24 Jun Chemistry C130/MCB C100A Prob
Chemistry C130/MCB C100A Problem Set 6 due March 14, 2014 (15 pts) Please turn in your homework at the beginning of class on Friday. Reading: Chapter 8 1. Energy microstates and multiplicity (3 pts) Consider a system of three independent, distinguishable molecules, each with energy levels En = n, where n is any non-negative integer. Consider two states of the system, A and B , that have the same energy E = 6. State A has one molecule in level 4 and two in level 1, while state B has one molecule each in levels 1, 2, and 3. Figure 1 below shows an example configuration for each state. Because the particles are distinguishable, permuting them between energy levels produces a different microstate. Permuting within an energy level, on the other hand, does not produce a new microstate. The multiplicity W of each state is therefore given by N! , W= i ni ! where the denominator is a product over all energy levels and corrects for permutations of molecules within the same energy level, ni ! = n1 !n2 !n3 ! · · · , i and N is the total number of molecules. In this problem, N = 3. A B 1 2 3 4 3 2 1 0 3 2 1 4 3 2 1 0 Figure 1: Microscopic configurations of each state in problem 5. The numbering of the particles is meant to distinguish them. (a) What is the definition of the energy multiplicity Wenergy ? (b) Calculate the relative probability of observing the system in state A versus state B . 1 (c) What is the change in entropy in going from state A to state B in units of kB ? (d) For this part of the problem, assume that the energy levels only range from n = 0 to n = 4, so that the five energy levels pictured in figure 1 are the only ones allowed. Does the entropy increase or decrease if we increase the energy of the system to the maximum possible value (E = 12)? (e) Now assume that n can be any non-negative integer again, so that the system has infinitely many energy levels. If we increase the energy of the system, does the entropy always decrease, always increase, or sometimes increase and sometimes decrease? 2. Energy distributions (3 pts) In quantum mechanics, a harmonic oscillator with angular frequency ω has evenly spaced energy levels n = ω (n + 1/2) , where n ranges from 0 to ∞. Suppose we have a system consisting of N quantum harmonic oscillators, all with the same angular frequency ω . (These could be the bonds in a protein or DNA molecule.) The total energy is the sum of the energies of the individual oscillators, N E= N ni i=1 = ω (ni + 1/2) , i=1 where ni is the quantum number of oscillator i. The total energy can be rewritten as E = E0 + ω N , where N = i ni is the sum of the quantum numbers. Fixing the energy E of the system also fixes N . This system is analogous to the rock-paper-scissors game we played in class, with N the total number of candies and N the number of students. (a) What is the average quantum number n of a harmonic oscillator in this system? (b) Why we are unlikely to observe the system with each oscillator in the average energy level in part (a), even if n is an integer? (c) Calculate the distribution p(n), where N p(n) gives the number of oscillators in energy level n, for a system with N = 9 and N = 3. (d) Suppose we bring our system into thermal contact with a heat bath at temperature T . The energy of the system is no longer constant since it can exchange energy with the bath. Now what is the distribution p(n) in terms of N ? (e) What happens to the distribution p(n) if the energy spacing of the system doubles but the temperature of the bath remains constant? 3. Energy multiplicity and heat transfer (2 pts) Consider two solutions that are isolated from their surroundings. One solution contains protein A and the other contains protein B . 2 (a) If protein A has a multiplicity of 300 and protein B has a multiplicity of 50, how many total microstates are available to the total system? (b) Suppose we take δE energy from protein solution B and deposit it in protein solution A. Explain why the multiplicity of protein A should increase while the multiplicity of protein B should decrease. (c) After the energy transfer in part (b), protein A has a multiplicity of 600 and protein B has a multiplicity of 10. Is this partitioning of energy more or less probable than the original one? Why? (d) Suppose we bring the two solutions into thermal contact so that they can exchange energy. The total energy remains constant, as required by the first law of thermodynamics, but now the two proteins can actively repartition the energy between them. What happened to the entropy of the total system after we allowed the solutions to interact? Explain. 4. Connections between energy and entropy (5 pts) Spin-1/2 particles, such as protons and electrons, can be either spin up (↑) or spin down (↓). The interaction of the protons in a biomolecule with an external magnetic field forms the basis of proton NMR and can be used to extract structural and dynamical information about biomolecules in solution. Suppose we have a model protein chain described by N connected beads on a lattice of M sites. The beads are identical spin-1/2 particles (like protons) and represent amino acids. (a) Assuming that E↑ = E↓ = 0 for each bead, what is the spin multiplicity Ws of this protein? (b) Now suppose our protein is digested by proteases, so that the N beads are no longer connected and can move freely on the lattice. What is the positional multiplicity Wr of this digested protein? Does the spin multiplicity change? (c) What is the total entropy of the digested protein? (d) Now suppose E↑ = 1 and E↓ = 0, so that it is energetically favorable for the beads to be spin-down. What is the spin multiplicity as a function of the energy of the digested protein? (e) For the system in part (d), express the total entropy S as a function of the parameters of the system. That is, find S = S (M, N, E ), where M is the total volume (or number of sites), N is the number of beads in the digested protein, and E is the total energy. (f) The derivative (∂S/∂M )N,E shows how the entropy of a system changes as its volume changes. What is the sign of this derivative for the digested protein? What does this tell you about the systems propensity to expand its volume? What thermodynamic variable might you associate with this derivative? (Hint: Our digested protein is like a gas. What thermodynamic variable regulates the expansion of a gas?) (g) Similar to part (f), the derivative (∂S/∂E )M,N is a measure of a systems propensity to accept or give up energy. What thermodynamic variable might you associate with this derivative? 3 5. Probabilistic definition of entropy (2 pts) To find the probability distribution that maximizes the entropy of a system at constant energy, we can use the method of Lagrange multipliers. Start with the following equation: L = −kB pi − 1 . pi ln pi + λ i i Here, pi is the probability of microstate i, and the sums are performed over all microstates available to the system. The optimal pi will maximize L . (a) What is the first term on the right-hand side? (b) What is the meaning of the second term on the right-hand side? (c) Find the pi that maximizes L . 4
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