Lecture 14 5.60/20.110/2.772 1 Why Ωworks for large N Derivation of the Boltzmann Distribution Law Partition Function • Why Ωworks for large N We have seen that a system will vary its degrees of freedom in order to maximize Ωand 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 NA? It turns out the most probable state at low N becomes even more likely at very high N. Consider: coin flips nHΩS =klnΩ4 ()1!0!4!4!!!==−=ΩnNnN0 3 4!1!3!4===Ω1.386k 2 6!2!2!4===Ω1.792k 1 4!3!1!4===Ω1.386k 0 1!4!0!4===Ω0 Then do for N = 10, 100, 1000 nN=4Ωmax01234ΩmaxnN=100050100ΩmaxnN=100005001000ΩmaxnN=100510Ωbecomes increasingly narrower as N↑. Compare numerically: 20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
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