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Interested in what I have learned in Isotopic geochemistry, I explored:

Calculation of the fractionation factor with a quantum chemical model

Based on the Urey equation, I calculated the fractionation factors of carbon isotopes between CO2 and CH4, oxygen isotopes between O2 and CO2, sulfur isotopes between hydrogen sulfide and sulfur dioxide. The vibration frequencies of molecules are provided by Gaussian 09, a quantum chemical model software.

Models of molecules I built with Gaussian, so cute :)

Prove the isotopic fractionation will not decrease the entropy

Once in class, the professor said that the isotopic fractionation follows specific rules like heavier isotopes tend to enter the molecules with stronger bonds, and this seems to create something "orderly" in a natural system. Since to be in order means a decrease in entropy of a system. We are wondering if naturally happened isotopic fractionation is at odds with the principle of increase of entropy?

Interested in this question, I learned statistical thermodynamics by myself and proved: isotopic fractionation will not decrease the entropy of a closed system. With the reaction going, the system's entropy will increase and reach a maximum. In the further, I hope to see if this maximum has something to do with the equilibrium of isotopic fractionation. More exploration to expect!

See my manuscript below. I will provide a typed version soon.