Gladstone–Dale relation
The Gladstone–Dale relation (J. H. Gladstone and T. P. Dale, 1864) is a mathematical relation used for optical analysis of liquids, the determination of composition from optical measurements. It can also be used to calculate the density of a liquid for use in fluid dynamics (e.g., flow visualization; Merzkirch 1987). The relation has also been used to calculate refractive index of glass and minerals in optical mineralogy (Mandarino 2007).
In the Gladstone–Dale relation, (n−1)/ρ = sum(km), the index of refraction (n) or the density (ρ in g/cm3) of miscible liquids that are mixed in mass fraction (m) can be calculated from characteristic optical constants (the molar refractivity k in cm3/g) of pure molecular end-members. For example, for any mass (m) of ethanol added to a mass of water, the alcohol content is determined by measuring density or index of refraction (Brix refractometer).
Mass (m) per unit volume (V) is the density m/V. Mass is conserved on mixing, but the volume of 1 cm3 of ethanol mixed with 1 cm3 of water is reduced to less than 2 cm3 due to the formation of ethanol-water bonds. The plot of volume or density versus molecular fraction of ethanol in water is a quadratic curve. However, the plot of index of refraction versus molecular fraction of ethanol in water is linear, and the weight fraction equals the fractional density (d; Teertstra 2005).
The Gladstone–Dale relation can be expressed as an equation of state by re-arranging the terms to (n−1)V = sum(kdm). The macroscopic values (n) and (V) determined on bulk material are now calculated as a sum of atomic or molecular properties. Each molecule has a characteristic mass (due to the atomic weights of the elements) and atomic or molecular volume that contributes to the bulk density, and a characteristic refractivity due to a characteristic electric structure that contributes to the net index of refraction.
The refractivity of a single molecule is the refractive volume k(MW)/An in nm3, where MW is the molecular weight and An is Avogadro's number. To calculate the optical properties of materials using the polarizability or refractivity volumes in nm3, the Gladstone–Dale relation competes with the Kramers–Kronig relation and Lorentz–Lorenz relation but differs in optical theory (Jaffe 1988).
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