The Gladstone–Dale relation is an empirical 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). The relation has also been used to calculate refractive index of glass and minerals in optical mineralogy. The relation is named after John Hall Gladstone and reverend Thomas Pelham Dale, who published first discussed it in 1863.
Expression In the Gladstone–Dale relation, ( n − 1 ) ρ = ∑ k m , {\displaystyle {\frac {(n-1)}{\rho }}=\sum km,}
n is the index of refraction of the mixture, ρ is the density of the mixture of miscible liquids, m is the mass fractions of each component molecule (summing to 1), and k is the specific refractivity of each component molecule (the light-bending ability for a given mass). The Gladstone–Dale relation applies to any unit system, so long as ρ and k uses the same units. In SI units the most common option is g/cm3. k is also known as the Gladstone–Dale constant as it holds constant for each molecule regardless of changes to its density (e.g. due to temperature changes). Its unit is the inverse of the density unit, e.g. cm3/g under the aforementioned choice of density unit. The same applies to mixtures of fixed composition, hence ( n − 1 ) ρ {\displaystyle {\frac {(n-1)}{\rho }}} are also given this name. With real gases, changing the temperature will end up changing the proportion of chemical species within it, making the value not constant (e.g. by dissociation of O2 into oxygen atoms) - hence the other name, Gladstone-Dale coefficient.
Examples
Alcohol and water Consider a mixture of ethanol and water in a ratio of m ∶ (1 − m). Although the mass is conserved on mixing, the volume of ethanol-water mixtures is smaller than the total volume of the pure liquids due to the formation of ethanol-water bonds. If one plots the volume V against the ethanol fraction m, the result is a quadratic-like curve; the density {{{1}}} is similarly a curve. However, the plot of the refractive index of the mixture n against m is linear.
Solids In the 1900s, the Gladstone–Dale relation was applied to glass, synthetic crystals and minerals. Average values for the refractivity of oxides such as MgO or SiO2 give good to excellent agreement between the calculated and measured average indices of refraction of minerals. However, specific values of refractivity are required to deal with different structure-types, and the relation required modification to deal with structural polymorphs and the birefringence of anisotropic crystal structures. In recent optical crystallography, Gladstone–Dale constants for the refractivity of ions were related to the inter-ionic distances and angles of the crystal structure. The ionic refractivity depends on 1/d2, where d is the inter-ionic distance, indicating that a particle-like photon refracts locally due to the electrostatic Coulomb force between ions.
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