Studies of Electrolytes in Dioxane Water Mixed Solvent

All praise goes to almighty Allah, without His endless mercy I could not reach so close to my destination, for the time being. Simultaneously I am remembering my prophet Hazrat Mohammad (SM), the supreme guide to knowledge and wisdom for mankind.

I have the pleasure to express my deepest gratitude and indebtedness to my reverend supervisor, Dr. Mahbub Kabir, professor, Department of chemistry, Jahangirnagar University, Savar, Dhaka, for his indispensable guidance, encouragement and suggestions throughout the period of my research.

I am particularly grateful to our honorable teacher Prof. Dr. Ilias Mollah, Chairman, Department of chemistry, Jahangirnagar University, Savar, Dhaka, for his kind co-operation fruitful discussions and spontaneous help out.

I am also grateful and indebted to Dr. Menarul Islam, Assistant Professor, Department of chemistry, Jahangirnagar University, Savar, Dhaka, for his indispensable guidance, encouragement and suggestions throughout the period of my research.

I also take this opportunity to express my gratefulness to all of my honorable teachers of the Department of chemistry, Jahangirnagar University, for their help and co-operation.

I would like to pay my respect and gratefulness to Md. Abdul Kuddus and Subir chowdhury, PhD researcher in physical laboratory for their utmost help to me during my thesis work.

I offer my grateful to Md. Mohibullah Khan Sowrav, Md. Rasedul Islam Ripon and Abul Kalam Azad for their utmost help to me during my thesis work.

I want to give thanks to all of my friends for their co-operation and help during the course of my lab work.

I offer my heartfelt thanks to my brothers for their constant inspiration. Finally I express my deepest gratitude and sincere thanks to my parents for keeping me in their good wishes forever.

 2.1   REAGENTS  AND  CHEMICALS 

A list of the chemicals is given below:

1.  Ferrous Sulphate    (Ana. R.)  :  B. D. H. England
2. Copper Sulphate (Ana. R.)       :  Ridel –dettean
3. Cadmium Sulphate (Ana. R.)   : B. D. H. England
4. Potassium chloride (R.)           : B. D. H. England
5. Dioxane  ( R.)                           : Ridel –dettean

Iron, cadmium and copper sulphates of analytical reagent grade were recrystallized twice from conductivity water at a temperature between 10 and 20  and then dried at room temperature. Reagent grade 1,4-dioxane  was fractionally distilled after being dried to remove traces of water. Potassium chloride used for the cell calibration was reagent grade and was recrystallized three times from conductivity water. It was then dried in vacuum at 200   for more than one week.

Dioxane-water mixtures were carefully prepared by using conductivity water (specific conductivity < 2  10^-6 Scm^-1 at 25  ). Solutions of different concentrations were carefully prepared by using dioxane – water mixture.

 2.2    CONDUCTIVITY   WATER 

Water used in conductance was deionized water purified by passing it through a double Barnsted ion- exchange column (organic materials exchange and mixed cation – anion exchange). The conductance of this water was monitored by measuring the conductance during the collection. The value was < 1  10^-6 Scm² at room temperature.

2.3   MEASUREMENT OF DENSITY, VISCOSITY AND DIELECTRIC CONSTANT 

The density of pure liquid and mixtures of liquids were measured by weighing a definite volume of the liquid in a density bottle. The viscosity of dioxane – water mixtures were measured experimentally by using Oswald viscometer. The dielectric constant of the solvent mixtures was interpolated from the published data.³⁶ The viscosity and dielectric constants of dioxane – water mixtures are summarized in    table 1.

2.3.1 MEASUREMENT OF DENSITY

The relative density is the ratio of the weight of a given volume of the substance to the weight of an equal volume of water at the same temperature and the density of the substance at temperature T   is equal to the relative density multiplied by the density of water at that temperature. Densities of liquids were measured by weighing a definite volume of the liquid in a density bottle. The densities of liquids   and different solutions were calculated by the equation.

            d_T   =    D_T

where

          w´ = the weight of liquid or solution, w   = the weight of water and

          = the density of water at T

2.3.2    MEASUREMENT OF   VISCOSITY 

According to the poiseuelle’s equation the ratio of the viscosity coefficients of the two liquid is given by

Where, t_{1 and} t₂ are time of outflow of water and liquid respectively.

Since the pressure p₁ and p₂ are proportional to the densities of the two liquids d₁ and  d₂ then

  = Consequently, once d₁, d₂ and  and  are known, determination of t₁ and  t₂  permits  the calculation of , the viscosity coefficient of the liquid  under consideration.

2.4   PREPARATION OF METAL SULPHATES   SOLUTIONS  

Solution of metal salts were prepared  measuring  required  amount  of  the compound  in an analytical  balance  and dissolving the solvent  to give  the appropriate  value  at  25 . Then solutions of different concentrations were carefully prepared by dilution of the stock solutions of metal sulphates at 25 , with increasing temperature, concentrations were changed. Solutions of different concentrations at different temperature were calculated by the equation

Where  and  are the concentrations of solution at temperature 25  and T   respectively. d₁  and  d₂   are  the  densities  of  solution  at  temperature 25  and  T   respectively.

