Table 1 gives the molar conductivities of MO sodium salt in water at 25.0 oC in the concentration range between 2.103 × 10− 4 and 9.255 × 10− 4 mol L− 1. Eq. (1) was used to determine the molar conductivity (Λ) as S cm2 mol− 1 for each measured solution of MO sodium salt.
Table 1
Molar conductivities of methyl orange (sodium salt) in water at 25.0 oC
[MO]/molL− 1

Λ/S cm2mol− 1

2.103 × 10− 4

76.27

2.310 × 10− 4

76.06

2.496 × 10− 4

76.01

2.822 × 10− 4

75.91

3.497 × 10− 4

75.66

4.203 × 10− 4

75.04

5.018 × 10− 4

75.50

5.617 × 10− 4

74.90

6.350 × 10− 4

74.34

7.076 × 10− 4

74.22

7.784 × 10− 4

74.91

9.255 × 10− 4

74.41

Λ = 103 k/C (1)
Where k is the conductivity of a measured solution of MO sodium salt (S cm− 1) and C is its molar concentration. According to the data in Table 1, the molar conductivity mostly decreases as the electrolyte concentration rises. Working with dilute solutions, inaccuracy is to be expected and cause a scatter points, wherefore, frequent measurements help to minimize these errors and ensure that the results are reproducible.
The DebyeHückelOnsager equation was used to analyse the data in Table 1. Given that the concept of complete dissociation forms the basis of this equation
Λ = Λo – (A + B Λo) C1/2 (2)
where Λo is the limiting molar conductivity of a measured solution of MO sodium salt at infinite dilution, while A and B can be expressed in terms of various constants that depend only on known quantities such as temperature, the charges on the ions, and the dielectric constant and viscosity of the solvent. For water at 25°C the constants A and B are 60.64 and 0.2299, respectively [38]. Eq. 2 is found to be very satisfactorily obeyed up to a concentration of about 2 × 10− 3 mol L− 1, where it assumes a linear relationship between Λ and C1/2 for aqueous solutions of uniunivalent electrolytes. An analysis using the linear least squares method was performed on the data in Table 1. The results of this analysis revealed that the MO sodium salt has Λo value of 77.93 ± 0.38 S cm2 mol− 1 and a slope value of 124.03. The corresponding correlation coefficient value was − 0.92. This difference from one shows the scattering of data that happens when working with dilute solutions.
For MO sodium salt, the calculated slope of Eq. (2) is 78.0 using the value of Λo as well as the values of the constants A and B. This value is consistent with the numbers that are typically used to describe an electrolyte ratio of 1:1 [39]. This finding suggests that MO sodium salt is a powerful electrolyte, and it also demonstrates that the monomer form of the dye's anion is present within the concentration range that is presented in Table 1, furthermore, it is in agreement with what Robinson and Garrett stated [4]. When Kendrick and Gilkerson [6] used the DebyeHückelOnsager equation on conductivity data over a relatively wide concentration range (1.0 × 10− 4 − 1.0 × 10− 2), they obtained a nonlinear plot, which was interpreted as a result of the formation of dimer forms of MO anion at higher concentrations and the limitation in the use of the DebyeHückelOnsager equation over 2 × 10− 3 mol L− 1. These authors assumed a value of 75 S cm2 mol− 1 for Λo of MO anion, while Robinson and Garrett [6] predicted a value of 82 S cm2 mol− 1. There is a possibility that the nonlinearity was also influenced by other factors, such as the adsorption of MO anion in the cell.
It is possible to calculate the value of the ionic conductivity for the MO anion at infinite dilution by making use of the reported value of the ionic conductivity of Na+ at infinite dilution (50.11 S cm2 mol1) [39], which is calculated as 27.82 S cm2 mol− 1.
The conductimetric method is also used in this study to determine the stability constant of the inclusion complex of MO anion with αCD at very low concentrations of the cyclodextrin, which is normally not feasible with the spectrophotometric method. For each αCD:MO anion ratio, the molar conductivity of the MO anion solution, Λ, was computed using an equation similar to Eq. (1). For each case under investigation, Λ declines as αCD:MO anion mole ratio rises, reaching a nearly constant value at a mole ratio higher than three. This behaviour was thought to be caused by the interaction of αCD and the MO anion. The resultant MO anion inclusion complex is less mobile than the free anion. This decrease in mobility of the MO anion can explain why the molar conductivity is going down. The binding of the MO anion to the cyclodextrin cavity as an inclusion process was assumed to result in a 1:1 inclusion complex [20, 22, 27, 28]. In this work, the analysis of the conductivity data of the system under consideration was performed utilizing the same binding stoichiometry. Eq. (4) represents a 1:1 binding of an MO anion by αCD molecule.
