1 Introduction

The scientific community [1,2,3,4,5] is fascinated by the optical characteristics, specifically the refractive index, of both pure liquids and their mixtures. This property serves various purposes, including assessing material purity and measuring solute concentrations in solutions. Moreover, it offers insights into molecular behavior and intermolecular forces by virtue of its correlation with the electronic polarizability of molecules [1]. Further, the refractive index of liquid binary mixtures serves as vital foundational information utilized in various fields such as process simulation, equipment design, solution theory, and molecular dynamics [6, 7]. These investigations offer valuable insights into molecular interactions, which facilitate numerous advancements in industrial, scientific, and technological domains [7, 8]. In order to understand the nature of molecular interactions between the mixture components, it is of interest to discuss the same in terms of excess value of refractive index rather than actual values. Many attempts have been made by researchers to understand the nature and degree of interactions in liquid mixtures using excess refractive index [9,10,11]. In addition to experimental measurement, the refractive indices of transparent mixtures can be predicted using “mixing rule” equations. These equations make use of the refractive index values of the pure components of the mixture. There are a numbers of recognized theoretical mixing rules like Eyring-John (E-J), Gladestone-Dale (G-D), Newton (Nw), Argo-Biot (A-B) and Heller (H), which can be used for predicting the refractive index of liquid mixtures [12, 13].

3-Bromoanisole, also known as m-bromoanisole, consists of a benzene ring substituted with a bromine atom and a methoxy group (-OCH3) at the meta position. The diverse reactivity and functional groups present in 3-bromoanisole make it a valuable compound in organic synthesis and various industrial applications [14,15,16,17]. The solvent properties of methanol (MeOH) make it valuable in industries ranging from fuel and oil to paints and pharmaceuticals [18,19,20,21,22]. Additionally, its role as a preservative is crucial in various products, including food items [23, 24]. Despite the relevance of these compounds, the detailed understanding of their molecular interactions, particularly under varying temperature conditions, is not well-documented in the literature. This study’s novelty lies in its comprehensive examination of the refractive index and excess refractive index of 3-bromoanisole and methanol mixtures across a range of temperatures, utilizing various theoretical mixing rules. By investigating the non-linear behavior of these optical properties, this work reveals critical insights into the molecular interactions, such as hydrogen bonding and dipole-dipole interactions, which are fundamental for predicting the behavior of similar systems in industrial applications. The use of Redlich-Kister polynomial modeling further enhances the understanding of these interactions, offering a novel approach to studying excess refractive indices in complex binary mixtures. This research contributes to the broader field by filling a gap in the existing knowledge base, providing new data and interpretations that could aid in the design and optimization of processes involving these mixtures.

This paper presents the experimentally determined refractive index of different compositions of 3-bromoanisole and methanol mixtures at four temperatures (293.15, 303.15, 313.15, and 323.15 K). The study utilizes different mixing rules to predict the refractive index of the binary mixtures. This work is part of a broader investigation into the molecular interactions of bromo-substituted anisole with primary alcohols, using physico-chemical, acoustic, and dielectric methods [25,26,27,28,29]. The primary aim is to examine how concentration and temperature variations affect the refractive index and its excess properties in the 3-BA and MeOH mixture, providing detailed insights into the molecular interactions within the system.

2 Materials and measurements

3-BA (extra pure, 99% purity) and MeOH (HPLC grade, 99.7% purity) were supplied by Spectrochem Pvt. Ltd. (India) and Ranbaxy Pvt. Ltd. (India), respectively. Total eleven different binary mixture samples were prepared by volume fraction of 3-BA in MeOH, and were converted into mole fraction (Xa) of 3-BA by relation reported in reference [30]. The measurement of refractive index of all the prepared binary solutions were carried out using Abbe’s refractometer with an accuracy of ± 0.001. The temperature was maintained using a constant temperature water bath with an accuracy of ± 0.1 K.

3 Evaluation of different parameters

3.1 Excess parameter

Information regarding the structural changes in binary mixtures can be accessed by studying the excess properties of the materials [31, 32]. The excess refractive index (\(\:{\text{n}}_{\text{D}}^{\text{E}}\)) of the materials were determined using the following relation,

$$\:{\text{n}}_{\text{D}}^{\text{E}}={\text{n}}_{\text{D}\text{m}}-\left({\text{n}}_{\text{D}\text{a}}{\text{X}}_{\text{a}}+{\text{n}}_{\text{D}\text{b}}{\text{X}}_{\text{b}}\right)$$
(1)

where, X is mole fraction and the suffixes a, b and m represent liquid a (3-BA), liquid b (MeOH) and mixture, respectively. These excess parameters were fitted to the Redlich-Kister (RK) Eqs. [33, 34],

$$\:\text{Q}={\text{X}}_{\text{a}}{\text{X}}_{\text{b}}\sum\:_{\text{i}}{\text{B}}_{\text{i}}{({\text{X}}_{\text{a}}-{\text{X}}_{\text{b}})}^{\text{i}}$$
(2)

where i = 0, 1, 2,…n. and B is RK coefficient. The coefficients Bi (where i = 0, 1, 2 and 3) and standard deviation (σ) of RK equation for liquid mixtures are reported in Table 1.

