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July 16th, 2019, 12:31 PM   #1
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Law of large numbers

I have a question in statistics. I am a Chemical Engineer (reaction chemistry research). In an example reaction experiment, the feed to a reactor consists of n compounds with mass fractions X1, X2, ... Xn so that:

X1+X2+X3+...+Xn=1

This feed goes into the reactor with constant temperature, pressure and all other conditions constant and the product composition is measured at the exit, e.g:- if the product has m components (m need not be equal to n, as reactions join and break molecules as per thermodynamic rules).

Y1+Y2+Y3+....Ym=1

My question is: if n is sufficiently large, will we have a fixed constant product composition Y1, Y2,.... Ym? (irrespective of the actual values of X1, X2, ..)

Can this be in the domain of "law of large numbers"? In this case, what is the definition of an "event"? Please let me know.

Thanks,
Murali
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July 16th, 2019, 01:38 PM   #2
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This looks like a chemistry question, not math.
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July 16th, 2019, 02:18 PM   #3
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Mechanical here, not chemical, but I have a few notes.

1) Obviously the product composition Yj cannot be totally independent of the reactants Xi, because both have to be stoichometrically balanced. Now, if you say, for example, that you will have a varying mixture of hydrocarbons, but you know that it will always be (a %) carbon, (b %) hydrogen, (c %) oxygen, (d %) nitrogen, etc., then it may be possible that Yj be pre-determined. This is just a minimal requirement, however.

2) It seems like the law of large numbers would apply not to having a large number of reactant species, but to things like having a large total number of molecules/atoms and having a long residence time in the reactor. These would ensure thorough mixing and complete reactions. When the molecules start breaking apart, they may recombine in any number of different ways. You know from thermodynamics that any reaction is possible, as long as the combined entropy of the products is greater than the combined entropy of the initial reactants. If you give everything enough time and energy, you can be reasonably sure the system will approach the most stable configuration (highest entropy). I'm guessing that what an appropriate time scale is, though, would probably have to be answered by empirical relations or some very sophisticated and possibly multi-scale simulations.

3) Even if you keep the mass fraction of the individual atoms fixed (stoichiometry), and wait a long enough time (2nd law of thermodynamics), don't forget the 1st law of thermodynamics. Different isomers and different combinations of the same atoms will still have different enthalpies of formation, and thus release different amounts of energy as they react. Unless your reaction chamber is in good thermal contact with a heat sink, changing the reactants will result in different outlet temperatures, and what constitutes the most stable products can change with temperature.
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July 16th, 2019, 08:48 PM   #4
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Thanks a lot for your reply.
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