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Show that the mean square uctuation in any quantity A can be written in terms of its mean




This is about physical chemistry;Attachment Preview;pchemps1.pdf Download Attachment;Chem 120B;Problem Set 1;Due: September 10, 2014;1. (i) Show that the mean square uctuation in any quantity A can be written in terms of its mean and;mean square values, (A)2 = A2 A 2. (Here, as usual, the uctuation in A is dened as;A = A A.);(ii) Generalize this result to the case of two different uctuation quantities, i.e., show that A B =;AB A B.;2. Consider a system of N molecules contained in a xed volume V. Imagine dividing this total volume;into a large number M of microscopic subvolumes v (so that M = V /v), and let ni be the number;of molecules within subvolume i. (This notional division does not prevent molecules from moving;among subvolumes, it is just a way to keep track of their spatial distribution.) As molecules move;within the container, the values of n1, n2,..., nM uctuate, while the total number;M;N=;ni;i=1;remains constant.;What is the average number of molecules within a single subvolume, ni? Make your argument;carefully, using the fact that all subvolumes are statistically equivalent, i.e., they undergo the same set;of uctuations. Write your answer for ni in terms of v and the total density 0 = N/V.;3. The notation;3;i=1 xi;is shorthand for a sum over the quantities x1, x2, and x3;3;xi = x1 + x2 + x3.;i=1;(i) Write out the double summation;3;3;Aij;i=1 j=1;explicitly in terms of the quantities A11, A12, etc.;(ii) Let S =;3;i=1 xi.;Write the quantity S 2 in terms of x1, x2, and x3.;(iii) Now consider the case Aij = xi xj. Using your results from parts (i) and (ii), show that;3;3;Aij = S 2;i=1 j=1;and therefore that;2;xi;i;(iv) Write out all terms in the sum;3;2;i=1 xi;=;xi xj.;i;j;and show that it is in general different from;3;2;i=1 xi).;4. Consider a dilute solution with a total number Ntot of solute molecules contained in a total volume;Vtot. The total number density of solutes is thus 0 = Ntot /Vtot.;We will focus on a limited region of the solution, marked by a dashed line in the sketch below.;Chem 120B, Fall 2014;1;cell i, hi = 1;cell j, hj = 0;(Lattice cells and solute molecules not drawn to scale);This observation region has a volume V. As solutes move across its boundary, the number N of solute;molecules inside the observation region uctuates about an average value N.;In order to examine uctuations in N, it is useful to imagine dividing the observation region into;microscopic cells of a cubic lattice, each with volume v. Let ni be the number of solute molecules in;cell number i in a given measurement. We will take the solution to be sufciently dilute that nding;two solutes in the same cell is negligibly unlikely. In other words, either ni = 0 or ni = 1. Assume;as well that uctuations in different cells are uncorrelated.;(i) Estimate the size of typical uctuations in N by calculating the root mean square deviation =;(N)2, where N = N N is the deviation of N from its average value. The correct answer;can be written solely in terms of N, 0, and v. (Hint: Note that n2 = ni.);i;(ii) Show that (N)2 = N in the limit of very low concentration, 0 v;1.;(iii) Calculate / N and comment on the size of uctuations in N relative to its mean when the;observation region is macroscopically large.;5. Consider the dilute solution of question 4 from a slightly different perspective: Any of the M cells;in the observation region should be occupied (ni = 1) with probability p1 = 0 v. The probability;P (n1, n2,..., nM) of nding the system in a particular conguration of the observation volume (in;which the values of n1, n2,..., nM are all specied) is thus pN (1 p1)M N.;1;(i) Show and/or explain these two facts, p1 = 0 v and P (n1, n2,..., nM) = pN (1 p1)M N.;1;(ii) Let W (N) be the number of congurations of the observation volume when N solutes are present.;Calculate the probability P (N) of observing a given value of N, in terms of p1, W (N), M, and N.;(iii) Later in the course, we will show that, when N and M are large;ln W (N) M [ ln + (1) ln(1)];where = N/M is the fraction of occupied cells. Using this fact, write ln P (N)/M solely in terms;of and p1.;(iv) Using a computer, plot your result for P (N) as a function of the fraction of occupied lattice cells,. On a single graph, show results for p1 = 0.1 and the cases M = 10, M = 100, M = 1, 000, and;M = 10, 000.;(v) Comment on the trends evident from your graph and your expectations for a macroscopic observation volume (with M 1024).;Chem 120B, Fall 2014;2


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