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Practice Final 1) An electron in a hydrogen atom sees a potential that goes as -Eh/r where r is the distance from the proton to the electron and Eh is a constant. You are to determine the probability that hydrogen is in its first excited state as a function of temperature. Provide a graph. Begin by carefully explaining how you will approach the problem and then go ahead and solve it. Things you will need to know: The constant Eh is called a Hartree and is equal to 27.2 eV. The quantized energy levels for the electron in this potential are given by En = -Eh/(2n2) where n is a quantum number equal to 1, 2, 3, … and each level is n2 degenerate. Explain why we do not need to consider the electronic contributions to the heat capacity for atoms and molecules at common temperatures. 2) An Einstein solid is a model of a crystal developed, obviously, by Albert Einstein. It assumes a crystal of N atoms each with 3 degrees of vibrational freedom (vibrations in the x, y, and z directions). Derive an expression for the solid’s internal energy, entropy and heat capacity as a function of temperature. What is the internal energy at 0 K? Explain why this is the case. 3) Consider a system of N non-interacting spins in a magnetic field of strength B (often measured in units of Tesla). The spins can be either parallel to the magnetic field or antiparallel (i.e. up or down). In the former case the energy of the spin is given as +µ Β and in the later case as -µ Β, where µ is the Bohr magneton (check the inside cover of your book for it’s value in Joules/Tesla). Derive an expression giving the internal magnetic energy, heat capacity and entropy for this system as a function of field strength and temperature. Plot these expressions.