CHAPTER 4 – Physical Transformations of Pure Substances.
I.
Generalities.
A.
Definitions:
1.
A phase of a substance is a form of matter that is uniform
throughout in chemical composition and physical state.
e.g., gas phase, liquid phase, and various solid phases of different
atomic structure.
2.
Phase transition is the spontaneous conversion of one phase into
another, occurring at a characteristic temperature for given pressure.
e.g. At 1 atm, ice is the most stable phase below 0° C, but liquid is
most stable above 0°C. That means ice has a lower Gibbs energy
per mole than the liquid at temperatures below 0°C, etc.
3.
Transition temperature Ttr is temperature at which the two phases
are in equilibrium and each have the same molar Gibbs energy.
4.
This chapter will concern itself with thermodynamics of phase
behavior, not kinetics.
E.g., one can lower a liquid below its freezing point and yet not see
the onset of the phase transition because of the slowness of the
molecules in getting reorganized into an ordered phase. This would
be called supercooling in this case.
5.
A persistent phase which is thermodynamically not the most stable
phase at the prevailing conditions is called a metastable state.
e.g. diamond is an allotrope of carbon that is thermally less stable
than graphite at normal pressures, but kinetically the phase
transformation is too slow to perceive.
6.
The central themodynamic quantity for treating phase behavior is the
chemical potential µ:
µ = (∂G/∂n)p,T
Useful for open systems, where amount n of substance is changing.
Tells how G changes with n (moles) in each phase.
7.
µ of perfect gas:
Now since G(p2) = G(p1) + nRT ln p2/p1
θ
if p1 = 1 bar (standard pressure) = p
and p2 = our working pressure p
θ
then G(p) = G° + nRT ln p/p
1
θ
Therefore: µ = µ° + RT ln p/p
Shows variation of chemical potential with pressure.
8.
Chemical potential of a real gas (the fugacity):
θ
ideal gas: µ = µ° + RT ln p/p
real gas: replace pressure p with the fugacity f.
f = γp
(γ is the fugacity coefficient, γ ≈ 1 in most cases)
f can be viewed as the "effective pressure".
µ = µ° + RT ln f/p
θ
θ
= µ°+ RT ln p/p + RT ln γ
where RT ln γ term is the effect of intermolecular forces and, hence,
deviations from ideality
γ ! 1 as p ! 0
γ is a measure of non-ideality and can be directly related to:
- Z (compression factor)
- VdW coefficients
- virial coefficients
B.
Use of µ:
pure substance
Phase equilibrium exists when µV = µl
When µV > µl, will be a net flow of vapor
molecules into liquid.
µV < µl, vice versa.
Transition T is temp at which µV = µl.
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II.
Phase diagrams for pure substances.
A.
General features:
Figure 4-4
1.
Phase boundaries = lines on phase diagram where two phases
coexist in equilibrium.
2.
Triple Point = unique point where solid, liquid and vapor can coexist
simultaneously in equilibrium.
µs = µl = µV
e.g., H2O
T3 = 273.16K, @ p = 4.58 Torr
Also T3 is the lowest p at which liquid phase can exist.
3.
Critical Point: critical temperature Tc + pc point. Tc is temperature
above which cannot form a liquid at any pressure.
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4.
Vapor pressure = the pressure of the vapor phase in equilibrium
with its corresponding liquid.
e.g., the liquid-vapor phase boundary from triple point to critical
point thus shows how vapor pressure varies with T (governed by
Clausius-Clapeyron equation)
e.g., the solid-vapor phase boundary thus shows how sublimation
vapor pressure varies with T.
I2(s) "! I2(g)
5.
Boiling Temperature = T at which vapor pressure of liquid equals the
surrounding pressure.
Tb = normal boiling point = boiling temperature when p = 1 atm.
Line from T3 to Tc shows the boiling point of the pure substance as
function of pressure above the liquid.
6.
Melting (or freezing) temperature = temp at which solid and liquid
coexist in equilibrium.
Tf = normal melting pt. = melting temp when pressure = 1 atm.
III. Examples of phase diagrams of pure substances.
A.
CO2 - a typical substance
CO2 cannot exist as liquid at
room pressure, since 1 atm
is well below the triple point pressure
of 5.11 atm
Thus solid CO2 sublimes
(so-called “dry ice”).