2.5   CONDUCTIVITY   MEASUREMENTS  

The electrical conductivity of the solutions was measured at 50/60 Hz by the conductance linear – bridge method. A conductivity  cell  which  is designated  for  solutions of  low conductivity  was used  throughout  this study.  The conductivity cell was calibrated using 0.1 and 0.01 decimal solutions of purified fused potassium chloride. The cell constant was determined to be 0.610 cm^-1 using a standard potassium chloride   solution and exhibited no tendency to change with the two standard solutions. The solution in cell was magnetically stirred at a rate of 1.2 revolutions per second.  The experimental error of the conductivity measurement was less than 1%. All the measurements were carried out in an electrically grounded thermostat bath containing distilled water. Temperature was controlled and measured with sensitively of 0.05 .

2.6 TABLES  FOR  THE  PHYSICAL  PROPERTIES  OF  DIOXANE – WATER MIXTURESAND  EQUIVALENT  CONDUCTIVITY OF  DIFFERENT  SOLUTIONS  

The physical properties  of  dioxane – water  mixtures are summarized  in  Table  1  and  the equivalent  conductance at different concentrations of  ferrous  sulphate,  copper  sulphate and cadmium sulphate  solutions  at  different temperatures  are listed  in Table  2.1  to  2.12.

Table – 1:  Physical properties of dioxane – water mixture.

              5% Dioxane              10% Dioxane              15% Dioxane           20% Dioxane

Table-2.1:  Equivalent conductance (Λ) of FeSO₄   in 5% (w/w) dioxane – water mixture at different temperature.

 Table-2.2:  Equivalent conductance (Λ) of FeSO₄   in 10% (w/w) dioxane – water mixture at different temperature.

 Table -2.3:  Equivalent conductance  (  Λ )  of  FeSO₄   in  15%  (w/w)  dioxane – water  mixture at different  temperature.

Table -2.4:  Equivalent conductance (Λ) of FeSO₄ in 20% (w/w) dioxane – water mixture at different temperature.

mol  dm^-3           Λ

Scm² equ^-1        Λ

Scm² equ^-1          Λ

Scm² equ^-1          Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

80.15

90.25

100.23

110.75

4.00 10^-4

72.52

81.54

90.75

99.50

6.00  10^-4

66.50

75.50

84.10

92.00

8.00  10^-4

63.75

73.25

81.50

89.10

12.00  10^-4

61.24

70.50

78.00

86.05

                                                      Table –2.5:  Equivalent conductance (Λ) of CuSO₄   in 5% (w/w) dioxane – water mixture at different temperature.

mol  dm^-3

Λ

Scm² equ^-1           Λ

Scm² equ^-1           Λ

Scm² equ^-1           Λ

Scm² equ^-1      25     30        35     402.00  10^-4 113.95127.98141.87155.99

4.00 10^-4

107.20

124.18

137.21

151.15

6.00  10^-4

103.53

118.01

131.05

142.35

8.00  10^-4

101.75

113.10

125.75

137.05

12.00  10^-4

98.50

109.50

122.08

131.92

 Table-2.6:  Equivalent conductance (Λ) of CuSO₄ in 10% (w/w) dioxane – water mixture at different temperature.

mol dm^-3

Λ

Scm² equ^-1          Λ

Scm² equ^-1           Λ

Scm² equ^-1           Λ

Scm² equ^-1     25     30        35     402.00  10^-4

101.95

113.98

126.10

138.10

4.00 10^-4

96.50

109.90

120.08

132.85

6.00  10^-4

91.25

105.05

114.60

124.98

8.00  10^-4

88.30

101.00

109.80

118.10

12.00  10^-4

84.50

96.00

103.00

112.90

 water mixture at different temperature.

mol dm^-3

Λ

Scm² equ^-1         Λ

Scm² equ^-1            Λ

Scm² equ^-1         Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

89.96

99.98

112.15

122.75

4.00 10^-4

83.50

95.87

107.00

116.98

6.00  10^-4

78.25

90.00

98.75

108.25

8.00  10^-4

74.10

83.15

93.10

102.85

12.00  10^-4

70.30

78.50

87.00

95.00

 Table-2.8:  Equivalent conductance (Λ) of CuSO₄   in 20% (w/w) dioxane – water mixture at different temperature.

mol  dm^-3        Λ

Scm² equ^-1           Λ

Scm² equ^-1          Λ

Scm² equ^-1            Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

79.15

88.10

99.50

107.90

4.00 10^-4

71.25

82.90

93.00

102.10

6.00  10^-4

67.20

74.70

83.25

93.30

8.00  10^-4

64.10

67.12

78.50

85.50

12.00  10^-4

59.00

62.80

72.00

77.10

 Table-2.9:  Equivalent conductance (Λ) of CdSO₄   in 5% (w/w) dioxane – water mixture at different temperature.

mol dm^-3

Λ

Scm² equ^-1         Λ

Scm² equ^-1          Λ

Scm² equ^-1        Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

118.25

131.70

141.00

152.50

4.00 10^-4

110.50

125.50

135.25

145.75

6.00  10^-4

106.00

119.20

127.00

139.25

8.00  10^-4

102.06

113.00

122.20

134.50

12.00  10^-4

97.15

109.00

118.00

129.30

 Table-2.10:  Equivalent conductance (Λ) of CdSO4   in 10% (w/w) dioxane –water mixture at different temperature.

