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Unformatted text preview: YOUR NAME HERE March 4, 2009 Chem 120B Midterm #1 Definitions and Useful Formulas: Inverse temperature: = 1 k B T Boltzmann distribution: P ( ) = e E Q Partition function: Q = summationdisplay e E Equilibrium averages: ( E ) = parenleftbigg ln Q parenrightbigg N,V , ( E 2 ) = parenleftbigg 2 ln Q 2 parenrightbigg N,V = k B T 2 C V Equipartition principle: ( E ) = 1 2 k B T per clasical degree of freedom with quadratic energy. Entropy: S = k B ln W, S = k B summationdisplay P ( ) ln P ( ) First and second laws of thermodynamics: dE = dw + dq, dS dq T Gaussian integration: integraldisplay  dx e x 2 / (2 2 ) = 2 2 , 1 2 2 integraldisplay  dx x 2 e x 2 / (2 2 ) = 2 Questions on this exam concern a long chain molecule, composed of N + 1 particles linked by N bonds, which can exchange energy with a heat bath at temperature T . Quantum mechanical effects can be ignored throughout. 1 1. Consider first a model in which consecutive particles in the chain are connected by springs and can move continuously in threedimensional space, as sketched below. . . . r 1 r 2 r 3 r 4 ... r N1 r N (The light gray curve represents the segment of this molecule whose particles and bonds are not explicitly shown.) The bond vector r j pointing from particle # j to particle # j + 1 has components x j , y j , and z j along x, y, and zaxes of the laboratory frame. The length of the j th bond is therefore  r j  = radicalBig x 2 j + y 2 j + z 2 j . This potential energy of this model chain molecule has independent contributions from each bond: U = N summationdisplay j =1 1 2 a  r j  2 , where the bond stiffness a is a positive constant. The total energy is therefore E = K + U, where K is the kinetic energy for N + 1 particles of mass...
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 Spring '08
 Geissler
 Physical chemistry, Equilibrium, pH

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