Table 1 Comparison of experimental and literature values of refractive index (nD) for MeOH at different temperatures
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3.2 Mixing rules

Various mixing rules for predicting the refractive index of binary mixtures are given below,

Eyring-John (E-J) [35],

$$\:{\text{n}}_{\text{m}}={\text{n}}_{\text{a}}{{\text{V}}_{\text{a}}}^{2}+{\text{n}}_{\text{b}}{{\text{V}}_{\text{b}}}^{2}+2{\left({\text{n}}_{\text{a}}{\text{n}}_{\text{b}}\right)}^{1/2}{\text{V}}_{\text{a}}{\text{V}}_{\text{b}}$$
(3)

Gladestone-Dale (G-D) [12],

$$\:{\text{n}}_{\text{m}}-1=\left({\text{n}}_{\text{a}}-1\right){\text{V}}_{\text{a}}+\left({\text{n}}_{\text{b}}-1\right){\text{V}}_{\text{b}}$$
(4)

Newton (Nw) [12],

$$\:{{\text{n}}_{\text{m}}}^{2}-1=\left({{\text{n}}_{\text{a}}}^{2}-1\right){\text{V}}_{\text{a}}+\left({{\text{n}}_{\text{b}}}^{2}-1\right){\text{V}}_{\text{b}}$$
(5)

Argo-Biot (A-B) [12],

$$\:\:{\text{n}}_{\text{m}}={\text{n}}_{\text{a}}{\text{V}}_{\text{a}}+{\text{n}}_{\text{b}}{\text{V}}_{\text{b}}$$
(6)

Heller (H) [36],

$$\:\frac{({\text{n}}_{\text{m}}-{\text{n}}_{\text{a}})}{{\text{n}}_{\text{a}}}=\frac{3}{2}\left(\frac{{\left(\frac{{\text{n}}_{\text{b}}}{{\text{n}}_{\text{a}}}\right)}^{2}-1}{{\left(\frac{{\text{n}}_{\text{b}}}{{\text{n}}_{\text{a}}}\right)}^{2}+2}\right){\text{V}}_{\text{b}}$$
(7)

Where in Eq. 3 to 7, n is the refractive index and V is the volume fraction. Suffix m, a and b represent a mixture, 3-BA and MeOH respectively.

The Root Mean Square Division (RMSD) values for the above mixing rules are determined by the following equation [12];

$$\:\text{R}\text{M}\text{S}\text{D}={\left[\frac{1}{\text{p}}\sum\:{\left({\text{n}}_{\text{e}\text{x}\text{p}}-{\text{n}}_{\text{c}\text{a}\text{l}}\right)}^{2}\right]}^{1/2}$$
(8)

Where, nexp, ncal and p represent the experimental refractive index, calculated refractive index and total number of concentrations respectively.

3.3 Electronic polarizability (αe)

The electronic polarizability (αe) correlates the refractive index (nD) and density (ρ) and is determined by the following Lorentz-Lorentz formula [3, 5],

$$\:{{\alpha\:}}_{\text{e}}=\:\frac{3{\text{V}}_{\text{m}}}{4{\pi\:}{\text{N}}_{\text{A}}}\left(\frac{{\text{n}}_{\text{D}}^{2}-1}{{\text{n}}_{\text{D}}^{2}+2}\right)$$
(9)

Where Vm is molar volume, NA is Avogadro number.

4 Results and discussions

The experimental values of refractive index of pure methanol at different temperatures are compared with the literature values in Table 2. The agreement between the experimental and literature values are found to be satisfactory. The literature values of refractive index for 3-BA are not available for comparison. The measured values of refractive index over entire range of mixture composition of 3-BA and MeOH at different temperatures are reported in Table 3.

Table 2 Experimental values of refractive index (nD) as function of mole fraction of 3-BA (Xa) at different temperatures and at atmospheric pressure
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Table 3 Coefficients of Redlich–Kister equation (Bi, i = 0, 1, 2, 3) and standard deviation (σ) for the binary mixtures of 3-BA + MeOH at different temperatures
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The ratio of speed of light in a vacuum relative to that in the considered medium provides the refractive index of the material. The variation of measured refractive index for a binary mixture of 3-BA and MeOH at different temperatures is shown in Fig. 1. The value of nD is found to increase nonlinearly with an increase in the weight fraction of 3-BA. This behavior is observed at all the measured ranges of temperatures. Nonlinear behavior of refractive index against the mole fraction of 3-BA indicates contribution of electronic polarization in molecular interaction [40]. Moreover, the refractive index of the studied mixture solutions found to decrease with rise in temperature, indicating that the light propagates at higher velocity when temperature increases [2].

