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Unformatted text preview: Lecture 14 5.60/ 20 .110/2.772 1 Why works for large N Derivation of the Boltzmann Distribution Law Partition Function Why works for large N We have seen that a system will vary its degrees of freedom in order to maximize and thus S. A system has a higher probability of being in a state due to it being more probable. This allows us to simply count states and see which one is more likely. The lattice model of mixing gases had only N=8 particles. Is this approach still justified when we look at a larger number of particles, like N A ? It turns out the most probable state at low N becomes even more likely at very high N. Consider: coin flips n H S =kln 4 ( ) 1 ! ! 4 ! 4 ! ! ! = = = n N n N 0 3 4 ! 1 ! 3 ! 4 = == 1.386k 2 6 ! 2 ! 2 ! 4 = == 1.792k 1 4 ! 3 ! 1 ! 4 = == 1.386k 0 1 ! 4 ! ! 4 = == 0 Then do for N = 10, 100, 1000 n N=4 max 1 2 3 4 n N=4 max 1 2 3 4 max n N=100 50 100 max n N=100 50 100 max n N=1000 500 1000 max n N=1000 500 1000 max n N=10 5 1 max n N=10 5 1 becomes increasingly narrower as N . Compare numerically: 20.110J / 2.772J / 5.601J Thermodynamics of Biomolecular Systems Instructors: Linda G. Griffith, Kimberly HamadSchifferli, Moungi G. Bawendi, Robert W. Field Lecture 14 5.60/ 20 .110/2.772 2 ( ) ( ) 45 ! 8 ! 2 ! 10 10 , 2 252 ! 5 ! 5 ! 10 10 , 5 = = = = 5.6X more likely ( ) ( ) 20 29 10 5 ! 80 ! 20 ! 100 100 , 20 10 1 ! 50 ! 50 ! 100 100 , 50 = = = = 10 9 X more likely! Even though the process is totally random: If the number of trials N is large enough, the composition of the outcomes becomes predictable with great precision. This allows us to better predict the most probable state! maximizing = maximizing S Derivation of the Boltzmann Distribution Law Microscopic definition of entropy: j t i j p p k S ln 1 = = What probability distribution (set of p j s) maximizes S? = j p S for all j constraint that probabilities sum to 1: 1 1 1 = = = = t j j j j dp p Utilize Lagrange multipliers to solve this problem. We add the constraint to the equation we are trying to maximize with a multiplier, . Then when we maximize the resulting equation the value of is determined. i.e., solving the set of equations: 1 = = t j j j dp p S for all j 20.110J / 2.772J / 5.601J Thermodynamics of Biomolecular Systems Instructors: Linda G. Griffith, Kimberly HamadSchifferli, Moungi G. Bawendi, Robert W. Field Lecture 14 5.60/ 20 .110/2.772 3 Plug in definition of S (pj): ln 1 = = t j j t j j j j p p p k p Take the derivative: ( ) = = = + 1 1 ln 1 ln...
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This note was uploaded on 11/11/2011 for the course BIO 20.010j taught by Professor Lindagriffith during the Spring '06 term at MIT.
 Spring '06
 LindaGriffith

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