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Phase Transformations of
Pure Substances
Phase Diagrams
Phase Stability
-Temperature Dependence
-Pressure Dependence
Classification of Phase Transitions
Surface Tension
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Thermodynamically Stable Phases
Usually, only one phase of a given sub-
stance is stable at any given temperatureand pressure.
At some conditions of temperature andpressure, two or more phases may exist in
equilibrium.A slight change in temperature or pressure
will favor one phase over others. Theconversion of one phase to another is aphase transition.
Phase transitions occur with a decrease(spont.) or no change (equil.) in Gibbsenergy.
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Example Problem
The standard Gibbs energy of formation of
metallic white tin (a-tin) is 0 at 25 oC andthat of nonmetallic gray tin (b-tin) is +0.13 kJ
mol-1at the same temperature. Which is the
thermodynamically stable phase at 25o
C? Solution:The thermodynamically stable
phase is the one with the lower Gibbs
energy, which would be the a- (white) tin at
25 oC. Note: At lower temperatures, the nonmetallic
gray tin becomes the stable form. The tran-sition is known among metallurgistsas the tin disease.
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G-T Phase Diagrams
This is similar to how Fig.5.1 should have looked.
(m/T)p=(Gm/T)p= -Sm The straight lines show that
S is approx. constant withtemp. (Really, they curve.)
Since S is positive for all
phases of all substances,the slopes are all negative.
Note that the gas phasehas the steepest slope;
the solid phase, the leaststee .
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p-T Phase Diagrams
At left is a generic
p-T phase diagram. For any of the phase
boundary lines , theslope is given by
dp/dT = DS/DV =DH/TDV For the transitions
s->l, l->g, and s->g,DS > 0
For l->g and s->g,DV > 0, while fors->l, DV is almostalways > 0
This explains thepositive slopes.
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Whats In a Phase Diagram
A one-component phasediagram represents the
situation when only thatcomponent is present.
Thus the vapor-pressure curveplots the pressure ONLY of the
vapor (no air) Water filling a container at 50
torr WILL NOT EVAPORATE. If some of the liquid is removed
without decreasing the size of
the container (or letting any airin) vapor will form in the spaceabove the liquid until thepressure of the vapor reachesthe equilibrium vapor pressure
(left).
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Phase Diagrams of ParticularSubstances: H2O, CO2 , He
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Example Problem
What is the minimum pressure at whichliquid is the thermodynamically stablephase of water at 25 oC?
Solution:Start in the liquid region of thephase diagram for water at 25 oC and dropdown in pressure until the vapor pressureline is reached. This occurs at 23.8 torr.Below this pressure liquid water cannotexist at 25 oC .
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Criteria of Equilibrium
At equilibrium, thechemical potential of
a substance is the
same throughout asample, regardless
of how many phases
are present.
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Temperature Dependence
(m/T)p = -Sm As the temperature is raised, the chemical
potential of a pure substance decreases, always.
When a phase transition occurs, the relative
values of the chemical potentials of the various
phases are modified.
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Pressure and Melting Point
(m/p)T = Vm. The greater the molarvolume, the more steeply mrises with pressure.
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Effect of Pressure on Vap. Pres.
(a) and (b) in the figureat left show two ways of
determining the effect of
applied pressure on
vapor pressures.
For a perfect gas,
p=p* exp[Vm
Dp/RT]
shows how the vapor
pressure increases when
the applied pressure
increases.
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Slopes of Phase Boundaries
When two phases aand bare in equi-
librium,
ma(
p,T) = mb(
p,T)
dm= -SmdT + Vmdp
-Sa,mdT + Va,mdp =
-Sb,mdT + Vb,mdp
(Vb,m- Va,m)dp =
(Sb,m- Sa,m)dT
dp/dT = DtrsS/DtrsV
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The Solid-Liquid Boundary dp/dT = DS/DV =
DH/TDV (the Clapey-ron equation) appliesto any two phases aand b.
For instance, inmelting, dp/dT =DfusH/TDfusV
If DH and DV areapprox. constant,p.p* + (T-T*)
(DfusH/T*DfusV)
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The Clausius-Clapeyron Equation The Clausius-Clapeyron
equation applies theClapeyron equation tothe special case wherephase bis the gasphase, and the gas is
assumed perfect. dp/dT = DvapH/TVg=DvapH/T(RT/p)
d(lnp)/dT = DH/RT2
lnp2= lnp1+ DH 1 - 1R T1 T2 Ifpis known at one
temp it can be found atanother temp.
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Liquid-Vapor and Solid-Vapor
The Clausius-Clapeyron equation
applied to both
vaporization and
sublimation. For sublimation, sub-
stitute DsubH for DvapH
The assumptions/
approximations are: DV.Vgand
Vg.RT/p
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Determiningpat a Given T How do you arrive at the value 23.8 torr for the vapor
pressure of water at 25 oC?
Some possible ways: Read it off ap-T phase diagram (if a diagram that can beread to that precision is available).
Look in a handbook or other reference for vapor pressuretables for water.
Calculate from a known vapor pressure at some other
temperature (C-C eqn, see next slide). Calculate from DG value for the liquid-to-gas transition.
This last approach can be used for water at 25 oC From the Table 2.6, DfG
o= -228.57 kJ/mol for gaseouswater at 25 oC and -237.13 kJ/mol for liquid water at 25 oC.
Therefore DtrGo= + 8.56 kJ/mol for vapn of water at 25 oC.
From DtrGocan be found Ktr , since DtrG
o= - RT ln Ktr= This gives ln Ktr= (+8.56 kJ/mol)/(-2.479 kJ/mol) = -3.453 Therefore Ktr= 0.0317 But for a vapn rxn, Ktris simply equal top/p
o (po= 1 bar)
So we arrive atp= 0.0317 bar = 3.17 kPa= 23.8 torr
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Phase Rule For a system at equilibrium,
F = C - P + 2 where C = the number of components (1 so far in
this chapter), P = the number of phases present,and F = the number of degrees of freedom(thenumber of intensive variables such as temp, pres, or molfrac that can be changed without disturbing the number ofphases in equilibrium)
For a one-component system the phase rulebecomes
F = 3 - PWhen only one phase is present bothpand T are
independently variable. (An area on thep-Tdiagram)When two phases are present there is only onep
possible for a given T. (A line on thep-T diagram) Three phases can be present (triple point) but
there is no variation allowed inpor T.
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Changes in Various Functions
Most functions(volume, enthalpy,entropy) change dis-continuously at a
phase transition. A few (chemical
potential) changecontinuously.
Cpgoes4and back.Do you see why?
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Second-Order Phase Transitions
The previous slide illustrated first-order phasetransitions, in which the chemical potential was acontinuous function with a discontinuous first
derivative. A second-order transition is one in which the first
derivative of mwith respect to temperature is con-tinuous but the second derivation isdiscontinuous.
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Lambda Transition
The transitionshown at left is
that of the fluid-
superfluid tran-
sition in liquid
helium.
It is named for
the shape of
the heat
capacity curve.
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Surface Tension
Surface tension, orsurface energy, is
the force per unit
length (along h) orenergy per unit
area to increase
the surface area of
a liquid.
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Bubbles, Cavities, Droplets
The pressure on theconcave side of an
interface,pinis
always greater than
the pressure on the
convex side,pout.
pin=pout+ 2g/r
This is the Laplace
equation.
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Capillary Rise
The pressureexerted by a
column of liquid
of density rand
height h is p= rgh
This matches
the pressure
from 2g/r.
Therefore,
h = 2 rgr