Fig. 1
figure 1

Plots of refractive index as a function of mole fraction of 3-BA in MeOH at different temperatures

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Fig. 2
figure 2

Plots of excess refractive index (\(\:{\text{n}}_{\text{D}}^{\text{E}}\)) against mole fraction of 3-BA in MeOH at different temperatures. Continuous line shows the fitted line of Redlich–Kister equation

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The excess refractive index for the studied system is determined and fitted to Redlich–Kister type polynomial equation [41], fitted parameters are reported in Table 1. In Fig. 2, it can be seen that the excess refractive index value, \(\:{\text{n}}_{\text{D}}^{\text{E}}\) for the binary system of 3-BA and MeOH are positive over the entire concentration range and temperature. Positive contribution of \(\:{\text{n}}_{\text{D}}^{\text{E}}\) values are the consequence of the reduction of the light speed in the binary mixtures in comparison to their pure component. This phenomenon arises due to the formation of more compact or packing structures of the mixture and strong intermolecular interactions, which can be attributed to the hydrogen bonding between the unlike molecules in the mixture system [1, 42].

The electromagnetic theory of light is the basis of the mixing rules of refractive index which treats the molecules as dipoles or assemblies of dipoles by an external field [43]. Since the mixture is composed of constituents belonging to different classes of compounds in which various molecular interactions are present. The most important mixing rules which are suitable for predicting the refractive index in 3-BA + MeOH binary mixture under consideration have been tested. The experimental indices obtained for the binary mixtures at 303.15 K are compared with the predicted results by the mixing rules proposed by E-J, G-D, Nw, A-B and He in Table 4.

Table 4 Calculated values of refractive index using different mixing rules, as a function of volume fraction of 3-BA (Va) in MeOH at T = 293.15 K
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The RMSD values for the various mixing rules at different temperatures are reported in Table 5. As RMSD values indicate that the refractive index for mixtures are predicted with high accuracy for the system under consideration. A close look of the table reveals that the values of RMSD for G-D and A-B relations are found to be identical when volume additively is assumed. Refractive index values predicted from Nw relation show good agreement with the experimental values. While RMSD values for E-J, G-D, A-B and He relations are relatively higher. The applicability of the mixing rules for predicting refractive index has also been emphasized by other researchers [13, 44, 45].

Table 5 The RMSD values of different mixing rules at different temperatures
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The experimental values of refractive index at optical wavelength are further utilized to determine the electronic polarizability (αe) of the system. The polarizability measures the ability of the molecules to deform and orient in the presence of the electric field [46]. In general, the imposed electric field, E displaces the electrons in an atom to produce a dipole moment, µ. The induced dipole moment is proportional to the electric field, µ = αeE [1]. The correlation of refractive index with the density provides electronic polarizability, which is evaluated from Eq. 9 for entire mixture concentration of 3-BA and MeOH and at all temperatures. The determined values of αe for MeOH is 5.17 × 10− 22 cm3mol− 1, increase linearly with an increase of 3-BA mole fraction and reach to αe for 3-BA, 25.44 × 10− 22 cm3mol− 1 at 293.15 K temperature. This linearly increasing behavior of electronic polarizability is observed at all measured temperatures. Generally, the electronic polarizability increases as volume occupied by electrons increases [1], and in this study, the values of αe for 3-BA is found about 4.92 times higher than MeOH. The 3-BA molecules in comparison to MeOH, are larger and have less electronegative atoms. Therefore, charge distribution takes place easily, and that is why the electronic polarizability of the mixtures increases on increasing 3-BA mole fraction in the system [1]. Similar linear behavior of polarizability was observed by Moosavi et al. [3] in an aqueous solution of 1, 2-etanediol, 1, 3-propanediol, 1, 4-butanediol, and 1, 5-pentanediol at 298.15 K temperature. They suggested that the increase of electronic polarizability of molecules is owing to the fact that the structure of a molecule becomes more complicated, its electron cloud becomes more distributed and decentralized [3]. Furthermore, perusal of Fig. 3 reveals that the linear trend of αe is practically temperature independent. Since refractive index is measured in the optical region, the polarizability should not include orientational effects, and thus the polarizability should not depend on temperature over a small temperature range [47].

Fig. 3
figure 3

Plot of electronic polarization (αe) versus mole fraction of 3-BA in binary solution of 3-BA + MeOH at different temperatures

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5 Conclusions

Experimentally determined refractive index values of different concentrations of the binary mixture of 3-BA and MeOH, at 293.15, 303.15, 313.15 and 323.15 K temperatures, are reported. Nonlinear variation of refractive index for binary mixtures of 3-BA + MeOH suggests the contribution of electronic polarization in molecular interaction. As the temperature increases, the refractive index of the examined mixture solutions declines, suggesting that light travels faster under elevated temperatures in the liquid medium. The \(\:{\text{n}}_{\text{D}}^{\text{E}}\) values for the binary system of 3-BA and MeOH are positive over the entire concentration range and temperature, indicating the strong intermolecular interactions, attributed to the H-bonding between the unlike molecules in the mixture system. The refractive index of the studied binary mixture is determined by using various mixing rules such as E-J, G-D, Nw, A-B and He. Newton (Nw) model shows good agreement with the experimental values of refractive index. The electronic polarizability of the mixtures increases with increasing 3-BA mole fraction.