phase transitions physics 313 professor lee carkner lecture 22
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Phase Transitions
Physics 313Professor Lee
CarknerLecture 22
Exercise #21 Joule-Thomson Joule-Thomson coefficient for ideal gas
= 1/cP[T(v/T)P-v] (v/T)P = R/P = 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
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 (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)
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 (vf -vi)/ (hf - hi) ] 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 streamm1 = m2
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/nj)dnj
Chemical Potential We can simplify with
and rewrite the dU equation as:
dU = -PdV + TdS + jdnj
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 = dU1/T1 -(/T1)dn1 + dU2/T2 -(/T2)dn2
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 - [(1/T1)-(2/T2)]dn1
Equilibrium Consider the equilibrium case
(1/T1) = (2/T2)
Chemical potentials are equal in equilibrium• •
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