Problem 1:This problem concerns a collection of N identical harmonic oscillators (perhaps anEinstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,and so on.a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving1-xthe formula by long division. Prove it by first multiplying both sides of theequation by (1 – x), and then thinking about the right-hand side of the resultingexpression.b) Evaluate the partition function for a single harmonic oscillator. Use the result of(a) to simplify your answer as much as possible.c) Use E = -дzto find an expression for the average energy of a single oscillator.z aBSimplify as much as possible.d) What is the total energy of the system of N oscillators at temperature T?

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Learning Goal: To derive the expression for the work done by an expanding gas, W pAV, and to understand how it follows from the expression WFAz for mechanical work In thermodynamics, positive work is defined to be the work done by a system on the exterior world. In classical mechanics, the converse is true: One always considers the work done on a system by the outside world to be positive. For example, suppose you push a large block with a certain force of Figure 101 Gas Part C What is AV, the increase in volume of the gas? Express the increase in volume in terms of Ar and other given quantities. >View Available Hint(s) - ΑΣΦ AV - PAAx Submit Previous Answers Request Answer * Incorrect; Try Again Part D Complete previous part(s) Part E Complete previous part(s) Part F Complete previous part(s) F 02-23-2
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Parts A-c
(c) Determine the average (internal) energy of the system, ⟨?⟩.
3. The specific Helmholtz Free energy f is related to the specific internal energy u as: ƒ (T, a) = u — Ts where the "natural variables" of f are temperature and specific volume. a. Expand the differential off in terms of partial derivatives with respect to the natural variables of f b. Using the result from a. and applying the 1st Law of Thermodynamics, what are af T and of la? θα c. From the equality of mixed partial derivatives, show that Op ari la əs θα
A molecule has states with the following energies: 0, 1ε, 2ε, 3ε, and 4ε, where ε = 1.0 x 10-20 J. Calculate the probability that a molecule is in the ground state (with zero energy) for a collection of molecules in thermal equilibrium at T = 300 K. Provide your answer as a number in normal form to 3 decimal places (in the form X.XXX). It is a good idea to keep 4 decimal places during your calculation, then round to 3 decimal places for your submitted answer. Hint: note that this molecule has a finite number of states so you must take a finite sum, do not use expressions for infinite sums. Also note that your calculations for this problem will be useful for the next two problems, so keep them.
We have a quantum mechanical system that has two states, one at energy 2ev and another at energy 3ev . We put this system in an equal superposition of these two states. Enter the frequency in Hz (cycles per second) of the resulting oscillation of this system. (Please give your answer to three significant figures. If you were entering a number 1.23 times 10 to the power 45, you could type 1.23e45.)
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