THE SECOND LAW OF THERMODYNAMICS Entropy Entropy and the

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THE SECOND LAW OF THERMODYNAMICSEntropy
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Entropy and the direction of time

• Microscopically the eqs. of physics are time
  reversible ie you can turn the arrow of time
  around.!!!
• Macroscopically there is an arrow to time whats
  going on ie processes are irreversible
• Key is understanding entropy

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Irreversibility

• The one way character of irreversible processes
  is so pervasive we take it for granted ie if you
  wrap your hands around a hot cup of Java you
  would be astonished if your hands became
  cooler
• The system of (coffee hands) obeys energy
  conservation but you dont expect that the flow
  of heat energy in the system to be from cold to
  warmer ie from hands to cup
• If an irreversible process occurs in a closed
  system, the entropy of the system always
  increases, it never decreases

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New Quantity Entropy

• Changes in energy do not violate the highly
  unlikely process of a popped balloon air
  turning into a fully expanded balloon
• There is a quantity that called entropy such that
  if an irreversible process occurs in a closed
  system the
• entropy increases S

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Arrow of time and Free expansion

• One doesnt expect the gas undergoing free
  expansion to spontaneously go back to the left
  hand volume ie the other side..
• Entropy is not a conserved quantity however it is
  a state variable as are pressure, p, temperature,
  T, internal energy, E etc.

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Since Entropy is a state variable can calculate
it by knowing only the initial and final states

• 1) In terms of systems temp and energy gain or
  loss as heat
• 2) Counting the ways in which the atoms of
  molecules that make up the system can be arranged

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Well what is it
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Example, free expansion of nitrogen gas from
initial to final volumewhat is entropy?
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• Isothermal process done in a reversible manner
  has the same initial and final states as an
  irreversible process undergoing free expansion
  since S is a state variable all we need are
  initial and final states
• To find the entropy of an irreversible process in
  a closed system, replace that process that
  connects the initial and final states, calculate
  the entropy for the reversible one and you have
  the entropy for the irreversible one

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Consider this irreversible process

• What is the reversible process that takes the
  system from initial to final state?
• If we can calculate the entropy change of the
  reversible process that takes the system from
  initial to final state then we have the entropy
  of the irreversible process..

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Prove that S is a state variable

• That is it can be defined as a function of other
  state variable and each equilibrium state has a
  unique value.
• Prove for the case of an ideal gas going through
  a series of reversible processes
• Did not specify a particular reversible process
  when we integrated, therefore the integration
  holds for all reversible processes that take a
  gas for some initial state i to some final state f

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Change Entropy for an Ideal Gas

• Consider an ideal gas undergoing a thermodynamic
  process
• Apply the First Law of Thermodynamics

It is useful to consider infinitesimal changes
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Change Entropy for an Ideal Gas
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Change Entropy for an Ideal Gas
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Change Entropy for an Ideal Gas

• Change in Entropy is independent of process (none
  was specified)
• Assumed a reversible process. Any irreversible
  process would have larger change in entropy
  hence, the inequality
• Equation is an equation of state since it
  includes only state variables (V, T) and requires
  knowledge of only the end points of the process.

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Puzzle the 2nd Law of Thermo

• If entropy always increases then
• what happens if you reversibly transfer
• heat back to reservoir? Doesnt
• entropy decrease ?
• Yes but ..

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• If a process occurs in a closed system the
  entropy of the system increases for irreversible
  process and remains constant for reversible
  processes. IT NEVER DECREASES.

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Second LAW

• If a process occurs in a closed system the
  entropy of the system increases for irreversible
  process and remains constant for reversible
  processes.
• IT NEVER DECREASES.

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The Carnot Engine Cycle

• Ideals engine, all processes are reversible and
  now wasteful energy transfers occur ie due to
  friction and turbulence
• Two adiabats and two isotherms
• Highest possible efficiency (Carnot Theorem)

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Can consider a T vs. S plot

• Entropy change
• Energy change

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T vs S Diagrams

• Isothermal processes are horizontal lines
• Adiabatic processes (isentropic processes) are
  vertical lines
• Area under the curve is heat exchanged
• Area enclosed in a cycle is work done (net heat
  exchanged equals work done because the change in
  internal energy of the substance is zero)

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Calculate the Change of Entropy for Melting Ice

• 10 g of Ice at 0 oC melts to form a puddle of
  water. What is the increase of entropy of the
  water?

so
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Microscopic explanation of Entropy

• Begin with the number of ways the energy of the
  molecules can be distributed the multiplicity
  of the number of states
• Apply Boltzmanns entropy equation
• Include Stirlings approximation

where
Use only for very large numbers
See section 21-7 for more examples
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Example of Statistical Approach to Entropy

• Consider 10 identical molecules to be distributed
  among 3 cells (1/3 of the container)
• Possible distributions are
• 10,0,0 (3 times) i.e, 10,0,0 / 0,10,0 / 0,0,10
• 9,1,0 (6 times) i.e, 9,1,0 / 9,0,1 / 1,9,0 /
  0,9,1 / 0,1,9 / 1,0,9
• 8,2,0 (6 times)
• 8,1,1 (3 times)
• 7,3,0 (6 times)
• 7,2,1 (6 times)
• 6,4,0 (6 times)
• 6,3,1 (6 times)
• 6,2,2 (3 times)
• 5,5,0 (3 times)
• 5,4,1 (6 times)
• 5,3,2 (6 times)
• 4,4,2 (3 times)
• 4,3,3 (3 times)

Calculate the entropy associated with the
distribution 7,2,1
where
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Statement of the Second Law and Heat Engines

• It is impossible to construct a heat engine
  operating in a cycle that completely converts
  heat to an equal amount of work
• (Kelvin - Planck statement)
• Always some heat that goes out unused..

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Heat Engine Efficiency

• operates through a thermodynamic cycle
• is reversible
• efficiency of a heat engine

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Summary of the Laws of Thermodynamics

• First Law Conservation of Energy
• You cannot build a perpetual motion machine of
  the first kind. (You cannot get more energy out
  than you put in).
• In other words, YOU CANT WIN!
• Second Law Law of increasing entropy or
  unidirectional flow of thermal energy
• You cannot build a perpetual motion machine of
  the second kind. ( You cannot build a machine
  that is 100 efficient).
• In other words, YOU LOSE!