## 2.4 The Kirkwood factor

The Kirkwood correlation factor (g) provides information about orientation of the electric dipoles in polar liquids [9]. The modified form of Kirkwood correlation factor that is, effective Kirkwood correlation factor (geff) given by [10, 11] was estimated by the equation:

$$\frac{4{\Pi }\text{N}}{9\text{K}\text{T}}\left(\frac{{{\mu }}_{1}^{2}.{ {\rho }}_{1}. {{\Phi }}_{1}}{{\text{M}}_{1}}+\frac{{{\mu }}_{2}^{2}.{ {\rho }}_{2}.{ {\Phi }}_{2}}{{\text{M}}_{2}}\right){\text{g}}^{\text{e}\text{f}\text{f}}=\frac{\left( {\epsilon }_{sm}-{{\epsilon }}_{{\infty }\text{m}}\right)(2{\epsilon }_{sm}+{{\epsilon }}_{{\infty }\text{m}} )}{{\epsilon }_{sm}{({{\epsilon }}_{{\infty }\text{m}}+2)}^{2}}$$

2

Where “geff” is the effective Kirkwood correlation factor for the binary mixtures.

In Eq. (2) N, K, T, \({\epsilon }_{sm} and {{\epsilon }}_{{\infty }\text{m}}\) ,\({{ \mu }_{1}}^{2}, {{\mu }_{2}}^{2}, {\rho }_{1}\),\({ \rho }_{2}\), \({M}_{1},\) \({M}_{2}\), \({\varPhi }_{1}, {\varPhi }_{2},\)are Avogadro number, Boltzmann constant, temperature, static dielectric constant of mixture and dielectric constant at high frequency, dipole moment, density, molecular weight, mole fraction of liquid 1 and 2 respectively.

## 2.5 Bruggeman factor (fBM)

Molecular interactions in polar binary mixtures can be confirmed from the Bruggeman mixture formula. The static dielectric constant\({ (\epsilon }_{sm}\)) of two binary mixtures is related to this formula with the volume fraction of solute\({ (\epsilon }_{sB}\)) that proves the interaction between the solute (\({\epsilon }_{sB}\)) and solvent (\({\epsilon }_{sA}\)) of the mixtures by the equation [12]:

$${f}_{BM=}\left[\frac{{\epsilon }_{sm}-{\epsilon }_{sB}}{{\epsilon }_{sA}-{\epsilon }_{sB}}\right]{\left[\frac{{\epsilon }_{sA}}{{\epsilon }_{sm}}\right]}^{\frac{1}{3}}=1-V, \left(3\right)$$

In above equation (V) is the volume fraction. According to this equation a linear relationship is expected in Bruggeman factor **(**fBM) and (V), deviation from this linear relation indicates presence of molecular interaction and hydrogen bonding in the mixtures.

2.6. Molar refraction, Atomic polarization, Permittivity at higher frequency, Polarizabilty, Solvated radii, molecular polarization and deviation in molar refraction

From experimental densities (d) and refractive indices (n) data of pure liquids and mixtures, the molar refraction (Rm) was calculated by the equation [13]:

$${\text{R}}_{\text{m}}=\left(\frac{{\text{n}}^{2 }-1}{{\text{n}}^{2}+2}\right){\text{V}}_{\text{m}}={\text{P}}_{\text{A}}+{\text{P}}_{\text{E}}={\text{P}}_{\text{T}}={\text{P}}_{\text{D}}$$

4

Where “n” is the refractive index of the liquid and \({\text{V}}_{\text{m}}\) = (M/d) is molecular volume, in this ‘M’ and ‘d’ is the molecular weight and the density of the pure liquids respectively. The right hand side of Eq. (4) is equal to the summation of both atomic polarization (PA) and electronic polarization (PE) and that is equal to total polarization (PT) or distortion polarization (PD).

The atomic polarization (PA) was calculated from the refractive indices (n) of pure substances and mixtures by the equation [14]:

$${\text{P}}_{\text{A}}=1.05 {\text{n}}^{2}$$

5

The permittivity at higher frequency\({ (\epsilon }_{\infty })\) is the square of the refractive index and it was calculated by the equation:

$${ {\epsilon }}_{{\infty }}={\text{n}}^{2}$$

6

Where (n) is the refractive index of the binary mixtures.

