The molar volume of copper soaps derived from various edible oils in 20% methanol-benzene mixture was calculated from density data and is recorded in Table 3. It has been revealed from Table 3 that the values of molar volume first decreased then there was a slight increase followed by a decrease in a linear manner showing straight line. This may be attributed to the fact that above CMC and in the higher soap concentration region, the degree of aggregation of molecules became dissimilar to that of below CMC. The plots of molar volume, against the soap concentration c are characterized by an intersection of convex curve and a straight line at a definite soap concentration which corresponds to the CMC of the soap. At the CMC, hydrocarbon chain structure of soap molecules allows extensive contact between adjacent chains of fatty acids of varying composition, possibly accompanied by changes in the vibrational and rotational degree of freedom of methylene group.

The values of CMC so determined are in good agreement with the values obtained from the density measurements. The molar volume of copper soaps derived from various edible oils in 20% methanol-benzene mixture follows the order:

Table 3
molar volume of copper soaps derived from various edible oils in 20% and 40% methanol-benzene mixture.

The comparison of the results indicates that the values of the molar volume for CM are higher than that of CSo which is due to the presence of longer fatty acid content present in the system.

Flow characterization of solution of soap solutions in terms of viscometric measurements has been employed as a tool to find out CMC of copper soaps. The viscosity, η of copper soaps in non-aqueous solvents of varying composition has been determined at a constant temperature i.e. 30 ±0.1⁰C. The value of viscosity η for all the copper soaps (i.e.CM₂₀, CG₂₀, CSe₂₀, and CSo₂₀) was found to increase with the increase in soap concentration. The increase in viscosity with the increase of soap concentration may be due to the increasing tendency of soap molecules to associate in the form of micelles. The existence of micelles of surfactants in organic solvents and mixed solvents has been reported by number of workers [29Khan, S.; Sharma, R.; Sharma, A.K. Ultrasonic studies of Cu (II) soap derived from seed oil of pongamiapinnata (Karanj), in non-aqueous binary and ternary systems at 298.15K. Malaysian J. Chem., 2017, 19(2), 99-110., 30Sharma, A.K.; Saxena, M.; Sharma, R. Acoustic studies of copper Soap- urea complexes derived from groundnut and sesame oils. J. Phy. Studies., 2017, 21(4), 4601-4606.].The plots of viscosity η against the soap concentration c are characterized by an intersection of two straight lines corresponding to the CMC of the soap Fig. (1). The change in the slope of the curve below and above CMC, however, suggests that there is a phenomenal change in the micellar agglomeration below and above CMC. At the CMC, hydrocarbon chain structure of soap molecules allows extensive contact between adjacent chains possibly accompanied by the change in vibration and rotational degree of freedom of methylene group. CMC follows the order:

The above trend clearly demonstrates that the association of soap molecules occurs at lower concentration in copper soaps of mustard and groundnut soaps in methanol-benzene as compared to sesame and soyabean soaps. It may also be pointed out that the viscosity of the soap solutions decreases with an increase in the chain length of the acid in the soap. This may be due to an increase in the size of the micelles with an increase in the number of carbon atoms in the soap. The viscosity of the soap has been found in the order:

The difference in the viscosities of copper soap solutions, in varying composition of methanol-benzene mixture is mainly due to difference in the viscosities of solvent mixture. Thus, with the increase in the number of carbon atoms in the hydrophobic chain, viscosity, η and CMC increased and then decreased. The viscosity results confirm that there is no appreciable aggregation of the soap molecules below the CMC whereas there is a marked change in the aggregation at this soap concentration.

Fig. (1) Plots of viscosity v/s concentration of copper soaps derived from various edible oils in 20 and 40 % methanol- benzene mixture.

The ratio of change in viscosity to the original viscosity of the solvent is called the specific viscosity η_sp [31Sharma, A.K.; Saxena, M.; Sharma, R. Ultrasonic studies of Cu (II) soaps derived from groundnut and sesame oils. Tenside. Surf. Det., 2017, 55(2), 127-134.
[http://dx.doi.org/10.3139/113.110544] ]

(3)

Where η and η₀ are the viscosities of solution and solvent, respectively.

