RealFluidsLect30&31ME501F2011

RealFluidsLect30&31ME501F2011 - Purdue University...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lectures 30 and 31: Behavior of Real Gases and Liquids: The Virial Equation of State and the Van der Waals Fluid Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana Lucht@purdue.edu , 765-494-5623 (Phone) November 30 and December 2, 2011
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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics • Virial equations of state. • Configuration integrals. • Evaluation of virial coefficients using the Lennard-Jones potential. • Discussion of the second virial coefficient. • Configuration integral for the Van der Waals fluid. • Evaluation of thermodynamic properties for the VDW fluid.
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Patterned Border Template 3 Purdue University School of Mechanical Engineering The Virial Equation of State • The virial equation of state for a real gas arises out of the pressure relation developed for the grand canonical partition function: ln B Pk T  • The grand canonical partition function (for a single species) is given by: where  ,, 00 exp / exp / N Ni B B N i NE k T Ek T          exp / B kT  • The canonical partition function for an assembly with N particles is given by:     , e x p / B i QT N E 
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Patterned Border Template 4 Purdue University School of Mechanical Engineering The Virial Equation of State • For independent, indistinguishable particles in the dilute limit: But: • Substituting for the grand canonical partition function we obtain:  ,, ! N Z QN T N      0, 0, , exp / exp 0 1 iB i QT E k T   1 ln 1 , , N N B P kT 
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Patterned Border Template 5 Purdue University School of Mechanical Engineering The Configuration Integrals • For systems with more than one particle ( N >1) the energy of the system can be separated into the energy of the individual particles and the energy associated with intermolecular forces between the particles: • The term is the configurational energy. The partition functions Q ( N,T, ) are then written    * 12 ** () , , , ,....... , ii N i i independent part ET N E r r r EE     , ,....... , N rr r      1 2 ,, e x p / exp / exp , ,....... , / Ni B i iin dpa r t B N B all quantum states QN T E kT Ek T r r r k T  
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Patterned Border Template 6 Purdue University School of Mechanical Engineering The Configuration Integrals • We have already dealt with the first term in the final summation involving individual, independent particles. The first term gives us: • The configurational energy part of the summation is replaced by an integration over all possible positions for the individual particles:  * exp / !
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RealFluidsLect30&31ME501F2011 - Purdue University...

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