PHYS 3341 PRELIM 1. There is one bonus problem at the end of this test. Please complete all of the other problems.

Transcription

1 PHYS 3341 PRELIM 1 Prof Itai Cohen, Fall 2014 Friday Oct. 17, 2014 Name: Read all of the following information before starting the exam: Put your name on the exam now. Show all work, clearly and in order, if you want to get full credit. Circle or otherwise indicate your final answers. There is one bonus problem at the end of this test. Please complete all of the other problems. Bonus points will be awarded for the bonus problem. However your score cannot exceed 100 points. Problem # Score 1 /40 2 /60 Bonus /10 Total /100 It is your responsibility to make sure that you have all of the pages! Good luck!

2 1 1. Free Energy (30 points) A system described by the van der Waals equation of state is taken through phase space as shown by the dashed arrows in figure (A). The P-V diagrams for the system at different temperatures are shown in figure (B). Please only consider the isotherms for T 1 and T 2. The other curves are there to help clarify. a. (20 pts) Carefully draw diagrams depicting the Gibbs free energy per mole g vs. volume V for the three points on the P T diagram (P 2,T 2 ),(P 1,T 1 ),(P 1,T 2 ). Consider the ordering of the energies for the different minima and the exact volumes at which they occur. b. (10 pts) Mark the stable states with an X and the metastable states with an 0 on each diagram.

3 2 c. (10 pts) In your own words, describe the differences between the processes of spinodal decomposition and nucleation and growth. In a process of spinodal decomposition, it is energetically favorable for the system to phaseseparate into gas and liquid, and the process occurs rapidly throughout the fluid. For nucleation and growth, there is an energy barrier between the initial state and the ground state. Separation proceeds slowly, through microscopic clusters that fluctuate past this energy barrier, and then growth of separated fluid around them. 2. Polymer (35 points) We examine a polymer, as we have seen in class. A polymer is a long molecule, such as a protein or DNA. It has many degrees of freedom because it is flexible. One end of the polymer is attached to a substrate. To the other end, which is free to move, a dipolar marker is attached. We will model the polymer as a random walk in thre dimensions, that has x = y = z = 0 x 2 = y 2 = z 2 = 1 3 Nl2. where x,y,z is the position of the dipolar marker, N is the number of atoms in the polymer and l is the distance between two atoms. With the help of a uniform electric field gradient, a constant force can be exerted on the dipole, so that its energy is E = ax. The constant a is positive and proportional to the marker s dipole moment and the electric field gradient. The polymer aligns itself along the field, and so Nl x N Nl, and the energy Nal < E < Nal. a. (10 pts) Write down the number of microstates (configurations) Ω(E) for which the energy of the polymer is between E and E +δe. (hint: approximate it as a Gaussian; assume Nal E 0) Since E = ax, we know E = ax = 0, E 2 = a 2 x 2 = 1 3 Na2 l 2, so we can write Ω(E) e 3E2 2Na 2 l 2. Since there are 6 N configurations overall, we have Ω(E)dE 6 NdE 3 al 2πN e 3E 2 2Na 2 l 2.

4 3 b. (10 pts) Find the entropy of S of the polymer at energy E. S = k B lnω N ln6 3E2 2Na 2 l [ ] 3 2 ln ln[al] 2πN c. (10 pts) The polymer is immersed in a solution at temperature T. If the polymer and the surrounding liquid are in thermal equilibrium, what is the (average) energy Ē of the polymer? what are the fluctuations of the energy? We have lnω E = β = 1 kt where T is the temperature of the solution. this yields with fluctuations characterized by the variance Ē = Nl2 a 2 3kT σ 2 = Nl 2 a 2. d. (10 pts) Next, we ll treat the solution as an ideal gas with N s particles. The polymer is rapidly stretched to its full length, so that x = Nl while the temperature of the solution remains T i = T. At this point, what is the energy of the polymer? What is the energy of the solution? (hint: are they related?) What is the total energy of the system? We know for the polymer, E = ax, and so E p = Nal. For the solution, we can just use the formula for the ideal gas, E s = 3 2 N skt i. There are unrelated - when we pulled the polymer rapidly we took them out of equilibrium. The total energy is just E t = E p +E s = 3 2 N skt i Nal.

5 4 e. (10 pts) After stretching it, we allow the polymer and solution to reach equilibrium again. What is the final energy of the system? What is the final temperature of the system? Since the system is isolated, there is no change in energy, and so E f = 3 2 N skt i Nal. But we also know that E f = E f s +E f p, and we can plug in the energy formulas for the solution and polymer at equilibrium, For the solution, we can just use the formula for the ideal gas, E f = 3 2 N skt f Nl2 a 2 3kT f (1) which we solve to find T f = E f 3kN s ( a2 l 2 NN s E 2 f ) f. (10 pts) Will the final temperature be higher or lower? Why? In stretching the polymer we reduced the total energy of the system (since its energy is minimal when it is stretched along the field) and so the final temperature will be lower.

6 5 BONUS ROUND 3. Extra Credit Reading (10 points) Answer only one of the following two questions. a. (10 pts) The Diverse World of Liquid Crystals: Describe two specific changes that can take place during a phase transition that involves a liquid crystal. Give an example which is described in the article on the practical uses of liquid crystals. b. (10 pts) Phase-transition dynamics in the lab and the universe: Describe what symmetry breaking has to do with phase transitions and why it is relevant for the structure of the universe. Explain why the universe does not undergo completely equilibrated phase transitions. By signing here, I certify that I have read the Diverse World of Liquid Crystals article and that I am willing to forfeit my scores on the entire exam if it is determined through oral examination that I did not read the article. You must sign here to get the extra credit points.

7 Scrap Page (please hand these pages in with the test packet) 6

8 Scrap Page (please hand these pages in with the test packet) 7

9 Scrap Page (please hand these pages in with the test packet) 8

10 9 Possible Useful Formulas C v = T n! n n e n 2πn Γ(n+1) n n e n 2πn ( ) N p n q N n 1 e (n Np)2 /(2Npq) n 2πNpq π dxe ax2 = 2 a 1/2 0 dxxe ax2 0 0 cosh(x) = ex +e x 2 d d cosh(x) = sinh(x) dx tanh(x) = sinh(x) cosh(x) = 1 2 a 1 π dxx 2 e ax2 = 4 a 3/2 PV = NkT de = TdS pdv df = SdT pdv ( ) ( ) S U = T V T V β = lnω E X a = 1 lnω β x a Ē = lnz β ( ) lnz p = kt V T,N ( ) lnξ N = kt µ V,T sinh(x) = ex e x cosh(x) = sinh(x) dx d dx tanh(x) = 1 cosh 2 (x) E = d 2 NkT 2 dh = TdS +Vdp dg = SdT +Vdp ( ) S C p = T T S = klnω pv = kt lnξ p = 1 lnω β V ( ) lnz S = klnz kβ β ( ) lnz µ = kt N T,V ( ) lnξ S = klnξ kβ β E = ρ E = B ǫ o t B = 0 H = J +ǫ E t 108 = v swallow (unladen) = African or European? P V,N V,µ