mol dm^-3

Λ

Scm² equ^-1           Λ

Scm² equ^-1          Λ

Scm² equ^-1             Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

106.50

119.75

128.00

139.25

4.00 10^-4

99..00

112.50

121.20

132.10

6.00  10^-4

94.10

106.25

116.50

126.00

8.00  10^-4

91.00

101.30

110.15

120.20

12.00  10^-4

86.30

96.00

106.00

116.50

 Table-2.11:  Equivalent conductance (Λ) of CdSO₄ in 15% (w/w) dioxane – water mixture at different temperature.

mol dm^-3

Λ

Scm² equ^-1          Λ

Scm² equ^-1          Λ

Scm² equ^-1         Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

93.50

103.00

111.50

124.75

4.00 10^-4

86.00

94.75

103.10

115.60

6.00  10^-4

81.20

89.30

97.30

109.00

8.00  10^-4

77.00

85.00

92.00

103.75

12.00  10^-4

73.25

82.15

88.40

98.00

 Table-2.12:  Equivalent conductance (Λ) of CdSO₄   in 20% (w/w) dioxane – water mixture at different temperature.

mol  dm^-3

Λ

Scm² equ^-1          Λ

Scm² equ^-1          Λ

Scm² equ^-1         Λ

Scm² equ^-1

25

30

35

40

2.00  10^-4

82.10

92.25

102.30

112.75

4.00 10^-4

74.50

83.50

92.75

101.50

6.00  10^-4

68.50

77.50

85.10

93.70

8.00  10^-4

65.15

74.25

82.50

90.15

12.00  10^-4

60.25

71.50

79.00

87.05

 3.1  DATA  TREATMENT 

For  the treatment  of  the data the Fuoss-Onsagar  equation in  the form  for  an associated  symmetrical  electrolyte was used .

          Λ = Λ₀ – S(cγ) ^1/2  + Ecγ log(cγ)  + Jcγ – K_A (cγ) f_±² Λ                 (3.1)

Where Λ₀ is the limiting equivalent conductivity, c is the molar concentration, γ is the degree of dissociation, S is the Onsager limiting slope, E and J are theoretical coefficients, and f_± is the mean ionic activity coefficient.

For   the actual analysis of the association, the Fuoss“ y – x ” method  was used.

Introducing   some new variables defined by Fuoss by the equation:

           Λ^׳ = Λ + S (cγ)^1/2 – Ecγ log (cγ)                                                  (3.2)

          Δ Λ = Λ^׳ – Λ₀                                                                                (3.3)

          y = Δ Λ / c γ                                                                                  (3.4)

 and     x =  f_±² Λ                                                                                      (3.5)

The equation (3.1) is transformed into a simple form

         y = J – K_A x                                                                                    (3.6)

Equation (3.6)  implies  that  if  the variables  x  and  y  are known, the  values  of  J  and K_A can be determined  by a plot of  y against  x ( the  y–x  plot).The coefficient, J is  a  function  of  ion size parameter  and is used  for  the evaluation  of  ‘a’  the  closest  distance  of  approach  between  the  oppositely  charged  ions .

In  order  to  compute  y,  values  of  γ  and  Λ₀  are needed.  The terms in  E  and  J  are of opposite  sign, if  they  are  both neglected  the  conductance  equation  reduces to

         Λ = γ (Λ₀ – Sc^1/2 γ ^1/2)

This  is the modified  form  of  the  Oswald  dilution  law  used  formerly  for  analysis  of  conductance  data.^37,38

 The  starting  approximation  for  γ  is  obtained  by  the  classical  Onsager  equation :

           = Λ / [Λ₀   – S (c. Λ / Λ₀   )^1/2]                                             (3.7)

Λ₀  is  estimated  from the intercept  on  y  axis  of the   plot  of  Λ  vs  √c.  S  is  Onsager  limiting slope  of  the  above  plot  and  the  observed  ‘S’  value  is used .  Λ₀ is used to compute E, Λ, Λ’ and x.

 For  the initial calculation  of  x, the  mean  activity  coefficient  f_±²  is estimated  by  the following equation.³⁹

–log f_±²   =                                (3.8)

 Here Ka .2 for dilute solution.⁴⁰

Trial values  of  Λ₀ are  used  in  ΔΛ and  y  until correct one is found, the “correct”⁴⁰ one  is  taken  as the  value  which  linearize  the ‘y-x’ plot. Too small  trial  value  gives  a curve  which is  concord upwards, while  too large a  Λ₀  value  leads  to  a   ‘y-x’  plots which is concave downwards. From the ‘y-x’ plot we get Λ₀,K_A, a through J.

A  series of  calculations  are  done  for  a set  of  Λ₀   values  using  equation  (3.7)   and  (3.2)  to calculate  a fitting   parameter for the  y-x line  for each Λ₀  originally  using  1  or  2  Λ₀ unit  until  increments.  Then  Λ₀  is selected  which  gives  the  best fit  and the whole calculation is repeated  with  smaller increment  of  Λ₀ This  process is continued until  the increment  are  only 0.01 Λ₀   units.  Then for higher approximation, γ_i ( i= 2 , 3 .  .  . )  is calculated by the equation

           γ_i = Λ/ ( Λ₀ – S ( cγ_i-1 )^1/2 + Ecγ_i-1 log cγ_{i-1   +}  J cγ_i-1)                      ( 3.9 )

where, on the right hand side, the  first approximation for γ from  (3.7)  and the values just obtained for Λ₀ and  J are used. The calculation is repeated using equation (3.9) until γ (in) = γ (out) i.e., γ_n-1 = γ_n. This values of γ is used to calculated Λ’ i. e., Λ₀  from equation  (3.2). For any small difference in Λ₀ values between the two, a second time Λ₀ fixation is carried out using equation (3.9) and (3.2).

