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Partition Function

Partition Function. Physics 313 Professor Lee Carkner Lecture 24. Exercise #23 Statistics. Number of microstates from rolling 2 dice Which macrostate has the most microstates? 7 (1,6 6,1 5,2 2,5 3,4 4,3 total = 6) Entropy and dice

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Partition Function

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  1. Partition Function Physics 313 Professor Lee Carkner Lecture 24

  2. Exercise #23 Statistics • Number of microstates from rolling 2 dice • Which macrostate has the most microstates? • 7 (1,6 6,1 5,2 2,5 3,4 4,3 total = 6) • Entropy and dice • Since the entropy tends to increase, after rolling a non-seven your next roll should have higher entropy • Why is 2nd law violated?

  3. Partition Function • We can write the partition function as: Z (V,T) = Sgi e -ei/kT • Z is a function of temperature and volume • We can find other properties in terms of the partition function (dZ/dT)V = ZU/NkT2 • we can re-write in terms of U U = NkT2 (dln Z/dT)V

  4. Entropy • We can also use the partition function in relation to entropy • but W is a function of N and Z, S = Nk ln (Z/N) + U/T + Nk • We can also find the pressure: P = NkT(dlnZ/dV)T

  5. Ideal Gas Partition Function • To find ideal gas partition function: • Result: Z = V (2pmkT/h2)3/2 • We can use this to get back our ideal gas relations • ideal gas law

  6. Equipartition of Energy • The kinetic energy of a molecule is: • Other forms of energy can also be written in similar form • The total energy is the sum of all of these terms • = (f/2)kT • This represents equipartition of energy since each degree of freedom has the same energy associated with it (1/2 k T)

  7. Degrees of Freedom • For diatomic gases there are 3 translational and 2 rotational so f = 5 • Energy per mole u = 5/2 RT (k = R/NA) • At constant volume u = cV T, so cV = 5/2 R • In general degrees of freedom increases with increasing T

  8. Speed Distribution • We know the number of particles with a specific energy: Ne = (N/Z) ge e -e/kT • We can then find dNv/dv = (2N/(2p)½)(m/kT)3/2 v2 e-(½mv2/kT)

  9. Maxwellian Distribution • What characterizes the Maxwellian distribution? • The tail is important

  10. Maxwell’s Tail • Most particles in a Maxwellian distribution have a velocity near the root-mean squared velocity: vrms = (3kT/m)1/2 • We can approximate the high velocities in the tail with:

  11. Entropy • We can write the entropy as: • Where W is the number of accessible states to which particles can be randomly distributed • We have no idea where an individual particle may end up, only what the bulk distribution might be

  12. Entropy and Information • More information = less disorder I = k ln (W0/W1) • Information is equal to the decrease in entropy for a system • Information must also cause a greater increase in the entropy of the universe • The process of obtaining information increases the entropy of the universe

  13. Maxwell’s Demon • If hot and cold are due to the relative numbers of fast and slow moving particles, what if you could sort them? • Could transfer heat from cold to hot • But demon needs to get information about the molecules which raises entropy

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