The value of specific viscosity η_sp of copper soaps derived from various edible oils in 20% methanol-benzene mixture was recorded as shown in Table 4. The results show that specific viscosity η_sp of the soap solution increases with the increase in soap concentration. This increase may be due to the increasing tendency of the soap molecules to form aggregates with increasing soap concentration. The values of the CMC so obtained are in good agreement with the values obtained from the plots of viscosity ηagainst the soap concentration c.

Table 4
Specific viscosity ηsp of copper soaps derivedfrom various edible oils (in methanol-benzene mixture).

The reciprocal of co-efficient of viscosity is called fluidity [32Khan, S.; Sharma, R.; Sharma, A.K. Acoustic studies and other Acoustic Parameters of Cu(II) Soap derived from non- edible Neem oil (Azadirectaindica), in Non-aqueous media at 298.15 Acta. Ac united Ac, 2018, 298, 277-283.
[http://dx.doi.org/10.3813/AAA.919170.] ].

(4)

The fluidityϕ of copper soap solution in 20% methanol-benzene mixture decreases with an increase in soap concentration. The plots of fluidity ϕ against soap concentration (c) show an abrupt change at a point corresponding to the CMC of the soap Fig. (2). A perusal of the CMC data shows that it is dependent on the solvent composition and values are in good agreement with the values obtained from η v/s c, and η_sp v/s c plots.

Fig. (2) Plots of fluidity v/s concentration of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture.

The results of the viscosity of the solutions of copper soaps in 20% methanol-benzene mixture have been interpreted in the light of equations proposed by Einstein, Thomas and Vand.

Einstein Equation [33Einstein, A. A New determination of molecular dimensions. Ann. Phys., 1906, 19, 289-306.
[http://dx.doi.org/10.1002/andp.19063240204] ]:

(5)

Thomas equation [34Thomson, D.J. Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles. J. Colloid Sci., 1965, 20, 267-277.
[http://dx.doi.org/10.1016/0095-8522(65)90016-4] ]:

(6)

Vand equation [35Vand, V. Viscosity of solutions and suspensions; theory. J. Phys. Colloid Chem., 1948, 52(2), 277-299.
[http://dx.doi.org/10.1021/j150458a001] [PMID: 18906401] ]

(7)

Where V, c, Q, η,η ₀ and η_sp are the molar volume of the soap, concentration of the soap, interaction coefficient, viscosity of the solution and viscosity of the solvent, respectively. The values of molar volume V evaluated from the plots of η_sp v/s c (Einstein equation), η_sp/c η v/s c (Thomas Equation) and 1/c v/s 1/ log η/η₀ (Vand equations) was below and above CMC. For our referred system, the plots corresponding to Einstein, Thomas and Vand equation was obtained as an intersection of two straight lines Figs. (3 and 5). Therefore the value of molar volume enumerated from these equations show a slight difference but the trend remains unaltered irrespective of the type of equation applied. Due to irregular trend, Thomas equation is not fit for copper soap for both the solvents i.e.20% methanol and 40% methanol-benzene mixture.

Fig. (3) Plots of ηspv/s concentration of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Einstein Equation).

Fig. (4) Plots of {(η/η0)-1}/c v/s concentration of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Thomas Equation).

Fig. (5) Plots of 1/ {log (η/η0)} v/s 1/c of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Vand Equation).

It was observed that the values of molar volume below and above CMC(i.e.V₁& V₂) enumerated from Einstein equation follow the order Table 5.

Table 5
values of molar volume obtained from different equations for copper soaps derivedfrom various edible oils in 20% and 40% methanol-benzene mixture.

Interestingly Moulik’s equation fits equally well to the soap solutions both below and above CMC. The equation is [36Moulik, S.P. Surface potential of aqueous electrolyte solutions. J. Phys. Chem., 1968, 72(2), 74-78.
[http://dx.doi.org/10.1021/j100847a014] ]:

(8)

Where M and K are constants. Evaluated values of M and K from the plots of (η/η ₀)² v/s c²are shown in Table 6.

