Class IX Legendre Transforms

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Class IX Legendre Transforms

• 1. Legendre Transforms
• 1.1 Fundamental Equations
• UU(S,V,N1,N2,,Nt) Enthalpic Representation
• SS(U,V, N1,N2,,Nt) Entropic Representation

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1.2. Extensive and Intrinsic Parameters

• S,U,N,V -gt Extensive Parameters
• Intensive parameters arise as derived concepts.
  (pressure, temperature, µ)
• In practice, its much more convenient to measure
  the intensive parameters. It may be better to
  treat systems by looking at the derived concepts.
  (Because of convenience of measurement)
• It is also much more convenient to control the
  intensive parameters in the laboratory.
• Therefore, a conversion between intensive
  extensive parameters is useful.
• This transformation between extensive and
  intensive variables is a matter of convenience.
  The energetic and enthalpic formulation has all
  the info for a system, however, without the
  transform, thermodynamics would be almost
  unusably awkward.

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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
(A certain slope for every Y)
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
Note This is conventionally described as Y1/9 X2
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory
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1.3. Legendre transform theory

• Legendre transformation can be made with a subset
  of independent variables. (Only transform one
  variable while treating others as constants) See
  page 144 for special notation.

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1.3. Legendre transform theory
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2. Section 5.3 (Thermodynamic potentials)

• Legendre transform of fundamental eqns-gt called a
  thermodynamic potential
• 1. Helmholtz potential (Helmholtz free energy)
• 2. Enthalpy (H)
• 3. Gibbs Free Energy (G)
• 4. Grand canonical potential (UT,U)

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2. Section 5.3 (Thermodynamic potentials)

• 1. Helmholtz potential (Helmholtz free energy)
  (F) -gt Transform of U that uses T instead of S
• Using eqn 5.32 on pg 144., can get diff. form of
  F (dF)
• Recognize SdF/dT

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2. Section 5.3 (Thermodynamic potentials)

• 2. Enthalpy (H) -gt Transform of U that replaces V
  with P

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2. Section 5.3 (Thermodynamic potentials)

• 3. Gibbs Free Energy (G) -gt Transform of U that
  replaces S with T and V with P

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2. Section 5.3 (Thermodynamic potentials)

• 4. Grand canonical potential (UT,U) -gt
  Transform of U that replaces S with T and Nj with
  Uj

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3. Section 5.4 (Generalized Massiu functions)

• Legendre transforms of S rather than U
• 1. S1/T -gt Replace U with 1/T
• 2. SP/T -gt Replace V with P/T
• 3. S1/T,P/T -gt U with 1/T V with P/T
• Only the last one is not trivially related to the
  thermodynamic potentials. (See pg. 152)