| Results and discussion
The results of the best fit models that contain the type of species and overall formation constants along with some of the important statistical parameters are given in Table 1. The low standard deviation in log β values indicates the adequacy of the models. The small values of Ucorr (sum of squares of deviation in concentrations of the metal, ligand and hydrogen ions at all represented by the model). For an ideal normal distribution, the values of kurtosis and skewness should be three and zero, respectively. The kurtosis values in the present study indicate that the residuals form leptokurtic pattern and a few forms a platykurtic pattern. The values of the skewness given in Table 1 are between -0.59 and 3.42 for LPA and in table 2 are between -0.26 and 4.11 for MA. These data evince that the residuals form a part of normal distribution; hence, least squares method can be applied to the present data. The sufficiency of the model is further evident from the low crystallographic R-value recorded [35]. Thus, these statistical parameters support the best fit models which portray the metal-ligand species in DMF-water mixtures.
Table 1: Parameters of best fit chemical models of Pb(II), Cd(II) and Hg(II)- LPA complexes in DMF- water mixtures.
% of
DMF
|
log βmlh (SD)
|
pH-Range
|
NP
|
Ucorr
×108
|
2
|
Skewness
|
Kurtosis
|
R-factor
|
ML2
|
ML2H
|
ML3
|
Pb(II)
|
0.0
|
6.38(39)
|
15.03(06)
|
9.25(18)
|
7.0-9.1
|
30
|
9.44
|
1.52
|
-0.02
|
4.92
|
0.0162
|
10
|
6.90(92)
|
14.82(88)
|
9.97(04)
|
7.3-8.3
|
27
|
9.28
|
2.18
|
-0.01
|
2.90
|
0.0322
|
20
|
6.64(66)
|
14.80(83)
|
9.73(25)
|
7.7-8.9
|
31
|
6.67
|
3.92
|
1.23
|
5.02
|
0.0623
|
30
|
6.80(83)
|
14.94(95)
|
9.63(25)
|
7.6-8.9
|
30
|
2.61
|
2.93
|
-0.24
|
3.17
|
0.0491
|
40
|
6.79(74)
|
14.89(88)
|
9.72(21)
|
7.6-8.9
|
37
|
10.00
|
6.25
|
0.03
|
2.65
|
0.0372
|
50
|
6.79(76)
|
14.93(94)
|
10.15(21)
|
7.0-8.9
|
30
|
3.53
|
5.87
|
0.63
|
3.79
|
0.0223
|
Cd(II)
|
0.0
|
6.27(28)
|
14.65(67)
|
9.06 (09)
|
7.0-9.1
|
41
|
3.33
|
5.95
|
-0.52
|
3.00
|
0.0352
|
10
|
6.64(67)
|
14.62(63)
|
9.10 (17)
|
7.3-9.1
|
43
|
1.00
|
1.35
|
3.42
|
5.49
|
0.0511
|
20
|
6.52(55)
|
14.58(59)
|
9.42(05)
|
7.3-9.3
|
43
|
1.5
|
6.16
|
2.01
|
2.78
|
0.0302
|
30
|
6.55(56)
|
14.33(36)
|
9.30(21)
|
7.3-9.3
|
41
|
7.78
|
7.13
|
0.62
|
3.01
|
0.0301
|
40
|
6.55(57)
|
14.65(64)
|
9.53(07)
|
7.3-9.5
|
40
|
1.76
|
1.33
|
0.23
|
7.96
|
0.0141
|
50
|
6.45(47)
|
14.60(62)
|
9.13(24)
|
7.3-9.5
|
47
|
2.86
|
2.80
|
-0.61
|
4.23
|
0.0161
|
Hg(II)
|
0.0
|
6.88(89)
|
15.34(35)
|
9.42(07)
|
6.0-9.4
|
44
|
5.71
|
1.33
|
0.21
|
4.41
|
0.0242
|
10
|
6.37(39)
|
14.80(82)
|
9.52(21)
|
6.0-8.4
|
34
|
5.45
|
7.52
|
-0.59
|
4.41
|
0.0232
|
20
|
6.71(73)
|
14.93(91)
|
9.45(07)
|
6.0-8.4
|
26
|
1.54
|
2.00
|
0.13
|
2.53
|
0.0132
|
30
|
6.97(98)
|
14.45(46)
|
9.76(29)
|
6.0-8.4
|
21
|
4.44
|
9.76
|
-0.53
|
4.12
|
0.0223
|
40
|
7.05(07)
|
14.70(72)
|
9.85(03)
|
6.0-8.4
|
27
|
1.42
|
8.14
|
-0.31
|
4.12
|
0.0141
|
50
|
6.59(62)
|
15.06(07)
|
9.49(03)
|
6.0-8.4
|
22
|
1.11
|
8.85
|
-0.54
|
3.63
|
0.0052
|
Table 2: Parameters of best fit chemical models of Pb(II), Cd(II) and Hg(II)-MA complexes in DMF - water mixtures.
