l15 - 20.110J / 2.772J / 5.601J Thermodynamics of...

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Lecture 15 5.60/ 20 .110/2.772 1 Q vs. q for distinguishable vs indistinguishable systems Derivation of Thermodynamic Properties from Q: U, S, A, µ , P Examples Partition Functions for independent and distinguishable particles We want to generalize for distinguishable and indistinguishable particles. Let’s make it easier on ourselves by considering only independent subsystems, i.e., the particles do not interact. Then the energy of the whole system, written as, () n interactio ε + = ¦ i i j E can be simplified because the 2 nd term ( ε interaction ) is 0. This allows us to say: i i q Q = system whole 1. Distinguishable particles (and independent) A, B independent of each other, and labeled: A B ε i A i=1,2,…a ε m B m=1,2,…b B m A i j E + = ¦ = = a i A i A kT q 1 exp and ¦ = = b m B m B kT q 1 exp ¦¦ ¦ == = = + = = a i b m B m A i a i b m B m A i t j j kT kT kT kT E Q 11 1 exp exp exp exp Because the sums are independent of each other B A b m B m a i A i q q kT kT Q = = ¦ ¦ = = 1 1 exp exp Generalize for N independent and distinguishable particles: N q Q = 20.110J / 2.772J / 5.601J Thermodynamics of Biomolecular Systems Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
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Lecture 15 5.60/ 20 .110/2.772 2 2. Indistinguishable particles (and independent) Now: no A and B labels! B m A i j E ε + = where i=1,2,…t 1 , m = 1,2,…t 2 . () ¦¦ ¦ == = + = = 12 11 1 exp exp t i t m B m A i t j kT kT E Q Now: cannot factor out of sum due to indistinguishability: can’t separate sums WHY? particle 1 ε 1 =10 particle 2 ε 1 =167 Can’t be distinguished from the situation where particle 1 ε 1 =167 particle 2 ε 1 =10 So overcounting is present. Divide by 2! ! 2 2 q Q = In general, for N particles, divide by N! Deriving Thermodynamic Properties using Q All thermodynamic quantities can be calculated from the partition function The Boltzmann factor and partition function are the two most important quantities for making statistical mechanical calculations. If we have a model for a material for which we can calculate the partition function, we know everything there is to know about the thermodynamics of that model . Now we will relate our favorite thermodynamic properties to q, the partition function. This is our link between the microscopic and macroscopic descriptions. Using the convenient dummy variable β = 1/k b T to simplify things. ! N q Q N = 20.110J / 2.772J / 5.601J Thermodynamics of Biomolecular Systems Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
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Lecture 15 5.60/ 20 .110/2.772 3 Deriving Energy, U ¦ ¦ = = = = t j E j t j j j j e E Q E p E 1 1 1 β Use trick ¦ ¦ = = ¸ ¸ ¹ · ¨ ¨ © § j j E j E e E e
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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.

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l15 - 20.110J / 2.772J / 5.601J Thermodynamics of...

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