For the calculation of x at second time, the mean activity coefficient, f_± is estimated by means of the Debye-Huckel equation.

               –log f_±² = –                                                  (3.10)

where z (z = |z₊| = |z_–|  for a symmetrical electrolyte) is charge number;                 I,the ionic strength, and A and B, the theoretical coefficients.

where

              A =

and           B =

In the calculation of f_± by above equation, the initial ‘a’ is obtained from the initial J  is used.

Then a final “y-x” plot is made. From this we get final Λ₀,K_A and a_J through J.

Calculation   of a_J from J

In the equation for J from the relation J = Ϭ₁ Λ₀ + Ϭ₂, different trial values of   ‘a’ is used. The values of ‘a’ which fits the J equation is desired a_J value. The equation  (3.1)  is  a three- parameter equation in  Λ₀,  the equivalent  at  infinite dilution; J the coefficient  of the linear term,  and  K_A, the  association constant.  The symbols  in equation  (3.1) are the same as used  by Fuoss  in his monograph^38,41 adopted properly  to 2:2  electrolytes by the introduction  of valences  z.

The various coefficients   then   read   for homologues electrolytes

 where   | z₊ |  =  | z_–| = | z | = z.

             S = α Λ₀  + β  =   z³ Λ₀  +    z²

             E = E₁ Λ₀  – E₂ =   z⁶ Λ₀  –

         J = Ϭ₁ Λ₀ + Ϭ₂

            = {10¹² z⁶ [ h(b)  + 0.9074 + ln ( )]} Λ₀  +

                  z⁵ +  z³a

                   – z² (1.0170 + ln  )

 Where

          h(b) = (2b² +2b – 1) /  b³

        b =

           K =

       –log f_±² =

 The calculation of the various coefficients is   shown below:

We know

            k = √ (

where  k  is the  gas  constant  per  molecule i.e., R / N ,  T  the  absolute  temperature, and  n_i  the  number  of  any given  kind of    ions  per  cubic  centimeter  of  solution.

The  concentration  of  ions  c  in  gram ionic  weight  per liter  from  the  density .

            c =  1000

          Then

       k = √ ( )

Here, z₊   =   z _– = z   for homologues electrolytes.

         =  √ ( )

      = √ ( ) z √ c

Here, e = electronic charge = 4.80298  10^-10e.s.u.

           K = Boltzmann constant

               = 1.38054    10^-16 erg.deg^-1

          k = √ (   z √ C

     or, k  =   z √ c

           b  =    =   z²

                                   =   z²

        E₁ = 2.3026 (k² a² b² / 24 c)   =

        E₂ = 2.3026 (k a b β / 16 c ^1/2)

             = 2.3026

                ð E₂ =

Ϭ₁ =  ( k² a² b² /12 c )  [ h ( b )  + 0.9074  + ln (    )

      =      Z⁶ [ h  ( b )  +  0.9074 +  ln (  ) ]

        Ϭ₂ = α β + (11 β Ka / 12 c ^1/2) – (ka b β / 8 c ^1/2) [1.0170 + ln (  )]

     =  z⁵ +  z³a

     – z² (1.0170 + ln )

 Calculation of Ϭ₁   and Ϭ₂ 

Ϭ₁ and Ϭ₂   for dioxane – water mixtures at different temperatures.

For 5 % (w/ w) dioxane – water mixture.

When     T = 298 ⁰ K

        Ϭ₁ = 229394405.6 a + 7.6080 10 ¹⁴ a² – 1.2616 10²¹ a³ + 34.5832 ln a

                     + 654.8611

          Ϭ₂   = 1.4205 10¹⁰ a – 584.0591 ln a – 10667.5706

When     T = 303 ⁰K

      Ϭ₁ = 232579377.8 a + 7.6606 10¹⁴ a² – 1.2616 10²¹ a³ + 35.3059 ln a

                + 691.5675

          Ϭ₂   = 1.5899 10¹⁰ a – 658.2373 ln a –12024.6696

When   T = 308 ⁰ K

           Ϭ₁ = 237811681.6 a + 7.7463  10¹⁴ a² – 1.2616  10²¹ a³ + 36.5040 ln a

                   +691.5675

           Ϭ₂   = 1.7860  10¹⁰ a – 747.6947 ln a – 13663.0334

When T = 313 ⁰ K

           Ϭ₁ = 241274629.6 a + 7.8025   10¹⁴ a² – 1.2616   10²¹ a³ + 37.3042 ln a

                   + 706.8628

           Ϭ₂ = 2.0040 10¹⁰ a – 843.5160 ln a – 15417.0781

For 10 % (w/w) dioxane – water mixture.