% of
DMF
|
log βmlh (SD)
|
pH-Range
|
NP
|
Ucor4r
×108
|
2
|
Skewness
|
Kurtosis
|
R-factor
|
MX2
|
MX2H
|
MX3
|
Pb(II)
|
0.0
|
6.65(66)
|
11.61(62)
|
10.15(16)
|
2.8-7.8
|
44
|
2.11
|
16.41
|
4.11
|
2.90
|
0.0493
|
10
|
6.83(84)
|
11.53(54)
|
9.50(51)
|
3.6-5.8
|
48
|
0.66
|
5.85
|
0.05
|
4.66
|
0.0083
|
20
|
6.73(74)
|
11.30(31)
|
9.60(62)
|
3.6-5.2
|
48
|
6.66
|
10.59
|
0.60
|
3.18
|
0.0244
|
30
|
7.10(12)
|
12.10(12)
|
10.50(51)
|
3.9-5.1
|
44
|
2.00
|
3.46
|
-0.14
|
2.62
|
0.0132
|
40
|
6.71(73)
|
11.65(66)
|
10.33(34)
|
4.1-5.2
|
46
|
0.66
|
9.41
|
0.76
|
5.95
|
0.0292
|
50
|
7.08(09)
|
11.91(92)
|
10.80(82)
|
3.8-5.1
|
77
|
6.00
|
6.58
|
-0.26
|
3.69
|
0.0243
|
Cd(II)
|
0.0
|
6.76(77)
|
11.73(72)
|
10.32(33)
|
4.0-7.2
|
72
|
3.18
|
5.11
|
0.26
|
3.14
|
0.0154
|
10
|
6.65(66)
|
11.63(65)
|
9.34(35)
|
4.0-7.5
|
73
|
6.087
|
1.69
|
3.61
|
4.04
|
0.0265
|
20
|
6.41(43)
|
11.55(54)
|
9.33(36)
|
4.2-7.4
|
44
|
4.58
|
5.02
|
1.37
|
5.78
|
0.0226
|
30
|
6.33(35)
|
11.38(40)
|
9.43(44)
|
4.2-7.3
|
88
|
3.04
|
1.64
|
1.52
|
7.43
|
0.0197
|
40
|
6.76(77)
|
11.53(54)
|
9.84 (85)
|
4.2-7.4
|
88
|
6.84
|
9.64
|
0.25
|
5.39
|
0.0299
|
50
|
6.55(56)
|
11.32(33)
|
9.83(82)
|
4.2-7.4
|
90
|
4.21
|
3.82
|
-0.03
|
2.72
|
0.0221
|
Hg(II)
|
0.0
|
6.93(94)
|
12.75(76)
|
9.35(36)
|
4.2-6.4
|
100
|
7.22
|
3.92
|
1.14
|
5.11
|
0.0292
|
10
|
6.58(61)
|
12.27(29)
|
9.22(23)
|
4.4-6.4
|
22
|
24.54
|
23.67
|
0.05
|
5.82
|
0.0583
|
20
|
6.36(37)
|
12.18(19)
|
9.36(35)
|
4.2-6.5
|
34
|
8.50
|
8.33
|
1.34
|
6.36
|
0.0344
|
30
|
6.69(70)
|
11.68(67)
|
9.39(40)
|
4.2-6.4
|
36
|
10.52
|
3.82
|
1.23
|
3.21
|
0.0375
|
40
|
6.39(40)
|
12.14(15)
|
9.22 (23)
|
4.2-6.4
|
77
|
5.26
|
38.00
|
1.95
|
2.12
|
0.0456
|
50
|
6.25(31)
|
12.46(47)
|
9.12(13)
|
4.5-6.4
|
65
|
8.23
|
4.00
|
1.16
|
3.45
|
0.0323
| Effect of systematic errors on best fit model
In order to rely up on the best fit chemical model for critical evaluation and application under varied experimental conditions with different accuracies of data acquisition, an investigation was made by introducing pessimistic errors in the concentrations of alkali, mineral acid, ligand and metal (Table 3). Errors were introduced in the concentrations of the ingredients intentionally to find their effect on the perturbation of stability constants. If the concentrations determined and the experimental conditions maintained by the researcher were appropriate, any variations in the concentrations of the ingredients will affect the magnitudes and statistical parameters of the stability constants. Sometimes even the species shall be rejected. Hence, we have studied the effect of errors in the concentrations of the ingredients on the stability constants. The data show that the order of affecting the magnitudes of the stability constants is alkali > acid > ligand > metal. Some species, were even rejected when the errors were introduced in the concentrations of the components. This shows that any deviation in the stability constants (log β) and also results in the rejection of the species.
Table 3: Effect of errors in influential parameters on M(II)- LPA/MA complex stability
constants in DMF-water medium.
Ingredient
|
% Error
|
Log βmlh(SD)
|
120
|
121
|
130
|
Pb(II)-LPA 50 % v/v DMF-water mixture
|
Acid
|
0
|
6.79(24)
|
14.93(35)
|
10.15(23)
|
-5
|
8.67(44)
|
16.99(30)
|
12.86(27)
|
-2
|
7.28(37)
|
15.75(25)
|
11.11(21)
|
+2
|
6.44(12)
|
Rejected
|
9.22(18)
|
+5
|
5.99(11)
|
Rejected
|
Rejected
|
Alkali
|
-5
|
5.83(12)
|
Rejected
|
Rejected
|
-2
|
6.43(11)
|
Rejected
|
9.08(21)
|
+2
|
7.24(46)
|
15.76(27)
|
11.19(22)
|
+5
|
8.39(29)
|
Rejected
|
12.09(30)
|
Ligand
|
-5
|
6.74(45)
|
15.06(38)
|
10.46(24)
|
-2
|
6.74(36)
|
15.03(36)
|
10.35(23)
|
+2
|
6.74(29)
|
14.98(33)
|
10.21(21)
|
+5
|
6.73(24)
|
14.95(31)
|
10.10(21)
|
Metal
|
-5
|
6.74(32)
|
15.00(34)
|
10.28(22)
|
-2
|
6.77(27)
|
14.96(35)
|
10.20(23)
|
+2
|
6.80(22)
|
14.91(35)
|
10.10(24)
|
+5
|
6.81(20)
|
14.87(36)
|
10.02(25)
|
|
| 