Please show all work and write clearly so its easy to follow Consider a gas of N

Question

Please show all work and write clearly so its easy to follow

Consider a gas of N_0 noninteracting molecules enclosed in a container of volume V_0. Focus attention on any sub volume V of this container and denote by N the number of molecules located within this subvolume. Each molecule is equally likely to be located anywhere within the container; hence the probability that a given molecule is located within the subvolume V is simply equal to V/V_0. What is the mean number N of molecules located within V? Express your answer in terms N_0, V_0, and V. Find the relative dispersion (N - N)^2/N^2 in the number of molecules located within V. Express your answer in terms of N, V, and V_0. What does the answer to part (b) become when V

Explanation / Answer

Solution Part (a)

Back-up Theory

If a random variable X assumes values, x1, x2, ....., xn with respective probabilities p(x1), p(x2), ....., p(xn),

then its mean is given by E(X) = Sum of {xi.p(xi)}, summed over i = 1,2, ....., n

Now, to work out the solution,

in the given problem, X = N and X i.e., N can assume values 0, 1, 2, ....., N0.

Also, given that each particle has equal chance(probabilty) of V/V0 of being located anywhere,

P(N = k) = (V/V0)k

So, Nbar = Mean of N = {0xP(N = 0)} + {1xP(N = 1)} + {2xP(N = 2)} + ..... + {N0xP(N = N0)} = S, say

Then, S = 1(V/V0) + 2(V/V0)2 + ......... + N0(V/V0)N0 ......... .....................(1)

Then, S(V/V0) = 1(V/V0)2 + 2(V/V0)3 +......... + (N0 - 1)(V/V0)N0 + N0(V/V0)N0+1 ............. (2)

(1) - (2): S{1 - (V/V0)} = (V/V0) - N0(V/V0)N0+1

So, S = {(V/V0) - N0(V/V0)N0+1}/{V0 - V)/V0}, which on further simplification gives,

Mean N = S = V[V0N0 - N0VN0]/[V0N0(V0 - V)] ANSWER

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