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Material Type: Exam; Class: Thermodynamics and Statistical Mechanics; Subject: Physics; University: University of Colorado - Boulder; Term: Fall 2008;
Typology: Exams
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Physics 4230, Fall 2008, Final Exam Saturday 12/13/
You have 2 1/2 hours to finish this exam. You may use calculators, and two 8 21 x 11 sheets of paper, with your own notes written on both sides.
Express all answers in terms of given quantities.
Explain your reasoning clearly and concisely for full credit.
Good luck!
(b) (6 pts) Clearly state the fundamental assumption of statistical mechanics. (c) (6 pts) Consider a system in equilibrium with a reservoir at temperature T. The system and reservoir can exchange energy. In this case the probability for the system to be in a state s is given by the Boltzmann distribution, Ps = (1/Z)e−Es/kT^ , where Es is the energy of the state. This tells us that the system has a higher probability to be in states with lower energy. Explain how this is consistent with the fundamental assumption of statistical mechanics. (d) (6 pts) Now, let’s think about an example that illustrates how the second law of thermodynamics follows from the fundamental assumption of statistical mechanics. Consider two 2-state paramagnets each contain- ing the same number of dipoles – call these systems 1 and 2. Initially system 1 has all its dipoles pointing up (maximum positive M ), and system 2 has all its dipoles pointing down (maximum negative M ). Then, the two systems are brought into contact, so that they can exchange magnetization M. Describe what happens to the entropy of system 1 (S 1 ) and of system 2 (S 2 ) when the systems are brought into contact. Explain how this behavior follows from the fundamental assumption of statistical mechanics.
(a) (13 pts) Suppose we have N energy levels and k identical fermions. These fermions don’t have spin, so only one of them can go into each energy level. How many distinct ways are there to fill the N energy levels with the k fermions? Explain the reasoning behind your answer. (b) (12 pts) Now suppose we have N energy levels and k identical bosons. How many ways are there to fill the N energy levels with the k bosons? Explain the reasoning behind your answer.
Thermally insulated container
Gas
Piston
Constant pressure environment
FIG. 1: Illustration for part (b). The gas is maintained at the same pressure as the constant pressure environment.
FIG. 2: Phase diagram for problem 4. There is a phase transformation between phase 1 and phase 2 at the curved line in the P - T plane. Please note that the particular shape of the line is not important for either part of the problem – the questions asked are very general and apply to any phase transformation.
dP dT
where ∆S = S 2 − S 1 and ∆V = V 2 − V 1 are the discontinuous changes in entropy and volume as one crosses the line. The line is the graph of a function P (T ), which simply tells us, for a given temperature, what pressure we need to have in order to be at the phase boundary. dP/dT is the derivative of this function. Derive the Clausius-Clayperon relation.
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