Assume there is an "observational region" in a dilute solution with a volume V, and as solutes move across its boundary, the number N of solute molecules inside the observation region fluctuates. Divide V into M regions of volume v each with n particles. The solution is dilute enough that n=0 or 1 (there is no v with more than one particle of solute), and each cell is occupied (n=1) with probability rho=(rho_0)v. If W(N) is the number of configurations of the observation volume when N solutes are present, what is the probability P(N) of observing a given value of N, in terms of p, W(N), M, and N.

gaby131o 2022-09-25 Answered
Probability of Observing N particles in a given volume?
I'm having an issue with a probability problem concerning solutions.
Assume there is an "observational region" in a dilute solution with a volume V, and as solutes move across its boundary, the number N of solute molecules inside the observation region fluctuates.
Divide V into M regions of volume v each with n particles. The solution is dilute enough that n = 0 or 1 (there is no v with more than one particle of solute), and each cell is occupied ( n = 1) with probability p = ( ρ 0 ) v.
If W(N) is the number of configurations of the observation volume when N solutes are present, what is the probability P(N) of observing a given value of N, in terms of p,W(N),M, and N.
I know the probability P ( n 1 , n 2 , , n M ) of finding the system in a particular configuration in the observation volume is p ( N ) = p N ( 1 p ) M N ,, (Bernoulli Distribution), and since there are N particles in M spaces then the maximum number of configurations is M ! ( N ! ( M N ) ! )
I'm not sure where to go from here.
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Answers (1)

Nathalie Rivers
Answered 2022-09-26 Author has 7 answers
Step 1
If p(N) is the probability of finding a given configuration of N particles, and W(N) is the amount of configurations for a given N number of particles, then the probability of observing N particles total is the sum of the probability of finding each of the configurations that have N particles. That is, P ( N ) = p ( N ) W ( N ) = p N ( 1 p ) M N W ( N ) ..
Step 2
Every possible configuration has the same probability of occurring, so you just count how many of those meet your requisites (i.e., that they have N particles).
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