1) Consider a particle that has only three energy levels, E1 = 0, E2 = , and E3 = 10 . The particle is in equilibrium at temperature T.
a) Write out the partition function for this particle.
b) Find the probability that the particle is in the level with energy E3.
c) If the system contains N such particles (non-interacting), find N3, the number of particles in the level with energy E3.
d) Find the temperature Tc below which there are no particles in the level with energy E3 (i.e., N3 < 1); you may assume N >>1.
e) Find the mean energy per particle.
f) Find the heat capacity Cv of the system.
g) Find the behavior of Cv at high (kT >> ) and low (kT << ) temperatures.
h) Sketch Cv vs. T.
2) Consider a system of N particles, each of which can exist in either State 1 with energy E1 or State 2 with energy E2. There are n1 particles in State 1 and n2 particles in State 2 and both n1 and n2 are much greater than 1. This system is placed in contact with a heat reservoir at a temperature T. The system now looses one unit of energy to the heat reservoir such that n2 decreases by one and n1 increases by one. The heat reservoir is sufficiently large that its temperature remains unchanged.
a) Write down the multiplicity of the two-level system (i) before it looses one unit of energy, (ii) after it looses one unit of energy, and (iii) hence find the change in its entropy. [HINT: Before, there are n1 particles in State 1 and n2 particles in State 2; after, there are (n1 + 1) particles in State 1 and (n2 – 1) particles in State 2.]
b) Find the change in the entropy of the heat reservoir.
c) From your results for parts (a) and (b), find an expression for the temperature dependence of the ratio n2/n1. [HINT: The exchange of energy is a reversible process.]
d) Now, instead, find an expression for the temperature dependence of the ratio n2/n1 from what you know about Boltzmann probabilities for any given state with energy E and compare your result to part (c).
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PHYS 425/ APN 525 Thermal and Statistical Physics 12/10/20 FINAL EXAM
3) The mineral albite can convert into a mixture of the two minerals jadeite + quartz; one mole of albite converts into one mole of jadeite plus one mole of quartz. Depending on the temperature and pressure, either albite or jadeite + quartz is more stable. Using the data listed on pages 404 – 405 (and neglecting the compressibility):
a) Determine which form (albite or jadeite + quartz) is more stable at room temperature and pressure.
b) Determine the transition pressure, above which, the other form becomes more stable at room temperature.
c) Determine whether the transition pressure increases or decreases with increasing temperature.
d) Determine the slope of the P-T phase boundary between albite and jadeite + quartz at room temperature.
e) What is the slope of this phase boundary as T goes to zero and why?
f) Sketch the P-T phase diagram.