Phase Transitions Physics 313 Professor Lee Carkner Lecture 22

Exercise #21 Joule-Thomson  Joule-Thomson coefficient for ideal gas   = 1/c P [T(  v/  T) P -v]   (  v/  T) P = R/P   = 1/c P [(TR/P)-v] = 1/c P [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 

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

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

Clausius - Clapeyron Equation  Consider the first T ds equation, integrated through a phase change T (s f - s i ) = T (dP/dT) (v f - v i )  This can be written:  But H = VdP + T ds, so the isobaric change in molar entropy is T ds, yielding: dP/dT = (h f - h i )/T (v f -v i )

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

Changes in T and P  For small changes in T and P, the CC equation can be written:   or:  T = [T (v f -v i )/ (h f - h i ) ]  P 

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?

Mass Conservation   Rate of mass flow in equals rate of mass flow out (note italics means rate (1/s))   For single stream m 1 = m 2   where v is velocity, A is area and  is density

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  = h + ke +pe (per unit mass)

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:  in [ Q + W + m  ] =  out [ Q + W + m  ]  In the special case where Q = W = ke = pe = 0 

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

Open Mixed Systems  Consider an open system where the number of moles (n) can change   dU = (  U/  V)dV + (  U/  S)dS +  (  U/  n j )dn j 

Chemical Potential  We can simplify with   and rewrite the dU equation as: dU = -PdV + TdS +   j dn j    The third term is the chemical potential or:

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: 

Mass Flow   Consider a divided chamber (sections 1 and 2 ) where a substance diffuses across a barrier dS = dU/T -(  /T)dn  dS = dU 1 /T 1 -(   /T 1 )dn 1 + dU 2 /T 2 -(   /T 2 )dn 2

Conservation   Sum of dn’s must be zero:  Sum of internal energies must be zero:  Substituting into the above dS equation: dS = [(1/T 1 )-(1/T 2 )]dU 1 - [(  1 /T 1 )-(  2 /T 2 )]dn 1

Equilibrium  Consider the equilibrium case  (  1 /T 1 ) = (  2 /T 2 )   Chemical potentials are equal in equilibrium 