The molecular dipole polarizability (α) was calculated from the experimental densities and refractive indices of pure substances and mixtures using Lorentz- Lorentz formula [15]:

$$\left(\frac{{\text{n}}^{2 }-1}{{\text{n}}^{2}+2}\right)=\left(\frac{4}{3}\right){\Pi }\text{n}{\prime }{\alpha }$$

7

Where \({n}^{{\prime }}=\frac{N}{{V}_{m}}\), N is Avogadro’s number, \({V}_{m}\) is molar volume, and (n) is refractive index of the binary mixtures.

Considering spherical form of the solvated molecules, the solvated radii of the pure liquids and binary mixtures were calculated using the equation [16]:

$$\text{V}= \left(\frac{4}{3}\right){\Pi }{\text{r}}^{3}$$

8

From experimental dielectric constants and densities of pure substances and mixtures the molecular polarization (Pm) was estimated using the equation [17]:

$${\text{P}}_{\text{m}}={\text{V}}_{\text{m}}\left(\frac{{{\epsilon }}_{\text{s}}-1}{{{\epsilon }}_{\text{s}}+2}\right)$$

9

Where (**ε**s) and (\({\mathbf{V}}_{\mathbf{m}}\)) is the static dielectric constants and molecular volume of the binary mixtures.

The deviation in molar refraction was determined from the experimental density and refractive indices by the well-known equation [18]:

\({\varDelta }_{\text{R}}={\text{R}}_{\text{m}\text{i}\text{x}}-\sum _{\text{i}}{{\Phi }}_{\text{i}}{\text{R}}_{\text{i}}\) Where i= 1, 2, (10)

In above equation (\({\text{R}}_{\text{m}\text{i}\text{x}}\)) is the molar refractivity of the mixtures (Φi) and (\({\text{R}}_{\text{i}}\)) are volume fraction and molar refractivity of pure liquids 1 and 2 respectively.

2.7. Excess Parameters (excess density (d E ), excess refractive index (n E ), excess molar polarization (P m ) E and excess molar volume (V)E

From experimental densities, the excess density (dE) was calculated using the equation:

$${\text{d}}^{\text{E}}={\text{d}}_{\text{m}\text{i}\text{x}}-\left({{\Phi }}_{1}{\text{d}}_{1}-{{\Phi }}_{2}{\text{d}}_{2}\right)$$

11

Where \({\text{d}}_{\text{m}\text{i}\text{x}}\) are the values of densities of mixtures,\({\varPhi }_{1},\) \({\varPhi }_{2}\) d1, d2 are the mole fractions and densities of the first and second liquids respectively.

From experimental refractive indices (n), the excess refractive index (nE) was estimated by the equation:

$${\text{n}}^{\text{E}}={\text{n}}_{\text{m}\text{i}\text{x}}-\left({{\Phi }}_{1}{\text{n}}_{1}-{{\Phi }}_{2}{\text{n}}_{2}\right)$$

12

Where \({n}_{mix}\) are the values of refractive indices of the mixtures\(\text{a}\text{n}\text{d} {\varPhi }_{1},\) \({\varPhi }_{2},\) \({n}_{1}\) \(,{n}_{2}\) are the mole fractions and refractive indices of the first and second liquids respectively.

The excess molar polarization (Pm)E of the mixtures was determined by the equation:

$${P}_{m}^{E} ={\text{P}}_{\text{m}\text{i}\text{x}}-[{\text{P}}_{\text{m}1} .{\varPhi }_{1}+{\text{P}}_{\text{m}2} .{\varPhi }_{2 } ]$$

13

Where Pmix is polarization of mixtures, Pm1, Pm2, Φ1 and Φ2 are polarization and mole fraction of liquid 1 and 2 respectively.

From experimental densities, the excess molar volume (V)E was estimated using following equation [19]:

\({ \text{V}}^{\text{E} }= {{V}_{m}}_{\text{m}\text{i}\text{x}}-\sum _{\text{i}=1}^{\text{n}}{{\Phi }}_{\text{i}} {.\text{V}}_{\text{m}\text{i}} \text{W}\text{h}\text{e}\text{r}\text{e} \text{i}=1, 2\) , 3---- (14)

Where \({{V}_{m}}_{\text{m}\text{i}\text{x}}\) is the molar volume of the mixtures\({{\Phi }}_{\text{i}}\)represents volume fraction and \({\text{V}}_{\text{m}\text{i}}\) is the molar volume of the components 1 and 2 respectively.