4.3, 4.4
Phase Equilibrium
f =1
Determine the slopes of the
Relate
p
and
T
lines
at equilibrium between two phases
Lets consider the Gibbs function
dG = −ηdT + Vdp
Applies to a homogeneous system
An open system where a new phase may form or a
new component can be added
dG = −ηdT + Vdp + ∑∑
i
j
∂G
dnij
∂nij
i
j
Number of components
Number of phases
∂G
dG = −ηdT + Vdp + ∑∑
dnij
i
j ∂nij
∂G ∂η
∂G
μ=
∂n
Is the chemical
potential
Change in the Gibbs function of the
system with a change in the number of
moles of a given component or phase
dG = −ηdT + Vdp + ∑∑ μij dnij
i
constant temperature
constant pressure
dG = ∑ μ j dn j
j
One component
j
μ =g
closed system
One component
3 phases
The number of moles of a given phase can vary
under possible phase transitions
Total number
of moles
n = n1 + n2 + n3 = constant
dn = 0
For a closed system at
constant temperature
constant pressure
dG = ∑ μ j dn j = 0
j
Conditions for equilibrium between two phases 1 and 2
Thermal equilibrium
T1 ≠ T2
if
Heat would flow from one
phase to the other
Mechanical equilibrium
if
p1 ≠ p2
μ1 ≠ μ2
No equilibrium
p1 = p2
One phase would be expanding
at the expense of the other
Chemical equilibrium
if
T1 = T2
No
equilibrium
μ1 = μ2
A transfer on n moles would change
the Gibbs function
During a phase change heat is added
to (or remove) from the system
No change of temperature of pressure
Enthalpy change during the
phase transition is
Δh = L
L
Latent heat of the phase in transition
Latent heat of fusion
Lil = hl − hi
Latent heat of vaporization
Llv = hv − hl
Latent heat of sublimation
Liv = hv − hi
Lil = Liv − Llv
l
liquid
solid-liquid
phase transition
liquid-vapor
phase transition
solid-vapor
phase transition
i
ice
v
vapor
Phase change process
dh = Tdη + vdp
Constant pressure
Δh = L
Δh L
=
Δη =
T
T
Latent heat
Strong in liquid
molecular interactions
Weak in a gas
During vaporization
The majority of latent heat is used to
overcome the cohesive forces holding
the molecules together in the liquid
Latent heat of fusion is much less than the heat of vaporization
Density difference between solid and liquid is relative small
Determine the slopes of the
μ1 = μ2
G1 = G2
f =1
lines
dg1 = dg2
Δh L
=
Δη =
T
T
dg1 = −η1dT + v1dp
dg2 = −η2 dT + v2 dp
− η1dT + v1dp = −η2 dT + v2 dp
Δh
dp η 2 − η1 Δη
L
=
=
=
=
dT v2 − v1 Δv TΔv TΔv
Clapeyron equation
First latent heat equation
dp
L
=
dT TΔv
Evaluate the slope
f =1
lines
dp
Lil
=
dT T (vl − vi )
Solid-liquid
equilibrium line
Inverting
dT T (vl − vi )
=
dp
Lil
Variation of the melting point with pressure
Liquid-vapor equilibrium line
dp
Llv
=
dT T (vv − vl )
At the triple point
vv = 206 m 3kg -1
vl = 10 −3 m 3 kg -1
vv >> vl
dp Llv
≈
dT Tvv
dp Llv
≈
dT Tvv
dp Llv p
=
dT RvT 2
If we use the
ideal gas law
Clausius Clapeyron equation
The boiling point temperature
The temperature at which the vapor pressure is
equal to the atmospheric pressure
dT RvT 2
=
dp Llv p
Decrease of boiling point temperature with decreasing pressure
To integrate
Llv (T )
Lets assume
Llv
constant
∫ d (ln e ) = ∫
e2
T2
e1
T1
e
Llv
dT
2
RvT
Water vapor pressure
e2
Llv ⎛ 1 1 ⎞
ln = − ⎜⎜ − ⎟⎟
e1
Rv ⎝ T2 T1 ⎠
e1 e2
⎡ Llv ⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
e2 = e1 exp ⎢−
⎣ Rv ⎝ T2 T1 ⎠⎦
Saturation vapor pressure at
Remember we assume
de Llv
≈
dT Tvv
T2
Vapor phase obeys the ideal gas
Llv
Vapor-ice
equilibrium line
T1
constant
de
Liv
=
dT T (vv − vi )
integrating
vi = 1.091 × 10 −3 m 3kg -1
vv >> vi
⎡ Liv ⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
e2 = e1 exp ⎢−
⎣ Rv ⎝ T2 T1 ⎠⎦
To integrate Clausius clapeyron equation more precisely
Including the variation of
