phase transitions physics 313 professor lee carkner lecture 22
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Phase Transitions
Physics 313Professor Lee
CarknerLecture 22
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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
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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:
T = [T (vf -vi)/ (hf - hi) ] P
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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 streamm1 = m2
where v is velocity, A is area and 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 = 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:
in [Q + W + m] = out [Q + W + m] 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 + (U/nj)dnj
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Chemical Potential We can simplify with
and rewrite the dU equation as:
dU = -PdV + TdS + jdnj
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 -(/T)dn
dS = dU1/T1 -(/T1)dn1 + dU2/T2 -(/T2)dn2
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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
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Equilibrium Consider the equilibrium case
(1/T1) = (2/T2)
Chemical potentials are equal in equilibrium• •