E115_Problem_Set_8 - et al., Nature, 410 , 653-654) uses...

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e 115 Engineering Thermodynamics Fall 2009 University of California Problem set 8 Due: Wednesday, September 23 rd , in lecture 1) ( 15 points ) The Helmholtz free energy functional of a one-component solid at very low temperatures may be written, to very good approximation, as A ( T , V ) = U ( V ) π 2 5 nRT T Θ D 3 = U ( V ) 1 3 U D ( T ) where Θ D is a parameter known as the Debye temperature. It is possible to derive several thermodynamic parameters from this expression of the free energy. a) Calculate the entropy and C V as a function of temperature. ( 5 points ) b) Show that the internal energy may be written as U = U + U D ( 5 points ) c) Calculate an expression for the system pressure in terms of the Grüneisen parameter, γ = V Θ D ∂Θ D V ( 5 points ) 2. (10 points) This problem is intended to demonstrate how to develop a realistic isothermal equation of state for a solid. This is necessary to accurately analyze data for materials taken at very high pressures. For instance, the included paper (L. S. Dubrovinsky
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Unformatted text preview: et al., Nature, 410 , 653-654) uses such an equation of state to analyze the mechanical response of “the hardest known oxide.” In this problem, you will derive a simple relationship between the internal energy and the volume of a system. Define the bulk modulus as: B = − V ∂ P ∂ V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ T Now, consider the experimental observation that the derivative of the bulk modulus is a function of pressure. This allows us to approximate the bulk modulus as B ( P ) = B + B 1 P . For the purposes of this problem, consider the case where the free energy does not depend on the entropy, ∂ U ∂ S ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ V = Using these two thermodynamic observations, develop an expression for the free energy, U ( V ) , in terms of the parameters B and B 1 , and the volume at zero pressure, V ....
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This note was uploaded on 02/12/2011 for the course E 115 taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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