Phase Transitions

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Phase Transitions

• Physics 313
• Professor Lee Carkner
• Lecture 22

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Exercise 21 Joule-Thomson

• Joule-Thomson coefficient for ideal gas
• m 1/cPT(?v/?T)P-v
• (?v/?T)P R/P
• m 1/cP(TR/P)-v 1/cPv-v 0
• Can J-T cool an ideal gas
• T does not change
• How do you make liquid He?
• Use LN to cool H below max inversion temp
• Use liquid H to cool He below max inversion temp

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First Order Phase Transitions

• Consider a phase transition where T and P remain
  constant
• If the molar entropy and volume change, then the
  process is a first order transition

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Phase Change

• Consider a substance in the middle of a phase
  change from initial (i) to final (f) phases
• Can write equations for properties as the change
  progresses as
• Where x is fraction that has changed

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Clausius - Clapeyron Equation

• Consider the first T ds equation, integrated
  through a phase change
• T (sf - si) T (dP/dT) (vf - vi)
• This can be written
• But H VdP T ds, so the isobaric change in
  molar entropy is T ds, yielding
• dP/dT (hf - hi)/T (vf -vi)

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Phase Changes and the CC Eqn.

• The CC equation gives the slope of curves on the
  PT diagram
• Amount of energy that needs to be added to change
  phase

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Changes in T and P

• For small changes in T and P, the CC equation can
  be written
• or
• DT T (vf -vi)/ (hf - hi) DP

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Control Volumes

• Often we consider the fluid only when it is
  within a container called a control volume
• What are the key relationships for control
  volumes?

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Mass Conservation

• Rate of mass flow in equals rate of mass flow out
  (note italics means rate (1/s))
• For single stream
• m1 m2
• where v is velocity, A is area and r is density

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Energy of a Moving Fluid

• The energy of a moving fluid (per unit mass) is
  the sum of the internal, kinetic, and potential
  energies and the flow work
• Total energy per unit mass is
• Since h u Pv
• q h ke pe (per unit mass)

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Energy Balance

• Rate of energy transfer in is equal to rate of
  energy transfer out for a steady flow system
• For a steady flow situation
• Sin Q W mq Sout Q W mq
• In the special case where Q W ke pe 0

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Application Mixing Chamber

• In general, the following holds for a mixing
  chamber
• Mass conservation
• Energy balance
• Only if Q W pe ke 0

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Open Mixed Systems

• Consider an open system where the number of moles
  (n) can change
• dU (?U/?V)dV (?U/?S)dS S(?U/?nj)dnj

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Chemical Potential

• We can simplify with
• and rewrite the dU equation as
• dU -PdV TdS S mjdnj
• The third term is the chemical potential or

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The Gibbs Function

• Other characteristic functions can be written in
  a similar form
• Gibbs function
• For phase transitions with no change in P or T

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Mass Flow

• Consider a divided chamber (sections 1 and 2)
  where a substance diffuses across a barrier
• dS dU/T -(m/T)dn
• dS dU1/T1 -(m1/T1)dn1 dU2/T2 -(m2/T2)dn2

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Conservation

• Sum of dns must be zero
• Sum of internal energies must be zero
• Substituting into the above dS equation
• dS (1/T1)-(1/T2)dU1 - (m1/T1)-(m2/T2)dn1

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Equilibrium

• Consider the equilibrium case
• (m1/T1) (m2/T2)
• Chemical potentials are equal in equilibrium