water used in the preparation of solution is deterined in the usual way by plaing the ondutivity ell in the therostat aintained at 25 0 C. (72) The value of Λ o for AgCl is given by (AgCl) Ag Cl S 2 ol -1 at 25 0 C. The speifi ondutane of a saturated solution of AgCl in water is 3.41 x 10-6 S -1 at 25 o C and if 1.60 x 10-6 is deduted for the ondutane of the water, the value of k is 1.81 x 10-6 oh Hene solubility of AgCl ay be deterined by putting the values in eq. (71). k s x 1.81 x 10-6 /

36 12.4 Condutoetri Titration 1.31 X 10-5 ole/l at 25 o C. The deterination of the end point of a titration by eans of ondutane easureents is known as ondutoetri titration. In these titrations easureent of atual speifi ondutane of the solution is not required, and any quantity proportional to it is suffiient. The titrant is added by a burette and hange of ondutane as a funtion of added titrant is used to deterine the equivalene point. A ondutane ell, in whih solution of substane to be titrated is taken, onsists of two platinu eletrodes of large surfae area aross whih an alternating low-voltage potential is applied. Generally, potential in the range 5-10Vat Hz is eployed. The ondutane ell, therefore, is inorporated into one ar of a Wheatstone bridge type of iruit and the ondutane is easured by adjustent of a alibrated resistor to balane the bridge. Soe typial exaples of ondutoetri titration and interpretation of their urves are given in following paragraphs. In all ases, the equivalene points are loated at the intersetion of lines of differing slope. One of the requireents of these titrations is that the titrant should be at least ten ties as onentrated as the solution being titrated, in order to keep the volue hange sall Aid-Base Titration (i) Strong Aid with a Strong Base When a strong alkali, e.g., sodiu hydroxide is added to a solution of a strong aid, e.g., hydrohlori aid, the following reation ours: (H Cl - ) (Na OH - ) Na Cl - H 2 O In this type of reation, the ondutane first falls, due to the replaeent of the H ( 350) by the added ation Na ( 40-80) and after the equivalene point has been reahed, the ondutane rapidly rises with further addition of strong alkali due to large value of the hydroxyl ion ( 198). The two branhes of the urve are straight lines provided the volue inrease is negligible and their intersetion gives the endpoint. At the neutral point the ondutane of the syste will have a iniu value, fro whih the equivalene point of the reation an be deterined. A Condutane end point L of base (NaOH) B Fig. 11.(A) 36

37 In atual pratie the lines ay be slightly urved due to variation in teperature, heat of neutralization, interioni effet and slight inrease in the volue of the solution beause of the addition of titrant. Inspite of this, the infletion is sharp enough to get the end point. (ii) Strong Aid with a Weak Base The titration of a strong aid with a weak base ay be illustrated by the neutralization of dilute HCl by dilute NH 4 OH. H Cl - NH 4 OH NH 4 Cl - H 2 O The first branh of the graph (Figure.11B) represents the neutralization of the aid and its (i.e, replaeent of fast oving H ions by slow oving NH 4 ions). After the neutralization (end point) is opleted, the graph beoes alost horizontal, sine the exess aqueous NH 4 OH is not appreiably ionized in the presene of NH 4 Cl. As NH 4 OH is a weakly ionized eletrolyte it has a very sall ondutivity opared with that of the aid or its salt. A Condutane end point B L of base Fig. 11 (B) (iii) Weak Aid with a Strong Base Consider the titration of the weak aid like CH 3 COOH with strong base NaOH. (CH 3 COO - H ) (Na OH - ) (CH 3 COO - Na ) H 2 O In this titration, the shape of the urve will depend upon the onentration and the Ka of the aid (Ka ~ 1.8 x 10-5 ). The sodiu aetate salt fored during the titration tends to suppress the ionization of the aeti aid due to oon ion effet and therefore its ondutane dereases. The rising salt onentration will however tend to result in an inrease in ondutane. When the neutralization of aid is oplete, further addition of alkali produes exess of OH - ions. This exess OH - ion inreases the ondutane of the solution ore rapidly. Finding an aurate end point is diffiult in this type of titration. For oderately strong aids, the influene of the rising salt onentration is less pronouned. Diffiulty is also experiened in loating the end point aurately and generally titration of weak aid and oderately strong aids with a strong base are not reoended for ondutoetri titration. 37