$$\text{M}{\text{O}}^{}+\alpha CD\stackrel{K}{\rightleftharpoons }\alpha CD\cdot M{O}^{}$$
4
where αCD·MO− represents the 1:1 inclusion complex and K represents the stability constant of the inclusion complex at equilibrium. Furthermore, the activity coefficient for ionic species approaches one under the dilution conditions used in this study; in this case, K is essentially the thermodynamic stability constant.
If α is the fraction of the total MO anion that is unreacted with αCD, then this leads to the following equations at equilibrium:
[MO−] = α[MO−]t (5)
[αCD·MO−] = (1 – α) [MO−]t (6)
[αCD] = [αCD]t – (1 – α) [MO−]t (7)
Where [MO−]t, [αCD]t, [MO−], [αCD·MO−], and [αCD] represent the molar concentrations of total MO anion, total αCD, free solvated MO anion, 1:1 inclusion complex of MO anion with αCD, and free solvated αCD, respectively.
By using Equations (5) and (6), then K can be given by Eq. (8).
K = (1 α)/ α[αCD] (8)
The molar conductivities Λ can be calculated using an equation, which similar to Eq. (1)
Λ = 103 k/[MO−]t (9)
The assumed measured conductivity, k, after adding αCD is as follows:
k = km + kc (10)
where km represents the free MO anion's contribution and kc the complexed MO anion's contribution with αCD.
The molar conductivity (Λ) can also be related to α by the equation
Λ = αΛm + (1 − α)Λc (11)
where Λm and Λc are the molar conductivity of the free MO anion before the addition of αCD, and the molar conductivity of the complexed MO anion with αCD, respectively. In Eq. (11), Λ is approximated by the simple additivity rule. However, solutions containing MO anion and αCD·MO− complex with the common cation, Na+ can be considered as two electrolytes with a common cation at constant ionic strength. Kell and Gordon [40] demonstrated that, for such a system, the difference between the calculated molar conductance of the combination using the simple additivity rule and the actual molar conductance is negligibly tiny. The next two equations are obtained by manipulating Eq. (11), and then substituting in the proper expressions for K and αCD [40]
[αCD] = [αCD]t  [MO−]t (Λ m  Λ)/( Λ m  Λ c) (12)
K = (Λ m  Λ)/[ αCD]( Λ  Λ c) (13)
Equations (4) to (13) are similar to those given by Tadeka and yano [41]. Eq. (14) can be obtained by substituting Eq. (12) into Eq. (13)
K = a1a2/(a2a3[αCD]t  a1a3[MO−]t) (14)
where a1 = Λ m  Λ, a2 = Λ m  Λ c, and a3 = Λ  Λ c. An approximation of the value of Λc was derived from the correlation between Λ and [αCD]/[MO] data. This value, along with the known quantities [αCD]t, [MO−]t, Λm, and Λ, yields a value for [αCD] according to the Eq. (12). A subroutine is then used to estimate a value for K.
Figure 2 shows the effect of αCD on the molar conductivity of MO salt at four initial concentrations of MO salt at 25.0 oC. The inverse relationship between the molar conductivity and the mole ratio of [αCD]/MO− or [MO−] is evident, and when the mole ratio exceeds 3, the difference in molar conductivity becomes insignificant and the value of Λc can be estimated. Five data sets (mole ratio: 0.5, 1.0, 1.5, 2. 0 and 2.5) of the type shown in Fig. 2 were employed to determine K for each MO concentration, viz. 20 values. Three other temperatures (20.0 oC, 32.0 oC, and 40.0 oC) were also used in determining the values of the stability constant of the inclusion of MO anion inside αCD. In every case, the observed molar conductivity, Λ, decreases steadily with the mole ratio [αCD]/[MO−], and at a mole ratio of about 3 it stands to stay the same. A total of 80 calculated values of the stability constant K (20 values for each temperature) were calculated. Table 2 displays the average estimated stability constant values (K) for four distinct temperatures between 20 and 40 oC. Figure 3 shows the effect of temperature on the molar conductivity against the mole ratio of [αCD]/MO− at a specific concentration of MO.
Table 2
The calculated average values of Stability constant (K) for MO/ αCD complex at different temperatures
Temperature/oC