Table 6
values of viscometric parameters obtained from different equations of copper soaps derived from various edible oils in 20% and 40% methanol-benzene mixture.

The plots of (η/η ₀)² v/s c²show a break at a definite soap concentration corresponding to the CMC of the soap Fig. (6). The values of Moulik’s constant M and K have been obtained from the intercept and slope of these plots follows the order:

Unlike ‘M’, the stipulated values of ‘K’ follow the order:

Fig. (6) Plots of (η/η 0)2 v/s c2 of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Moulik’s equation).

The intrinsic viscosity [η] = limit (c→ 0) [η_sp] of the soap solution according to Gray and Alexander can be represented by the equation [37Gray, V.R.; Alexander, A.E. Viscosity and streaming birefrigence of aluminum soap solutions. J. Phys. Colloid Chem., 1949, 53(1), 9-23.
[http://dx.doi.org/10.1021/j150466a002] [PMID: 18112143] ]:

Fig. (7) Plots of (c/ηsp) v/s c of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Gray and Alexander equation).

(9)

The values of intrinsic viscosity and coefficient K’ are determined from the plot of reduced viscosity (c/η_sp) against the concentration c of the soap. The plot of (c/η_sp) v/s c is characterized by an intersection of two straight lines Fig. (7), thus Gray and Alexander equation fits well both below and above CMC. The sudden change in the trend of the plot at CMC clearly indicated that the change in the behaviour of the soap took place in the periphery of this concentration.

The viscosity data have also been interpreted in light of Jones-Dole Equation [38Jones, G.; Dole, M. The viscosity of aqueous solutions of strong electrolytes with special reference to Barium chloride J. Am. Chem. Soc., 1929, 51(1), 2950-2964.
[http://dx.doi.org/10.1021/j150466a002] ]:

(10)

For convenience, the equation may be expressed as:

(11)

(12)

Where the coefficient A and B refer to the solute–solute and solute–solvent interaction, respectively.

The plots of ψ/√c v/s √c was characterized by an intersection of two straight lines at a point corresponding to the CMC of the soap Fig. (8). The values of CMC are in close agreement with the values obtained fromη v/s c, η_sp v/s c, ϕ v/s c plots.

In view of the two intersecting straight lines forψ/√c v/s √c plots, it is reasonable to evaluate two values of each coefficient below and above CMC designated as A₁, B₁ and A₂, B_2, respectively, that are given in Table 6. The positive value of A suggests that there is a strong soap- soap interaction in this system and positive value of B indicates a strong alignment of solvent molecules with solute, which reveals the structure forming behaviour of the solvent.

Fig. (8) Plots of (ψ/√c) v/s √c of copper soaps derived from various edible oils in 20 and 40% methanol- benzene mixture (Jones – Dole equation).

From the data collected it is obvious that below the CMC, the values of coefficient B₁ (soap-solvent interaction) are larger than the values of coefficient A₁ (soap-soap interaction) which confirms that the molecules of the soap do not aggregate appreciably below the CMC and there is sudden change in the aggregation above the CMC. From this trend, it is clear that soap-solvent interaction is larger than the soap-soap interaction in dilute solution. The trend of these coefficients may be as follows:

From this trend it may be concluded that soap-solvent interaction is more pronounced below CMC. This may be ascribed to the favorable interaction between soap and solvent molecules at the premicellar concentration.

The value of molar volume forcopper soaps derived from various edible oils in 40% methanol-benzene mixture were calculated and recorded. The results show that the effect of soap concentration on the molar volume of copper soap solutions in 40% methanol-benzene mixture is almost similar to that of molar volume of copper soap solutions in 20% methanol-benzene mixture.

The plots of molar volume, V against the soap concentration is characterized by an intersection of convex curve and a straight line. Below CMC, a straight line followed by a convex curve in increasing trend is observed, and after CMC, there is a sharp decrease in molar volume.

