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Phase Transitions

Phase Transitions. Physics 313 Professor Lee Carkner Lecture 22. Exercise #21 Joule-Thomson. Joule-Thomson coefficient for ideal gas m = 1/c P [T( v/T) P -v] ( v/T) P = R/P m = 1/c P [(TR/P)-v] = 1/c P [v-v] = 0 Can J-T cool an ideal gas T does not change

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Phase Transitions

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  1. Phase Transitions Physics 313 Professor Lee Carkner Lecture 22

  2. Exercise #21 Joule-Thomson • Joule-Thomson coefficient for ideal gas • m = 1/cP[T(v/T)P-v] • (v/T)P = R/P • m = 1/cP[(TR/P)-v] = 1/cP[v-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

  3. 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

  4. 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

  5. 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)

  6. 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

  7. 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

  8. 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?

  9. 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

  10. 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)

  11. 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

  12. Application: Mixing Chamber • In general, the following holds for a mixing chamber: • Mass conservation: • Energy balance: • Only if Q = W = pe = ke = 0

  13. 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

  14. Chemical Potential • We can simplify with • and rewrite the dU equation as: dU = -PdV + TdS + Smjdnj • The third term is the chemical potential or:

  15. 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:

  16. 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

  17. Conservation • Sum of dn’s 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

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

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