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Statistical Mechanics Physics 313 Professor Lee Carkner Lecture 23
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Exercise #22 Shower 140 F mixed with 50 F to get 110 F water m h h h + m c h c = m s h s = (m h +m c )h s Define x = m h /m c and divide by m c x = (h s -h c )/(h h -h s ) x = (78.02-18.06)/(107.96-78.02) = 2
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Particle Statistics The microscopic properties of the molecules are all different even when the macroscopic properties are constant We need to be able to specify the parameters of a distribution and relate it to the macroscopic properties
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Particle Properties Particles are quasi-independent Necessary in order to thermalize Particles are indistinguishable Large numbers Larger numbers means better statistics Particle can only exist in specific states
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Particle Energies x = ½mv 2 x = p 2 x /2m We can use quantum mechanics to state the momentum as: where n x is the quantum number and h is Planck’s constant
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Energy The energy is then: and the quantum number can be written as: n x = (L/h)(8m x ) ½ In three dimensions the energy can be written as: = (h 2 /8mL 2 ) (n 2 x + n 2 y +n 2 z )
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Energy Levels How many different ways can this energy be achieved by a particle having different values for n x, n y and n z ? The number of quantum states for an energy level is its degeneracy (g)
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Distributions Number of quantum states generally much larger than number of particles in that level All quantum states have equal likelihood of being occupied Need a statistical relationship
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Distinguishable In general, number of ways in which particles can be distributed is: gNgN However, the particles are in general indistinguishable True number of ways for the distribution is less g i Ni /N i !
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Macrostates and Microstates A microstate a way which particles can be distributed to achieve a macrostate The probability of a macrostate depends on the number of microstates that could produce it Each macrostate has a probability given by: Called the thermodynamic probability or the number of accessible states
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Stirling’s Approximation ln (x!) = x ln x -x ln = N i ln (g i /N i ) + N Note that the ’s and the g’s are constant and that the N’s are the variables
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Equilibrium Population We want find the population at equilibrium Using the method of Lagrangian multipliers, we get: N i = g i e - i Energy level population is proportional to degeneracy and varies exponentially with energy
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Partition Function If we take the previous expression and sum over all levels we get: We can rewrite part of it as: Z = g i e - i Partition function is also called the sum over states and is related to T and V
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Lagrangian Multipliers We can now write as: It can be shown that: Where k is the Boltzmann constant We can combine these equations to write For equilibrium
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Entropy Equilibrium must be at highest entropy Specifically: We can also say that: S/ U) V
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