The values of CMC determined are in good agreement with the values obtained from density measurements. The values of molar volume V for 40% methanol-benzene mixture was observed lower than that of 20% methanol-benzene mixture which is due to the fact that the nature of solvent plays a significant role in ordering and compacting the scattered micellar units. It is further suggested that methanol occupies quite a different position in the palisade layers of soap anion exhibiting a different degree of aggregation in the mixed solvent of varying composition.

The effect of the soap concentration on the viscosity, specific viscosity and fluidity of copper soap in 40% methanol-benzene mixture of varying composition similar to that of 20% methanol-benzene mixture. It observed that the viscosity of copper soap in 40% methanol-benzene mixture slightly higher than 20% methanol-benzene mixture which is due to the difference in the viscosities of solvent mixtures. The values of the CMC for the above referred system are recorded in Table 7. It can be easily seen from the table that the CMC of these soap follows the order:

Table 7
Values of viscometric parameters obtained from different equations of copper soaps derived from various edible oils in 20% and 40% methanol-benzene mixture

This is an agreement with the fact that there is a decrease in the CMC values with the increase in the number of carbon atom in the hydrophobic chain. Viscosity of copper soap derived from various edible oils in 40% methanol-benzene follows the order:

The above trend clearly demonstrates that the viscosity of the soap solutions increases with an increase in the average molecular weight of the soap.

It has been found that Einstein and Vand’s equations are equally applicable to copper soap solution in 40% methanol-benzene mixture. But Thomas equation gives no plausible inference due to irregular data. The change in the values of molar volume below and above CMC may due to the fact that entrapping of solvent molecules below CMC is dissimilar to the entrapping of solvent molecules above CMC. From comparison of the result of copper soap in both the solvents, it maybe suggested that the values of molar volume was higher in 20% methanol-benzene mixture than that of 40% methanol-benzene mixture.

Furthermore, the viscometric data below and above CMC have also been interpreted in terms of Moulik’s equation and pertaining parameters M and K. The order of M and K below and above CMC follows the order:

Jones-Dole equation is also applicable to copper soap in 40% methanol-benzene mixture. The values of A₁, A₂ and B₁, B₂ was calculated and the trend observed similar to that of copper soap in 20% methanol-benzene mixture.

A review of the results shows that below the CMC, the value of coefficient B₁ is larger than the value of coefficient A₁. This trend indicates that below the CMC, the solute-solvent interaction is more pronounced than solute-solute interaction. The trend of these coefficient is shown below-

The order A₂ > A₁ is inconsonance with the expectations that solute-solute interaction becomes stronger after CMC as the micellar aggregation takes place. For all the copper soap, the positive values indicate a strong alignment of solvent molecules which reveals the structure forming the behavior of solvent.

Literature survey [39Tank, P.; Sharma, R.; Sharma, A.K. Studies of Ultrasonic and acoustic parameters of complexes derived from Copper (II) surfactant of mustard oil with N and S atoms containing ligands in non- aqueous media (benzene) at 303.15 K. J. Acous. Soc. Ind., 2017, 44(2), 87-99., 40Bhutra, R.; Sharma, R.; Sharma, A.K. Viscometric and CMC studies of Cu(II) surfactants derived from untreated and treated groundnut and mustard oils in non-aqueous solvent at 298.15 K J. Inst. Chemists (India), 2017, 90, 29-47.] reveals that the positive value of A suggests a strong solute-solute interaction. In our referred system, the values of A ₁ and A₂ are positive and they follow the orderA₁<A₂. Thus, it is suggested that solute-solute interaction becomes stronger after CMC as the micellar aggregation takes place as compared to before CMC.

Further A₁ and A₂ (solute-solute) interaction in 20% methanol-benzene mixture was found higher than in 40% methanol-benzene mixture. This may be accounted in terms of change in the mobility of the solute with a change in the dielectric constant of the medium. The positive value of B indicated a strong alignment of solvent molecules with solute, which reveals the structure forming behavior of the solvent.

The decrease in B values B₁(40%) < B₁ (20%) showed that solute-solvent interactions are influenced by the increase in the non-polar solvent content in the system below and above CMC.