3. In class we examined the random walk executed by a single molecule (or sailor). Here you will extend our results to the mutual diffusion of two molecules (or sailors). In our model both molecules wander among sites of a lattice with spacing . But one molecule is more mobile than the other: In each time interval At, molecule #1 undergoes an r-displacement of +6, while molecule #2 jumps a larger distance +me. Individually, their diffusion constants are thus Di = (/(2At) and D2 = (3/(2At). This scenario is sketched below for the specific case m = 2. X X In this problem you will analyze the separation between the two molecules X = x2 - 21 as they move about the lattice. (i) Compute the mean squared change in X after n steps, i.e., after a time t = nAt. You should find it useful to define AX, as the change in separation during the y step. Write your answer in terms of t, At, and the mean squared change in X during a single step, (AX}). (ii) List all the possible values of AX, and their associated probabilities. (iii) Using your result from part (ii), evaluate (A.X?) in terms of 41 and (2. (iv) Assemble all your results to show that the effective diffusion coefficient for the intermolecular separation is Deff = D1 + D2