Question

5. 4] Consider a box of volume Vo containing an ideal gas of No identical molecules. Consider a section of the box with volume V-V/3. For parts a)-d) of this problem. No is very large a) [0.5] At any given time, what is the probability of finding all No molecules in ? (There is no wall separating Vfrom the rest of the box.) b) [0.5] If you could measure the number of molecules in V, and you carried out such a measurement many times, what is the mean number you would find? What would be the standard deviation of such a measurement? c) [0.5] Now consider two situations: First, put all the molecules in V, and keep them there with a divider wall in place, as shown in the figure. Second, remove the divider and let the molecules fill the entire volume Vo. How much larger is the total multiplicity of the system in the second case than in the first? In other words, what is the ratio g(%)/g(V)? d) [0.5] What is the difference in entropy between the two situations, o(Vo) - o()? e) [2] Now let No4. Write down the probabilities to find 0, 1, 2, 3, or all 4 molecules in V. Check that your probabilities add up to 1

Answer 1

(a) Probability of finding one molecule in volume v - P(v) = (/v_0) = 1/3 Probability of finding no molecule in v P = (1/3)^No (b) Assuming the molecular are evenly distributed then no. of molecules proportionalto volume Average no. of molecules in V = N_0/3 since this probability distribution is binomial standard derivation sigma = (squareroot N_p^q) here N = N_0, P = probability of finding the molecule in volume v = 1/3 and q = 1 - p = 2/3 \r\n sigma = squareroot 2 N_0/3 (c) Multiplicity proportionalto volume(v) g(V_0)/g(v) = 1/3 (d) because sigma(v) = k/ng(v) sigma(v_0) - sigma(v) = k/ng(v_0) - k/n(gv) = k/n(g(v_0)/g(v)) sigma(v_0) - sigma(v) = kn(g(v_0)/g(v)) (e) In binomial distribution P_r(k, n, p) = n_c_k p^k(1 - p)^1 - k probability of finding zero particle in v_0 p(0) =^4C_0(1/3)^0 (2/3)^4 p(0) = 16/8! P(1) =^4C_1(1/8)^1 (2/3)^3 = 4 times 8/8! = 32/8! P(2) =^4C_2 (1/3)^2 (2/3)^2 = 6 times 4/8! = 24/8! P(3)