When T = 298 ⁰ K

          Ϭ₁ = 260113174.5 a + 8.1014 10¹⁴ a² – 1.2616 10²¹ a³ + 41.7575 ln a

                    + 792.0302

           Ϭ₂ = 1.3634  10¹⁰ a –596.9219 ln a – 10921.2563

When T = 303 ⁰K

           Ϭ₁ = 263920844.5 a + 8.1605 10¹⁴ a² – 1.2616   10²¹ a³ + 42.6771 ln a

                   + 809.6399

           Ϭ₂ = 1.5482  10¹⁰ a – 682.7763 ln a – 12494.5262

When T = 308 ⁰K

           Ϭ₁ = 269863663.1 a + 8.2519  10¹⁴ a² – 1.2616   10²¹ a³ + 44.1273 ln a

                   + 837.3855

           Ϭ₂ = 1.7445  10¹⁰ a – 778.2596 ln a – 14246.1641

When T  = 313 ⁰K

           Ϭ₁ = 273961169.5 a + 8.3143  10¹⁴ a² – 1.2616 10²¹ a³ + 45.1361 ln a

                   + 856.6996

           Ϭ₂ = 1.9471  10¹⁰ a – 874.8776 ln a – 16018.0675

 For 15 % (w/w) dioxane – water mixture.

When T = 298 ⁰K

           Ϭ₁ = 297351308.5 a + 8.6547  10¹⁴ a² – 1.2616   10²¹ a³ + 51.0382 ln a

                   + 969.7688

           Ϭ₂ = 1.3268  10¹⁰ a – 621.0914 ln a – 11384.2358

When T = 303 ⁰K

           Ϭ₁ = 302052241.8 a + 8.7301  10¹⁴ a² – 1.2616  10²¹ a³ + 52.2533 ln a

                   + 993.0615

           Ϭ₂ = 1.5073  10¹⁰ a – 711.1400 ln a – 13037.5636

When T = 308 ⁰K

           Ϭ₁ = 308761188.1 a + 8.8265  10¹⁴ a² – 1.2616  10²¹ a³ + 54.0038 ln a

                   + 1026.6268

           Ϭ₂ = 1.7160  10¹⁰ a – 818.5757 ln a – 15011.7124

When T = 313 ⁰ K

           Ϭ₁ = 313769502.1 a + 8.8978 10¹⁴ a² – 1.2616   10²¹ a³ + 55.3231 ln a

                   + 1051.9299

           Ϭ₂ = 1.9157 10¹⁰ a – 921.1828 ln a – 16897.1105

 For 20 % (w/w) dioxane – water mixture.

When T = 298 ⁰K

           Ϭ₁ = 343311231.6 a + 9.3073  10¹⁴ a² – 1.2616  10²¹ a³ + 63.3173 ln a

                   + 1205.3577

           Ϭ₂ = 1.3202 10¹⁰ a – 664.0975 ln a – 12196.3738

When T = 303 ⁰ K

           Ϭ₁ = 349086500.6 a + 9.3852    10¹⁴ a² – 1.2616   10²¹ a³ + 64.9217 ln a

                   + 1236.1713

           Ϭ₂ = 1.4986  10¹⁰ a – 760.1259 ln a – 13963.1386

When T = 308 ⁰K

           Ϭ₁ = 356840910.9 a + 9.4874 10¹⁴ a² – 1.2616  10²¹ a³ + 67.0969 ln a

                   + 1277.9571

           Ϭ₂ = 1.6935  10¹⁰ a – 868.4692 ln a – 15958.1212

 When T = 313 ⁰K

           Ϭ₁ = 362929469.5 a + 9.5631  10¹⁴ a² – 1.2616   10²¹ a³ + 68.8215 ln a

                   + 1311.0948

           Ϭ₂ = 1.8954  10¹⁰ a – 980.2454 ln a – 18016.1541

 3.2    SPECIMEN   CALCULATION

Specimen  calculation  of  ion – association  constant  of  ferrous  sulphate  in  5%      (w/ w)  dioxane – water  mixture at temperature  298 ⁰K   is  shown  below :

The starting approximation for     is obtained by the classical   Onsager equation.

                  = Λ / [ Λ₀ – S (c. Λ / Λ₀)^1/2]

Where Λ₀ is estimated from the Λ vs. c^1/2 plot in Fig. 1 is 128.8.  The Onsager limiting slope ‘S’ obtained from of   Λ vs. c^1/2 plot is 899.50.

This trial value Λ₀ is used to compute E,  and Λʹ.  These  values  are  used  to  calculate  y  and  x  by  using  equation   ( 3.4 ),  ( 3.7 ) and  ( 3.5 ).

Then the series of calculations are made for   a set of Λ₀values. For calculation of , Λ^׳ and    the  following  values  are  needed.

       T = 298 ⁰K, D = 74.38, η = 0.009664 poise,

       S = 899.50 and E₁, E₂

       E₁ =    Z⁶ =  2⁶  =  39.82

     E₂ =   Z⁵ =   2⁵ = 672.43

   And,      E = E₁ Λ₀ – E₂

Using the value Λ₀ = 128.8, E = 4455.82

       = 116.50 / [128.8 – 899.50 (2.00  10^{-4    1/2}]

        = 0.998271

     Λ^׳ = Λ + S (cγ)^1/2 – Ecγ log (cγ)

          = 132.50

and the  difference  between  Λ^׳ – Λ₀  =  3.69

if Λ₀ =  132.0  is used,   then    = 0.9704  and   Λ^׳  =  132.241168.

  and  Λ^׳ – Λ₀ =  0.24

To  fix  Λ₀,  the  difference  between ( Λ^׳ – Λ₀)  become  less  than  0.01.

