CHEM 120B - Fall 2007 - Chandler - Final

CHEM 120B - Fall 2007 - Chandler - Final - Name Chemistry...

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Unformatted text preview: Name December 14, 2007 Chemistry 120B Final Examination 1 Name Useful formulas • Probability of j th microstate for system at temperature T = ( k B β )- 1 , P j = e- βE j /Q, Q = X j e- βE j = e- βA , where A = U- TS, with U = h E i = X j E j P j and the entropy S as a function of U and mechanical extensive variables X is S ( U, X ) = k B ln W ( U, X ) where W ( U, X ) is the number of micro-states with energy U and X . When X stands for volume V and number of molecules N 1 and N 2 , d U = T d S- p d V + μ 1 d N 1 + μ 2 d N 2 where p is the pressure and μ i is the chemical potential of species i . • Constant volume heat capacity, C v = ( ∂U/∂T ) N,V • Gibbs-Duhem equation for a one-component system, d μ =- s d T + v d p where s is the entropy per molecule and v is volume per molecule. • Rate constant for a rare event, k = νe- βW * where ν- 1 is the typical time to successfully cross the transition region given that the reaction coordinate begins in the transition region and W * is the reversible work to move the reactant to the transition region. • Maxwell-Boltzmann distribution for velocity v in one dimension, φ ( v ) = r βm 2 π e- βmv 2 / 2 • Potentially useful integrals, Z ∞-∞ e- αx 2 d x = r π α , Z ∞-∞ x 2 e- αx 2 d x = 1 2 α r π α 2 Name (75 pts) 1. In this problem you are asked to derive thermodynamic properties of a magnetic material that is placed in a magnetic field. Assume the material contains N identical spins, that interactions between spins are negligible, and that the energy of the i th spin is- m i H where m i is the magnetic moment of that spin and H is the magnetic field. The magnetic moment can be ± m . The degeneracy of each of these single-spin states is one. (a) Determine the equilibrium average magnetization, h m i i , for one of the spins. Express your answer in terms of H, m , and β , where β- 1 = k B T is Boltzmann’s constant times temperature. 3 Name (b) Determine the mean magnetic energy of the total material. Express your answer in terms of H, m , β and N ....
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This note was uploaded on 09/20/2011 for the course CHEMISTRY 120 taught by Professor Head-gordon during the Spring '11 term at ASU.

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CHEM 120B - Fall 2007 - Chandler - Final - Name Chemistry...

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