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• •