Partition Function

Physics 313Professor Lee

CarknerLecture 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 2nd law violated?

Partition Function We can write the partition function as:

Z (V,T) = gi 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/NkT2

we can re-write in terms of U

U = NkT2 (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 (2mkT/h2)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/NA) At constant volume u = cV T, so cV = 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 finddNv/dv = (2N/(2)½)(m/kT)3/2 v2 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:

vrms = (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 disorderI = 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