exam1.Fall09Template

# exam1.Fall09Template - s 1 ,s 2 ,...,s N . Here, each...

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Name October 2, 2009 Chemistry 120B Hour Examination Useful formulas First law of thermodynamics: dE = dq + dw , where dE stands for diﬀerential change in internal energy, dq stands for diﬀerential heat ﬂow into system, and dw stands for work done on the system. Entropy, S = k B ln Ω, and reversible heat ﬂow: TdS = dq rev , where k B is Boltzmann’s constant, Ω is the number of microstates at energy E and T = ∂E/∂ Ω is temperature. Probability of j th microstate for a system in equilibrium in the canoncial ensemble at temperature T = ( k B β ) - 1 : P j = e - βE j /Q where E j is the energy of the j th microstate, and Q is the canonical partition function: Q = X j e - βE j Gaussian integral: Z -∞ dxe - αx 2 = r π α Maxwell-Boltzmann velocity distribution: Φ( v ) exp ( - βmv 2 / 2) = exp [ - βm ( v 2 x + v 2 y + v 2 z ) / 2] where m is the particle mass, and v is the magnitude of the vector with Cartesian components v x , v y , and v z . 1

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Name 1. Consider a magnet at temperature T composed of N non- interacting micromagnets with the microstates speciﬁed by the list of all the spin states for the micromagnets:

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Unformatted text preview: s 1 ,s 2 ,...,s N . Here, each micromagnet spin state has one of two values: s i = ± 1 , i = 1 , 2 ,...,N . The net energy of the system in one of these microstates is-h ∑ N i =1 s i , where h is a constant magnetic ﬁeld. 2 Name 2. For the magnet considered in Problem 1, the reversible work diﬀerential for the ﬁrst law of thermodynamics is dw rev = hdM with M denoting the average net spin, i.e., M = h s 1 + s 2 + ... + s N i . At small enough values of h , the equation of state for the magnet is βh = m , where m = M/N . For the questions that follow, assume the system is in this regime of small h . 3 3. Consider N identical classical particles of mass m in two dimensions, i.e., d = 2. These particles are at equilibrium, in a ﬁxed volume and at a temperature T . 4...
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## This note was uploaded on 01/03/2011 for the course CHEM 120B taught by Professor Geissler during the Spring '08 term at Berkeley.

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exam1.Fall09Template - s 1 ,s 2 ,...,s N . Here, each...

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