For this we put Λ₀= 132.21. Then  = 0.9686, Λ^׳ = 132.219586  and  ( Λ^׳ – Λ₀ )  = 0.009  which  is less than  0.01

Thus Λ₀= 132.21 is fixed.

Using  this  Λ₀ = 132.21 ,  , Λ^׳, y  and  x   are  calculated  for  each  c  and  Λ.

 Calculation of  y

       y =

        =

        = 49.48

Calculation of   x 

         x =    Λ

–log   =

             = 0.789

     x =    Λ = 91.90

Thus  the  value  of  y  and  x  are  calculated  in this  manner  for  each  concentration  and  equivalent  conductance.

Then the initial y vs. x  plot  is made  according  to  the  equation  y  =  J – K_A x  and  the  slope  and  intercept  give  K_A  and  J  respectively.

In this case, the initial value of K_A and J is 251 and 22979 respectively.

From  the  value of  J  the  values  of  a_J  are  obtained  as  follows:

           J = Ϭ₁ Λ₀ + Ϭ₂

Where

          Ϭ₁ = (k² a² b² /12 c) [h (b) + 0.9074 + ln ( )

      =      Z⁶ [h (b) + 0.9074 + ln (  )]

 Ϭ₂ = α β + (11 β Ka /12 c ^1/2) – (ka b β / 8 c ^1/2 ) [1.0170 +  ln ( ) ]

       =   z⁵ +  z³a

      – z² (1.0170 + ln  )

 For 5% (w / w) dioxane – water mixture at 298 ⁰K.

         Ϭ₁ = (229394405.6 a + 7.608005233  10¹⁴ a²– 1.26120645  10²¹ a³

                + 34.58317374 lna + 654.8660589)

        Ϭ₂ = 1.420509192 10¹⁰ a – 584.059164 lna – 10667.570558.

 If Λ₀ = 132.21, then

      J = 4.4510¹⁰ a + 1.0058 10¹⁷ a²– 1.67  10²³ a³ + 3988.18 lna +75912.27

In this equation, different trial values of ‘a’ is used.  The  values of  ‘a’  which  fits  the  J  equation  is the  desired  ‘a_J’  value.

J = 22979 is obtained from the slope of   y vs.  x plot.

When

        a = 16.50  10^-8 cm,   J = 22958

        a = 16.55  10^-8 cm,     J = 23002

        a = 16.52  10^-8 cm,    J = 22975

        a = 16.53  10^-8 cm,      J = 22979

Thus the initial value of ‘a_J’ desired as 16.53  10^-8 cm or 16.53 Å.

Then for higher approximation, γ_i  ( i= 2 , 3 .  .  . )  is calculated by the equation

           γ_i = Λ/ ( Λ₀ – S ( cγ_i-1 )^1/2 + Ecγ_i-1 log cγ_{i-1   +}  J cγ_i-1)

where, on the right hand side, the  first approximation for γ from  (3.7)  and the values just obtained for Λ₀ and  J are used. The calculation is repeated using equation (3.9) until γ (in) = γ (out) i.e., γ_n-1 = γ_n. At second time, Λ₀ is fixed in same way.

If Λ₀ = 132.17 is used in the calculation of  γ in the above equation, then the constant γ׳ = 0.9643 and Λʹ׳ = 132.176245 for Λ =116.50 and c = 2.0 10^-4. Then (Λʹ׳–Λ₀) = 0.006 i.e., less than 0.01. Thus final Λ₀ = 132.17 is obtained.

Using Λ₀ = 132.17, J = 22979, Λ= 116.50 and γ₁ = 0.9686, we get from the above equation:

               = 0.9643

                = 0.9643 using

Here,  =   = γ (constant).

Using this constant γ׳ (0.9643), Λʹ׳ = 132.176245 and y = 32.38 is obtained. Similarly different values of ‘y’are obtained for different Λ and c.

For the calculation of x at second time the equation of the mean activity coefficient is used.

               =  –

         = {1/2 (2.0  10^-4  2² + 2.0  10^-4  2² )}

                 = 0.02966439

            a = a_J (initial value) = 16.53 10^-8 cm.

            A =  =  = 0.552430962

            B =   =   = 33778893.3

               = –

             = 0.8830799

                  = 0.7798

                       x =    Λ = 0.7798  116.50

                                         = 90.85

Thus different x values are calculated for different Λ and c. Thus final “y – x” plot is made. From this plot final K_A and J are obtained.

The final a_J is obtained from final J. Using final Λ₀ = 132.17 and

            J = Ϭ₁ Λ₀ + Ϭ₂

              = 4.45  10¹⁰ a + 1.005    10¹⁷ a² – 1.67  10²³ a³+3986.8 lna + 75885.42

 Final J = 24263

Different trial values of ‘a’ is used in this equation.

when

             a = 17.55  10^-8 cm, J = 23870

            a = 17.95  10^-8 cm, J = 24219

            a = 17.98  10^-8 cm, J = 24263

Thus, the final a_J = 17.98 Ǻ is obtained.