Stability constant (K)

20.0

(5.01 ± 0.09) × 103

25.0

(4.36 ± 0.12) × 103

32.0

(3.15 ± 0.11) × 103

40.0

(2.51 ± 0.14) × 103

As the temperature rises, the calculated value of the stability constant, K, of the equilibrium in Eq. (2) decreases. This suggests that the binding of the MO anion to αCD is an exothermic process. The values of the thermodynamic parameters ΔHo and ΔSo of the inclusion process were determined according to the following linear thermodynamic equation (Van’t Hoff equation):
Log K =  ΔH/RT + ΔS/R (15)
where R is the molar gas constant. It is assumed that ΔHo and ΔSo are not significantly affected by temperature due to moderate temperature changes. This analysis utilized duplicated data (with at least 10% agreement in their average). The use of molarity as a concentration unit is often inconvenient when dealing with various temperatures at a constant concentration since the volume of the solution varies somewhat with temperature value. This issue was handled by incorporating a temperature adjustment to preserve the same species concentration for all usage temperatures. This was accomplished by testing the increase or decrease in solution volume inside the volumetric flask at each selected temperature relative to the flask's standard volume at 20°C, and then preparing each solution with a starting volume greater or less than 50 mL to be exactly 50 mL when thermal equilibrium was reached inside the thermostat at the selected temperature.
The data conformed to the linearity suggested by Eq. (15). The correlation coefficient is 0.986, and the estimated uncertainty in log K is within ± 0.06 (Fig. 4). From the plot in Fig. 4, the values of the standard thermodynamic parameters for the αCD/MO− complex are ΔHo = 27.35 kJ mol− 1 and ΔSo = 9.70 J K− 1 mol− 1. Although a thermodynamic study does not typically reveal the details of an inclusion process, some conclusions can be drawn from the results presented above. The spontaneous and exothermic nature of the inclusion process represented by Eq. (4) is evident. The magnitude of ΔHo for the inclusion process for MO− inside αCD is indicative of weak intermolecular forces. Notably, ΔSo is negative, indicating that methyl orange anion and αCD achieve a more ordered state after forming an inclusion complex. Since the inclusion process of a cyclodextrin can involve multiple steps [28], such as the breakdown of the water structure inside the cyclodextrin cavity and the removal of some water molecules from the cavity, the breakdown of the water structure around the part of MO− that will be incorporated into the αCD, the formation of hydrogen bonds between MO− and αCD, and the reconstruction of the water structure around the exposed parts of MO− after the inclusion process. It is difficult to comment quantitatively on the ΔHo and ΔSo values reported in this study.
The conductometer’s accuracy was assessed by a specific test using an aqueous solution of NaCl at 25 oC. A concentration of 5.65 × 10− 4 mol L− 1 NaCl gave a value of 125.13 S cm2 mol− 1 as a molar conductivity. The calculated value using the following derived equation for NaCl [38] is 124.37 S cm2 mol− 1.
ΛNaCl = 126.42–88.53 C1/2 + 89.5 C (1–0.2274 C1/2) (3)
Given that the difference between these two values is less than 1%, it was determined that the conductometer's accuracy was sufficient for the studies covered in this paper. The molar conductance of the same NaCl solution was determined in the presence of varying amounts of αCD over the range of mole ratios (αCD:NaCl) 0–15.0: 1. It was found that the molar conductivities were in the range 124.3125.2 S cm2 mol− 1. As a conclusion that was reached, the conductometer was unable to detect any difference in the molar conductivities induced by the action of αCD on the viscosity of the medium. The present study's conductivity measurements were not viscositycorrected. The viscosity effect is only visible at high cyclodextrin concentrations and requires high precision to measure conductivity [35, 37].