Some elements of statistical thermodynamics

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Some elements of statistical thermodynamics

• 9/10/03

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The energy is quantized

• How are the molecules distributed in these
  allowed states at a given T?
• A system has states with energies E1, E2, E3,,
  the probability that the system is in a state
  with energy j is

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Partition function
We will learn how to calculate pressure, energy,
heat capacity, in terms of Q
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An ensemble is a collection of systems, each has
same N, V, and T
Aj is the number of systems with energy Ej and
the total number of systems in the Ensemble is A
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What is Ej ?

• For the simple case of an ideal gas, is a sum of
  the molecular energies
• Ej e1 e2 e3 ..eN

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What is the fraction aj/A
In the limit of large A, aj/A pj
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The denominator is the partition function Q
where b 1/(kBT)
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Average ensemble energy
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Partition function for an ideal monoatomic gas

• eatomic etrans eelec
• q(v,T) qtrans (V,T) qelec(T) (atomic PF)
• Since

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qtrans(v,T)
The sum can be substituted by an integral and can
be solved analytically
Partition function for gonoatomic ideal gas
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Now we can calculate ltetransgt
For N molecules, ltegt Nltetransgt For n moles, N
nNA, and kBNA R
How much is the internal energy of an ideal
gas???
u ltegt (3/2)RT for one mole of a monoatomic
ideal gas
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For a diatomic ideal gas
Avg. trans. energy
Avg. rot. energy
ZPE
Avg.vibrational energy
See example 3-2
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Specific heat at constant volume,Cv

• Is the partial derivative of the average energy
  ltegt (we call it u, internal energy) wrt T, at
  constant N and V.
• Since we know ltegt, we can calculate Cv.
• How much is Cv for a monoatomic ideal gas?
• And for a diatomic?
• For an atomic crystal ?? (see example 3.3)

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Molar heat capacity cv for O2(g)
Experimental (dashed) and theoretical (solid)
curves for Cv/R
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Heat capacity of a crystal
High T limit in agreement with Dulong and Petit
experimental findings
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Molecular interpretation of heat and work
Avg. energy of a system at N, V, T
Since Ej Ej(N, V), at N fixed, dEj (dEj/dV)NdV
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Infinitesimal change in allowed energies
Infinitesimal change in the probability
distribution
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We can deduce an expression for P
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If we know Q we can calculate P
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The ideal gas EOS
How much is P for a diatomic ideal gas?? (see q
from example 3-2)
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System of independent, distinguishable atoms or
molecules
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System of independent, indistinguishable atoms or
molecules
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How can we compute Q in this case?

• Pauli exclusion principle no two electrons in an
  atom can have the same set of 4 quantum numbers.
• It applies to all particles of spin ½,3/2, 5/2,
  etc.
• These particles are called fermions (e-, proton,
  neutron)
• Particles with spin 0,1,2, etc, (called bosons)
  do not have this restriction.
• Example photons, alpha particles.

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Case of fermions

• Terms where two or more indices are the same are
  excluded from the sum. Therefore the direct
  evaluation of the sum is very difficult.

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Case of bosons

• Sum is equally difficult, especially when two or
  more indices are the same.
• If such terms were not present, we could compute
  the sum as

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Boltzmann statistics

• If the of quantum states available to any
  particle is gtgt N ( of particles) gt it will be
  unlikely for any two particles to be in the same
  state.
• Most of the q.m. systems have an infinite of
  energy states, but at a given T many are not
  available, because the energies are gtgt kBT (avg.
  energy of one molecule).
• If the of quantum states with E lt kBT, is gtgtN,
  then all e in the sum will have different indices.

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If Boltzmann statistics is followed
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If Boltzmann statistics is followed
See Table 3.1
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Energy of one molecule