Lecture 5. Whole new way to teach thermodynamics

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Classical thermodynamics getting a lot out of
simple sanity (and without messing with molecules)
Sanity Heat, temperature EQW C, KP
T (QT/ Qref) Tref ?(max) 1-Tcold/Thot dS
(dQ/T)rev function of state Entropy
coordinate or space for heat Equilibrium
principles for closed systems Smax (isolated),
Gmin (p,Vconst), Amin (V,Tconst) This is not
all we did not consider algebraic relationships
of thermodynamic quantities
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and a lot more from sanity DONT STUDY THESE
For the exam!!!
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NOT FOR HOMEWORK or EXAMS
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What if we add molecules with their microscopic
states into our consideration?
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• Statistical thermodynamics
• (a whole new way of thermodynamics)
• 2) Molecular statistics in biochemistry

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BOLTZMANN
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S k log W
Boltzmanns formula for Entropy W (omega)
multiplicity of system k - universal constant
1.3810-23 J/K (Boltzmann constant)
well get there
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Boltzmann distribution
Ratio of populations of molecular states i and j
with energy levels Ei and Ej in thermal
equilibrium. First we will derive (almost) this
distribution
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Outline of derivation

• Find number of ways to assemble system of N
  identical particles to a particular distribution
  into available energy levels Ei
• Maximize that number of ways under constraints
  that Etot E , Ntot N

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Odds exercise
1 2 3
Lets calculate our odds of getting this hand
from a poker deck (52 cards)
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of ways to sort cards by suit
(getting closer to Boltzmann)
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Back to Boltzmann
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Find number of ways to assemble system of N
identical particles to a particular distribution
into available states with energy levels Ei

• same as sorting cards by suit
• instead of suits we have energy levels
• instead of 52 cards we have N particles

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• Find a way to Wmax
• under sanity constraints
• S Ni N
• S NiEi E

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Here is what we want
W max
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Stirlings formula helps to differentialte W
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Method of undetermined multipliers(Lagrange
multipliers)
true if constraints are const in our case
they are things get real simple we only need
to find a and b
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Real close to Boltzmann

• a - make sure that S Ni N (first sanity
  constraint) this is easy
• b - is different, needs to be tied to reality

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We can get everything out of this!(under
favorable conditions)
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Molecular Partition Function
Master tool of statistical thermodynamics, but
we still dont know b Molecular Ek are energy
levels of isolated molecules
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Quick run to reality(details spared)

• take 1 mole of ideal gas (Ei mvi2/2) ,
  calculate ltEgt from q
• energy is related to work pdV
• obtain equation

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( well still get back to S k ln W )
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Molecular partition function, molecular because
Ej are energies of individual molecules (not the
whole system yet)
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2) Molecular statistics in biochemistry(Boltzmann
and DNA)
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Forms of DNA
A B Z
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Forms of DNA
A B Z
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DNA supercoiling topoisomerism

• Tw (bp)/(ltbp/turngt)
  Twist
• Wr (number of supercoils) Writhe
• Lk Tw Wr
  Linking number
• Lk const for each topoisomer
• s Wr/Tw (-0.006 on average)
  Superhelical density
• dG(topoisomer) dG(Tw) dG(Wr)
• dG(Wr) 1100RT/(bp)Wr2

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Topoisomerases relax superhelical strain (and
change Lk)
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DNA can be sorted by Wr(by gel electrophoresis)
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Relaxing ccDNA
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Different species relax to same products
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Limit product is independent of enzyme used
N-C enzyme
PMII ligase
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Gel profiles fit by Gaussian curves
boltzmann works
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Denaturation of repeat proteins