Now for FeSO₄  in  5%  (w/w)  dioxane – water  mixture  the  values   are  as  follows:

     Λ₀ = 132.17,    K_A = 268, J = 24263   and a_J = 17.98 Å.

The initial  y – x  plot for  5%  (w/ w)  dioxane – water  mixture  at  298 ⁰K  is  shown  in  Fig. 4.1. The final  y – x  plots for  5%  (w/ w)  dioxane – water  mixture  at  298 ⁰K  is  shown  in  Fig. 4.2  and other  plots  are  shown  in Fig. 5  to 15.

The  calculated  values  of   γ, γ,׳ Λ’, Λ’׳,  y , x , K_A ,  J  and  a_J  are  shown  in  appendix  Tables 10.1 to 12.16.

 Calculation of   a_K from K_A

a_k  are  calculated  from  the  slope  of  the  plots  of   log K_A  vs. 1/ D according to  the  equation

            K_A =    exp

From the log K_A vs.1/ D (Fig.17) for FeSO₄   in  5 %  (w/ w)  dioxane-water  mixture  at  298 ⁰K .

        Slope =   = 113.00

       Or,   =      = 113.00

     Or,     =   19.85 10^-8 cm

                  =   19.85 A˚

 CALCULATION OF   THERMODYNAMIC PARAMETER

Calculation of ΔH⁰

From the plot of log K_A vs. 1/ T (Fig.21)  for  FeSO₄  in  5%  ( w/w )  dioxane – water  mixture.

        Slope =    = – 945.90

      ΔH⁰      =   – 8.316  2.303  (– 945.90) Joule mol^-1

                   = 18115.64 Joule mol^-1

                  =   18.12 k Joule mol^-1

 Calculation of ΔG⁰

From equation (1.51),

            ΔG⁰ = – 2.303 RT log K_A

Here, T = 298⁰K, log K_A = 2.40

               ΔG⁰ = – 8.316  2.303  298  2.40

                      = – 13697.33 Joule mol^-1

                      = – 13.69   k J mol^-1

 Calculation of ΔS⁰

From equation (1.54), ΔS⁰ =

                                                 =

                              ΔS⁰₂₉₈ = 106.75 J K^-1 mol^-1

Calculation of   ΔG_t⁰ 

ΔG_t =   ΔG_t⁰_{( mixed  solvent)} –  ΔG_t⁰ _{( water )}

ΔG⁰ ₂₉₈⁰K for FeSO₄   in water = – 13.05 kJ mol^{-1 42}

ΔG⁰ ₂₉₈⁰K for CuSO₄   in water = – 13.24 kJ mol^{-1 42}

ΔG⁰ ₂₉₈⁰K for CdSO₄   in water = – 13.43 kJ mol^{-1 42}

 ΔG⁰₂₉₈⁰K_{( 5% dioxane )}   for  FeSO₄  = -13.69  kJ  mol^-1

ΔG_t⁰ = –13.69 + 13.05 kJ mol^-1

          = – 0.64 kJ mol^-1

          = – 640  J mol^-1

ΔG⁰_{t ( el )}  =  4.184 166  (  – 0.0127)   (  + )

For FeSO₄   in 5 %  dioxane-water  mixture

ΔG⁰_{t ( el )}  =  4.184 166  (  – 0.0127)   (  +  ) kJ  mol^-1

                = 0.914 kJ mol^-1

               =   914 J mol^-1

Where   r₁ = ionic radii of Fe²⁺ = 0.75 Ǻ    and

            r₂ = ionic radii of SO₄^2- = 2.30 Ǻ

 ΔG⁰_{t ( ch )}  =  ΔG_t⁰ –  ΔG⁰_{t ( el )}

                = – 640 – 914 = – 1554 J mol^-1

 Calculation of Hydrodynamic Radii

The hydrodynamic radius for a cation and anion is given by

          R⁺ =

        R^– =

To  calculate    and  ,  the  value  of    = 79.80 in water  at  25⁰C has  been  taken.^43,44 The Λ₀ values of  FeSO_4, CuSO₄  and  CdSO₄   in  water are 138.36, 135.43 and 141.65 respectively at 25⁰C^43,44  The anion transport  number  of  infinite  dilution  was  calculated  by  the  equation.

                    t⁰_– =     and    t⁰₊  =  (1 – t⁰_–).

 For FeSO₄ in 5 wt. % dioxane- water mixture at 25⁰C.

 t⁰_– =     =    =  0.58

  and    t⁰₊  =  ( 1- 0.58 ) =  0.42

Assuming  t⁰₊  to  remain  constant  with  solvent  composition   R⁺  and  R^–   could  be  calculated  at  different  solvent  compositions.

                 = 0.42    132.17 = 55.51

   = 132.17 – 55.51 = 76.66

              R⁺ =   = 3.06 Ǻ

          R^– =    = 2.21 Ǻ

 4. RESULTS   AND   DISCUSSION

The equivalent conductance (Λ) of ferrous sulphate, copper sulphate and cadmium sulphate solutions at different temperatures varying solvent compositions are listed in Table 2.1 to 2.12. Plots of Λ vs. √ C are shown in Fig. 1 to 3. The limiting Onsagar slope (S) was found from the slope of the resulting straight lines.  Extrapolation of Λ at √C = 0  gave  values  of  limiting  equivalent conductance (Λ₀) which  were  used  in  the  Fuoss-Onsagar  equation as a preliminary  data.