Llv Liv
⎛ ∂Δh ⎞
⎛ ∂Δh ⎞
⎟⎟dp
dΔh = ⎜
⎟dT + ⎜⎜
⎝ ∂T ⎠
⎝ ∂p ⎠
With
T
dh = c p dT
⎛ ∂Δh ⎞ dp
dΔh ∂Δh ⎛ ∂Δh ⎞ dp
⎟⎟
⎟⎟
= Δc p + ⎜⎜
=
+ ⎜⎜
dT
∂T ⎝ ∂p ⎠ dT
⎝ ∂p ⎠ dT
dLlv
= c pv − c pl
dT
The variation
of the Latent
heat of fusion
Kirchoff’s law
second latent
heat equation
dLil
= c pl − c pi
dT
small
Consider a system
Layer of liquid water overlain by a layer of water vapor
Vapor pressure equal to the saturation vapor pressure of the liquid
Condensation
decreases
Entropy
Evaporation
increases
Liquid is a “less
random” state
Atmospheric humidity variables
Last sections
In the atmosphere
Pure water vapor as the gaseous phase
under consideration
Mixture of dry air gases and
water vapor
e
Partial pressure of the water vapor
pd
Partial pressure of dry air
Total atmospheric pressure
p
The saturation vapor pressure with respect to liquid
The saturation vapor pressure with respect to ice
esi
es
dp Llv p
=
dT RvT 2
Clausius-Clapeyron equation
Integrating the Clausius-Clapeyron equation for atmospheric
water vapor
⎡ Llv
es = es ,tr exp ⎢−
⎣ Rv
⎡ Liv
esi = es ,tr exp ⎢−
⎣ Rv
⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
⎝ Ttr T ⎠⎦
saturation vapor
pressure with
respect to liquid
⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
⎝ Ttr T ⎠⎦
saturation vapor
pressure with
respect to ice
Reference pressure and temperature
es ,tr = 6.11 hPa
Triple point
Ttr = 273.16 K
Application of the Clausius-Clapeyron equation
to determining the saturation vapor pressure in
the atmosphere is not strictly valid
Dalton’s law of
partial pressures is
not strictly valid
The total pressure is not
the sum of the partial
pressures of two ideal gases
The condensed phase is under a total pressure that is
augmented by the presence of dry air
The condensed phase is not purely liquid water, but
contains dissolved air
Departure from ideal gas
When high accuracy
is needed
Less than 1%
7
Empirical values
es = a1 + ∑ an (T − Ttr ) n −1
n=2
Values of saturation
vapor pressure
Used to determine commonly used
atmospheric variables
e
H=
es
Relative
humidity
e
saturation vapor
pressure with
respect to liquid
Multiplied
by 100
e
Hi =
esi
Relative humidity
with respect to ice
saturation
Function only of
Partial pressure
of the water
vapor
percentage
T
If we compare
saturation vapor
pressure with
respect to liquid
saturation vapor
pressure with
respect to ice
es
esi
⎡ Llv
es = es ,tr exp ⎢−
⎣ Rv
⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
⎝ Ttr T ⎠⎦
⎡ Lil ⎛ Ttr
es
⎞⎤
= exp ⎢
⎜ − 1⎟ ⎥
esi
⎠⎦
⎣ RvTtr ⎝ T
⎡ Liv
esi = es ,tr exp ⎢−
⎣ Rv
⎛ 1 1 ⎞⎤
⎜⎜ − ⎟⎟⎥
⎝ Ttr T ⎠⎦
es (T )
>1
esi (T )
For subfreezing
temperatures!!
Ratio increases as the temperature decreases
Atmosphere saturated with
respect to liquid water
T(C)
Hi
0
-10
-20
-30
-40
1.0
1.10
1.22
1.34
1.47
H =1
Supersaturated with
respect to ice
Water mixing ratio
Ratio of the mass of
water vapor present to
the mass of dry air
Saturation mixing ratio
p >> e
wv
mv ρ v
e
wv =
=
=ε
md ρ d
p−e
ws
p >> es
mv
wv
e
qv =
=ε
=
md + mv
p − (1 − ε )e 1 + wv
ε = Mv M
es
ws = ε
p − es
wv
H=
ws
wv qv
e
H=
es
Are always
smaller than
0.04
qv ≈ wv
Given T,p,and one of the humidity variable all
the other humidity variables can be determined
d
Precipitable water
Water vapor path
∞
Wv = ∫ ρ v dz
0
Wv
The total mass of water vapor in
a column of unit cross-sectional
area extending from the surface
to the top of the atmosphere
Wv
If all the vapor in the
column were to be condensed
the depth would be
ρ
The relationship between precipitable water and specific humidity
∞
Wv = ∫ ρ v dz
dp
= − ρg
dz
mv
qv =
md + mv
1
Wv =
g
0
1
Wv =
g
∫
p0
p
qv dp
∫
p0
p
ρv
dp
ρa