This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 three-dimensional space, as sketched below. . . . r 1 r 2 r 3 r 4 ... r N-1 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 z-axes 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...
View Full Document