Download Thermodynamics and Statistical Mechanics Problems: Entropy, Energy, and Partition Function and more Exams Physics in PDF only on Docsity! Test II Thermo. & Stat. Mech., PHYS 4420 April 24, 2009 (20 points per problem) 1. Consider a model system of electrons with energy levels K,5,4,3,2,,0 εεεεε . The system is completely isolated from the rest of the universe; there are electrons in the system, and the total energy of the system is 3=N ε5=U . The only degeneracy of the energy levels is associated with the spins of the particles. (Electrons are spin=1/2 particles and they are indistinguishable.) (a) What is the entropy of the system subject to the above constraints? (b) What is the expectation value of the number of electrons in each of the single- particle energy levels? 2. Consider the two-dimensional photon gas (black body radiation) at temperature T , confined to an area A . (a) Obtain the density of modes )(νg . (b) Find the internal energy and the heat capacity of the system, and . ),( TAU ),( TACA (c) Find the free energy of the system, . ),( TAF (d) Obtain the entropy of the system, . ),( TAS (e) Obtain the pressure of the system, , i.e., the equation of state. ),( TAP (f) What is the average number of photons in the system at temperature T , ? ),( TAN ph To keep your expressions relatively compact, the following relationship will come useful: )()( 10 1 νζν ν Γ= −∫ ∞ − dx e x x , where ∑ ∞ = = 1 1)( l lν νζ . Also, 6 )2( 2πζ = , and 202.1)3( ≈ζ . 3. Consider independent localized (hence, distinguishable) particles. The energy levels of a single particle are N εε jj = , ∞= ,,2,1,0 Kj .The degeneracy of the j th level is . To avoid some subtlety associated with this system, focus only on the jjg 2= )2ln(k T ε< regime. (a) Obtain the partition function Z . (You must obtain an exact form for Z . No approximations should be involved for Z.) (b) Obtain the internal energy of the system. ),( TNU (c) Obtain the heat capacity of the system. From your result, obtain the low- temperature behavior ( ),( TNC 1/ >>kTε ) of . ),( TNC (d) Obtain the entropy . Obtain the low-temperature behavior of . ),( TNS ),( TNS (e) What is the expected number of particles on energy level , j ),( TNN j ? dacbwr S= fy, he*®
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