Download First Law of Thermodynamics - Modern and Statistical Physics - Exam and more Exams Physics in PDF only on Docsity! Statistical and Modern. Spring 2007. Pick 4 out of 6. 1) a) Starting with the first law of thermodynamics and the definition of pc and vc , show that p v T P U V c c p V T ∂ ∂ − = + ∂ ∂ . Here pc and vc are the specific heat capacities per mole at constant pressure and volume respectively, and U and V are the energy and volume of one mole. b) Use the above result plus the expression T V U p p T V T ∂ ∂ + = ∂ ∂ to find p vc c− for a van der Waals gas with equation of state ( )2 a p V b RT V + − = . Here a and b are constants. c) Use this result to show that as V → ∞ at constant p , you obtain the ideal gas result for p vc c− . 2) The rotational motion of a diatomic molecule is specified by two angular variablesθ and φ and the corresponding canonical conjugate momenta, ,p pθ φ . Assuming the form of the kinetic energy of the rotational motion to be ( ) 2 2 2 1 1 2 2 sinrot p p I Iθ φ ε θ = + a) Derive the classical formula for the rotational partition function , ( ),r T 2 2 ( ) IkT r T = h b) Calculate the Helmholtz free energy rotF . c) Calculate the corresponding entropy and specific heat. The following may be helpful e dx a dx ax ax a ax− −∞ ∞ = = − ∫ ∫ 2 2 π sin ( ) cot( ) / 3) Assume that the neutron density in a neutron star is 30.1/fm (that is 0.1 neutron per cubic Fermi). Assuming T=0 and ignoring any gravitational forces calculate the ratio of neutrons to protons to electrons. Hint: determine their Fermi energy. The electron, neutron and protons masses are .511 2MeV/c , 939.6 2MeV/c and 938.3 2MeV/c . The constant 1240MeV fmhc = . You should be able to work out "by hand" an approximate value. 4) A π µ− atom consists of a pion and a muon bound in a Hydrogen-like atom. a) What are the energy levels for such an atom compared to those for Hydrogen? b) π µ− atoms are produced in KL decays ( LK π µ ν→ − + ). If the KL has 0.8β = what are the minimum and maximum energies of the π µ− atom expressed in terms of the K , π and µ masses with 0mν = ? c) Approximately what fraction of KL decays will produce a π µ− atom (hint: use the Heisenberg uncertainty principle)? 5) a) You are familiar with the quarter-wave thin film coating that acts as a “reflection–reducer”. For the moment, let us look at a simpler thin film—the air gap between two pieces of glass such as you would find in a Newton’s rings experiment. Why do we get constructive interference in the reflected when the thickness is one-fourth of the wavelength of light or some odd multiple of a quarter wavelength? Why isn’t it constructive at one-half wavelength of the light? For assistance, I present two of the Fresnel equations (in two forms) for reflected light. cos cos tan( ) cos cos tan( ) cos cos sin( ) cos cos sin( ) t i i t i t t i i t i t i i t t i t i i t t i t n n r n n n n r n n θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ⊥ − − = = + + − − = = − + + P Where the parallel and perpendicular symbols refer to the plane of incidence, and i,t refer to incident and transmitted media, θ ’s are angles of incidence and transmission, and n’s are indices of refraction. b) In light of the previous, to get destructive reflection in a thin film-i.e.-a quarter-wave film, such as the one illustrated below, what condition must prevail among the indices of refraction for the three media (n0 may be taken as = 1.0 for air.) c) The destructive interference described in part b) will generally not be complete. Find the value of 1n as a function of 2n which gives completely destructive interference at normal incidence. n1 n2 n0
