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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 2nd law violated?

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

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

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
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

e = (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:
Ne = (N/Z) ge e -e/kT

We can then find
dNv/dv = (2N/(2p)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 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

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
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