 4.1  ION-ASSOCIATION   CONSTANTS

The ion – association constant (K_A), for the following reaction

          M²⁺ + SO₄^2- ↔ M²⁺. SO₄^2-   (M = Fe, Cu, Cd)

were  calculated  from  conductivity  data  according  to the  Fuoss-Onsagar  conductance equation.^40,41 The revised Fuoss-Onsgar equation for the equivalent conductance of the  associated  homologues  electrolyte  is given by

             Λ = Λ₀ – S(cγ) ^1/2  + Ecγ log(cγ)  + Jcγ – K_A (cγ) f_±² Λ

where     K_A = (1 – γ) / c γ²

The  values  of  Λ₀, K_A, J and a_J at  different  temperatures  and solvent  composition  are  summarized  in  table  3.1 to 3.3. Fig. 16 shows that the values of Λ₀ decreased with decrease dielectric constant (D) of the medium. The ion-association constants (K_A) values were found to increase with increase of organic portion of the solvent mixture. The K_A values of  CuSO₄ found to at each temperature  and at each solvent  composition are larger  than  those of  FeSO₄  and   the  K_A  values  of  FeSO₄  larger  than  those  of  CdSO₄. The trend of the all K_A values are shown below

SUMMERY

The conductance  of  three  homologous  electrolytes  namely  FeSO₄, CuSO₄ and CdSO₄ is  measured  in  dioxane-water  mixtures  at  varying  composition at different  temperatures. The ion-association constant (K_A) of the three electrolytes are found to increase with increase of organic portion in the solvent mixture and follow order at certain temperature.

                            CuSO₄ > FeSO₄ > CdSO₄

The values of K_A are found to increase with decreasing cation size. This result suggests that ion-association is cation size dependant. The K_A values also increase with increase of temperature.

The changes of standard Gibbs free energy of ion – association are negative which shows that   the association occurs thermodynamically spontaneously. The changes of standard entropies of ion – association are positive and decrease with increase of temperature. This phenomenon can be explained as the result of reduction of electric field of the associating ion. The positive values of standard enthalpy change reveals that the association is an endothermic process.

The change of standard Gibbs free energy  of  transfer (ΔG_t⁰) from water to mixed solvent are  always  negative  which suggests that the ion pair are in  a lower energy-state  in  aqua-organic  solvents than in water.The ΔG⁰_{t (ch)} values for all the three electrolytes are formed negative which indicates spontaneous solvation and specific ion–solvent interaction. The values of ΔG⁰_{t ( el )} are   positive  for  all  the  three electrolytes and  magnitude  decreases  with  increase  of  dielectric  constant  of  the  solvent  mixture.  The  lower  the  values  of  ΔG⁰_{t ( el )}  the  greater  is  the  electrostatic  interaction  between  ion  and  total  charges  on  the  solvent  molecules.

The walden products of all electrolytes in different solvent mixture are not constant which indicates that the stoke’s  model  of  continuous  solvent  structure  is  not  valid  for  the  system. The walden products (Λ₀η) for the three electrolytes follow the order

                             CdSO₄ > FeSO₄ > CuSO₄

      The walden products of CuSO₄ only show positive temperature coefficient almost every solvent composition except 10% of dioxane-water mixture. This result shows that CuSO₄ has the greatest    structure making character than FeSO₄ and CdSO₄.

The values of a_K  are closer to the distance of closest approach, a_J and the values  of  a_K  as  well  as  a_J for all the three electrolytes much larger than the sum of their lattice radii which indicates that  the solvent molecules enter  into  the  structure  of  the  ion – pairs  giving  rise  to solvation   and formation of solvent separated ion pairs. The  enhanced  electric  effect  due  to  the  double  charges on each  ion  and   the  small  size of Cu²⁺ gives  rise  to  considerable  polarization  to  a  number .

ABSTRACT

 Ion-association constant (K_A), distance of closest approach (a_J) and  thermodynamic parameters for several electrolytes  namely ferrous  sulphate, copper  sulphate  and  cadmium  sulphate  were  calculated  from their conductance data in  dioxane- water  mixtures  of  varying compositions  at  temperature of 25⁰C, 30⁰C, 35⁰C and 40⁰C. The values of ion-association constants are increased with increase of dioxane content and it also increased with increase of temperature. The standard Gibbs free energy, entropy and enthalpy for the reaction M²⁺ + SO₄^2- = M²⁺ SO₄^2- were calculated from the temperature dependence of   the ion-association constant. The standard free energy of transfer (G⁰_t) from water to dioxane–water mixtures was calculated. The  electrostatic  and  non-electrostatic  contributions  to  the  transfer  free  energy  change, the  free  energy  of   transfer  of  individual ions  in  different  solvent  mixtures  have  been  calculated. The solvated ionic radii   have also been calculated. The distance closest of approach (a_J) of each solvated ion has been calculated. The walden products of the electrolytes in different solvent compositions were evaluated