1. Statistical Mechanics Physics 313 Professor Lee Carkner Lecture 23

2. 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 

3. 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

4. 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

5. 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

6. 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 )

7. 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)

8. 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

9. 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 !

11. 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 

12. 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

14. 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

15. 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

17. 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

18. Entropy   Equilibrium must be at highest entropy   Specifically:  We can also say that:  S/  U) V 