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B.
Pure H2O phase diagram
Figure 4A.9
Special feature of pure H2O phase diagram:
Negative slope is unorthodox. Shows that:
Melting temp↓ as p↑.
Density of solid less than liquid phase.
(H2O expands as it freezes)
H-bonding network (see right)
Figure 4A.10
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C.
Helium:
Note: solid never in equilibrium
with vapor
TWO liquid phases:
normal and superfluid
Superfluid He flows without
viscosity.
IV. Thermodynamics of Phase Transitions of Pure Substances.
A.
Two phases α and β in equilibrium, then:
µα = µβ
B.
But µ depends on T and p:
1.
T-dependence:
(∂µ/∂T)p = -Sm
(molar entropy)
Now since always S>0, µ always ↓ as T↑
Slopes on right diagram are = -Sm
Heating a region always lowers its
Chemical potential
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2.
P-dependence
(∂µ/∂p) T = Vm
(molar volume)
shows chemical potential always goes up when pressure goes up.
Figure 4B.2
Typical subst.: Density of solid > liquid
So Vm of solid is smaller than liquid.
Increase in p causes
freezing T to go up.
C.
Water: Density of solid < liquid
So Vm of solid > liquid
Increase in p causes freezing T
to go down.
Derive equation for phase boundary lines in phase diagram (p vs. T):
First: see µ as function of p and T
dµ = (∂µ/∂p)T dp + (∂µ/∂T)p dT
so
dµ = Vmdp – SmdT
Now along a phase boundary line
dµα = dµβ
µα
and
µβ
remain equal, so:
(as p and T change)
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Therefore:
Vm,α dp - Sm,α dT = Vm,β dp - Sm,β dT
(Sm,β - Sm,α)dT = (Vm,β -Vm,α)dp
Finally - Clapeyron Equation:
dp ΔSm
=
dT ΔVm
= change in molar entropy/change in molar volume as
α→β
and since entropy change is enthalpy change/transition temp:
ΔHtr
dp
=
dT Ttr ΔVm
True for any phase transition, pure substance.
dp/dT is slope of coexistence line on p vs. T diagram:
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D.
The solid-liquid line:
Replace ΔSm with ΔHfus/T.
dp ΔHfus
=
dT Tf ΔVm
typical behavior on right
ΔHfus > 0
melting requires energy
(endothermic)
ΔVfus > 0 for most substances
(liquid less dense and occupies larger
volume than solid)
Now we understand H2O slope:
For H2O, ΔVfus < 0
Figure 4-16
solid occupies larger volume than liquid
dp/dT = +/- = (-)
negative slope exaggerated here
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E.
The liquid-vapor line:
Substitute ΔSvap with ΔHvap/T
dp ΔHvap
=
dT T ΔVm
ΔVm = Vm,vap - Vm,liq
↑≈0
≈ Vm,vap ≈ RT/p
ΔHvap
dp
=
dT T (RT / p)
ΔHvap
dp
=p
dT
RT2
(dp / dT) ΔHvap
=
p
RT2
and so finally we obtain:
dlnp ΔHvap
=
dT
RT2
Clausius-Clapeyron Equation
Integrated form:
$ ΔH $
''
1
1
vap
&& − ∗ )))
p = p exp & −
&
R % T T ()(
%
∗
Gives vapor pressure at two temperatures T and T*
F.
The solid-vapor line:
Follow all the same arguments.
dlnp ΔHsub
=
dT
RT2
↑sublimation
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V.
Types of Phase Transitions (Ehrenfest’s classification)
A.
First order - most melting, vaporization are examples.
1st derivative of µ vs T are discontinuous at transition.
Slope changes!
µ
T
e.g. Volume:
Similarly H:
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B.
Second-order - order-disorder transitions, ferromagnetism.
1st derivative of µ vs T is continuous.
2nd derivative is discontinuous.
µ
T
ΔV = 0 at transition:
V
T
Figure 4B.10
Cp
grows
before
trans
Τλ
T
gel → liquid crystal in lipid bilayers.
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Summary:
1st order
2nd order
Figure 4B.9
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Notes:
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