# Calculating the physical properties of isotopes

I am trying to determine the theoretical physical properties (namely melting and boiling temperatures) of isotopic substances. Harold Urey used Debye thermodynamics (which is quite dense) in his 1946 paper to calculate the physical properties of isotopic substances. Most of the equations appear to be for exchange rates of the isotopes within a given system. I would like to know which equations are available to determine isotope effects on physical properties of isotopologues and the assumptions that go with them.

• Besides being 'dense' what is the problem with Urey's work? Dec 14 '15 at 16:45
• He doesn't explain the calculations used for calculating physical properties, but rather selective enrichment of isotopes in various compounds.
– A.K.
Dec 14 '15 at 16:50
• It seems you were rather asking about isotopologues not isotopomers. Sep 24 '18 at 19:38
• @ mithoron You are correct.
– A.K.
Sep 24 '18 at 19:41

The Debye frequency of a crystal is a theoretical maximum frequency of vibration for the atoms that make up the crystal.

It seems safe to assume melting will be occurring at the Debye frequency.

It is quite possible that the authors of your paper used the Lindemann model, which was published in 1910, in conjunction with Debye frequency, which is unique to each topoisomer, to predict the melting point using the following equation of Lindemann:

$$T_m = \cfrac{4\pi^2 m \nu^2 c^2 a^2}{k_B}$$

$$m = \mathrm{atomic\ mass}$$ $$v = \mathrm{frequency}$$ $$k_B=\mathrm{Boltzmann\ constant}$$ $$T=\mathrm{temperature}\ \left(^\circ \mathrm K\right)$$ $$c = \mathrm{Lindemann\ constant}$$ $$a = \mathrm{atomic\ spacing}$$

Different diastereomers have different resonant frequencies which can be found through either Debye calculations or IR and Raman spectroscopy.

Substituting in the Debye expression for frequency (I copied this from Wikipedia; I didn't actually derive it myself) gives:

$$T_m = \cfrac{2\pi m (\theta^2_D k_B)c^2 a^2}{\hbar^2}$$

Where $\theta^2_D$ is the Debye temperature and $\hbar$ is Planck's constant.

Another way of doing it is by using a hard sphere model in combination with Debye/Lindemann (PDF link).