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   Since the entropy tends to increase, after rolling a non-seven your next roll should have higher entropy  Why is 2 nd law violated? 

Partition Function  We can write the partition function as: Z (V,T) =  g i e -  i/kT  Z is a function of temperature and volume   We can find other properties in terms of the partition function  (dZ/dT) V = ZU/NkT 2  we can re-write in terms of U U = NkT 2 (dln Z/dT) V 

Entropy  We can also use the partition function in relation to entropy  but  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 

Ideal Gas Partition Function  To find ideal gas partition function:    Result: Z = V (2  mkT/h 2 ) 3/2  We can use this to get back our ideal gas relations   ideal gas law

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)

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/N A )  At constant volume u = c V T, so c V = 5/2 R  In general degrees of freedom increases with increasing T 

Speed Distribution  We know the number of particles with a specific energy: N  = (N/Z) g  e -  /kT   We can then find dN v /dv = (2N/(2  ) ½ )(m/kT) 3/2 v 2 e -(½mv2/kT) 

Maxwellian Distribution  What characterizes the Maxwellian distribution?    The tail is important  

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

Entropy  We can write the entropy as:  Where  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 

Entropy and Information   More information = less disorder I = k ln (  0 /  1 )  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

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 