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I need to determine the purity of a solution and we need to discuss about the 'theoretical' refractive index of the substance that we are supposed to have.

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There is no single equation that accurately predicts the refractive index of a mixture. This reference discusses 9 approaches to estimating the refractive index of mixtures and provides the formula for each method. The simplest method is the Arago-Biot approach. It is reasonably reliable and is described by the following formula

$$n = n_1 \phi_1 + n_2\phi_2$$

where $n$ is the predicted refractive index of the mixture, $n_x$ is the refractive index of the $x^{th}$ component, and $\phi_x$ is the volume fraction of the $x^{th}$ component. The volume fraction of the first component is given by

$$\phi_1 = x_1 V_1 / \Sigma~ x_i V_i$$ Here $x_i$ is the mole fraction of the $i^{th}$ component and $V_i$ is the molar volume (the molecular weight divided by the density) of the $i^{th}$ component. The more ideal the liquids, the more accurate the predictions.

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The answer by ron is quite correct, multiple methods exist for binary mixtures, as his reference details quite nicely. However, the question posed was for a mixture of four substances. I am unaware of work in refractive index mixing rules beyond binary mixtures, a June 2018 search in SciFinder (American Chemical Society literature search engine) found 23 articles on refractive index mixing, but all seemed limited to binary mixtures. Hence, the question of which rule might be reasonably applicable to a quaternary mixture is unanswered in the literature, as far as I can see.

Naturally, one could try extending the existing rules to ternary or quaternary mixtures in some situations, the Arago-Biot rule mentioned above would be easy to change into a four-term sum, (as would Lorentz-Lorenz and several others), but a significant difficulty would be in finding the volume fractions (which requires partial molar volumes) in a quaternary mixture. Determining partial molar volumes is not hard in a binary mixture and can be found in many texts. Extension to ternary or quaternary mixtures is not at all common, but this reference might be of help.

The main point is, there is no apparent "theoretical" refractive index for a mixture of four components. Even doing this with two components requires that you know the partial molar volume data (these are not constants but vary with composition!), and take your pick of a dizzying variety of mixing rules. What is very certain is that the refractive index is NOT simply the median of the refractive indices of the components in any general sense. I suspect that the original post was from a bewildered student. This professor is bewildered by the question as well, and would suggest that the person who posed the question didn't appreciate the difficulty of the topic.

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