SMES1202

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A Statistical view of Entropy
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• 4 identical (and thus indistinguishable)
  molecules of a gas
• At any instant a given molecule will be in either
  the left or the right half of the box
• Because the two volumes are equal, the molecule
  has the same likelihood, or probability, of being
  in either half

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• In general, a given configuration can be achieved
  in a number of different ways
• The different arrangements of molecules are
  called microstates.
• The number of microstates associated with a
  configuration is called the multiplicity O of
  that configuration.
• Basic assumption of statistical mechanics All
  microstates are equally probable.

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• Configuration I and V least probable
• The system is in these configurations with
  probabilities of 1/16 or 6.25 each.

• Configuration III most probable
• The system is in these configurations with a
  probability of 6/16 or 37.5 of the time.
• The entropy is also the largest.

The state with higher entropy also has the higher
probability of occurrence.
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• State I can occur because N4 (four molecules) is
  extremely small
• Suppose N100.
• For a configuration with n150 and n250
• ?

• For a configuration with n1100 and n20
• ?

• Thus, a 50/50 distribution is more likely than a
  100/0 distribution by an enormous factor of 1 x
  1029

• Note if we could count, at one per second, the
  number of microstates corresponding to 50/50
  distribution, it would take 3 x 1012 years! (
  750x the age of the universe!)

• But N100 is still a small number!

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• For large N, say N 1024

No need to worry about this state ?

• For large N, nearly all the microstates belong to
  the configuration in which the molecules are
  divided equally between the two halves of the box.

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Another example
Two particles colliding

• Most likely result
• Fast one slows down
• Slow one speeds up
• Possible (but unlikely)
• Fast one speeds up
• Slow one slows down

Suppose probability of a fast particle becoming
faster after collision with a slower one
(i.e. one in ten chance)
Chance for 6 fast particles bouncing off 6 slow
particles, and becoming faster is
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Then the chance for 96 fast particles bouncing
off 96 slow particles, and becoming faster is

• Chance ?
• Trillions of particles involved
• The number of zeros would be more than cover the
  whole surface of the earth!

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• Austrian physicist Ludwig Boltzmann first pointed
  out the relation between multiplicity and entropy
  in 1877

k 1.38 x 10-23 J/K is the Boltzmann constant

• In calculating O, using a calculator, overflow
  will occur for N! for N greater than a few
  thousand
• Fortunately, there is a very good approximation,
  known as Stirlings approximation, not for N! but
  for ln N! ? exactly what is needed in
  Boltzmanns equation

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• Entropy can be viewed as a measure of molecular
  disorder or molecular randomness

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Organised and disorganised energy
Gas
Although the molecules in the gas phase have a
considerable amount of kinetic energy, they
cannot rotate the paddle wheel to produce useful
work ? The energy possessed by the molecules is
disorganised
Paddle wheel
Disorganised energy does not create much useful
effect, no matter how large it is.
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Molecules in the shaft rotating in the same
direction ? organised energy ? can lift weight,
generate electricity, etc..

• Work
• an organised form of energy
• Free of disorder or randomness ? free of entropy
• There is no entropy transfer associated with
  energy transfer as work.

• Thus, in the absence of friction, the process of
  lifting the weight by rotation of the shaft will
  not produce any entropy.
• Any process that does not produce a net entropy
  is reversible.

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• The paddle wheel work converted to internal
  energy of the gas (and T rises) ? higher
  molecular disorder or chaos
• Organised energy to disorganised energy
• Therefore, energy is degraded, and the ability to
  do work is reduced
• Molecular disorder is produced
• Associated with all this, there is an increase in
  entropy

• Heat is a form of disorganised energy
• The increase in the entropy in the cold body is
  greater than the decrease in entropy in the hot
  body? overall entropy increases
• The combined system is at a state of greater
  disorder at the final state
• The universe is getting more chaotic everyday

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• At absolute zero (T0 K), molecules are
  motionless ? ultimate order (and minimum energy)
• Therefore, the entropy of a pure crystalline
  substance at absolute zero temperature is zero
  since there is no uncertainty about the state of
  the molecules at that instant.
• ?The third law of thermodynamics
• The entropy determined relative to this point is
  called absolute entropy.

• Substances that are not pure crystalline is not
  zero at T0K.
• This is because more than one molecular
  configuration exist for such substances, which
  introduces some uncertainty about the microscopic
  state of the substance.

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The use of entropy (disorganisation, uncertainty)
is not limited to thermodynamics