Metr3210 clausius-clapeyron

Thermodynamics M. D. Eastin
Clausius-Clapeyron Equation
Cloud drops first form when the vaporization equilibrium point is reached
(i.e., the air parcel becomes saturated)
Here we develop an equation that describes how the vaporization/condensation
equilibrium point changes as a function of pressure and temperature
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid

Outline:
 Review of Water Phases
 Review of Latent Heats
 Changes to our Notation
 Clausius-Clapeyron Equation
 Basic Idea
 Derivation
 Applications
 Equilibrium with respect to Ice
 Applications

Homogeneous Systems (single phase):
Gas Phase (water vapor):
• Behaves like an ideal gas
• Can apply the first and second laws
Liquid Phase (liquid water):
• Does not behave like an ideal gas
Solid Phase (ice):
Review of Water Phases
αpddTcdq v +=
T
dq
ds rev
≥
vvvv TRρp =

Heterogeneous Systems (multiple phases):
Liquid Water and Vapor:
• Equilibrium state
• Saturation
• Vaporization / Condensation
pw, Tw
pv, Tv
wv pp =
wv TT =
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
Equilibrium States for Water
(function of temperature and pressure)

Equilibrium Phase Changes:
Vapor → Liquid Water (Condensation):
• Equilibrium state (saturation)
• Isobaric
• Isothermal
• Volume changes
wv pp = wv TT =
C
V
P
(mb)
Vapor
Solid
Tt =
0ºC
Liquid
Liquid
and
Vapor
Solid
and
Vapor
Tc =
374ºC
T1
6.11
221,000
T
B AC
A B C

Equilibrium Phase Changes:
• Heat absorbed (or given away)
during an isobaric and isothermal
phase change
• From the forming or breaking of
molecular bonds that hold water
molecules together in its different
phases
• Latent heats are weak function of
temperature
Review of Latent Heats
constantdQL ==
C
V
P
(mb)
Vapor
Solid
Tt =
0ºC
Liquid
Tc =
374ºC
T1
6.11
221,000
T
L
L
L
Values for lv, lf, and ls are given
in Table A.3 of the Appendix

Water vapor pressure:
• We will now use (e) to represent the
pressure of water in its vapor phase
(called the vapor pressure)
• Allows one to easily distinguish between
pressure of dry air (p) and the pressure
of water vapor (e)
Temperature subscripts:
• We will drop all subscripts to water and
dry air temperatures since we will assume
the heterogeneous system is always in
equilibrium
Changes to Notation
vvvv TRρp =
iwv TTTT ===
TRρe vv=
Ideal Gas Law for Water Vapor

Water vapor pressure at Saturation:
• Since the equilibrium (saturation) states are very important, we need to
distinguish regular vapor pressure from the equilibrium vapor pressures
e = vapor pressure (regular)
esw = saturation vapor pressure with respect to liquid water
esi = saturation vapor pressure with respect to ice
Changes to Notation

Who are these people?
Benoit Paul Emile Clapeyron
1799-1864
French
Engineer / Physicist
Expanded on Carnot’s work
Rudolf Clausius
1822-1888
German
Mathematician / Physicist
“Discovered” the Second Law
Introduced the concept of entropy

Basic Idea:
• Provides the mathematical relationship
(i.e., the equation) that describes any
equilibrium state of water as a function
of temperature and pressure.
• Accounts for phase changes at each
equilibrium state (each temperature)
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
V
P
(mb)
Vapor
Liquid
Liquid
and
Vapor
T
esw
Sections of the P-V and P-T diagrams for
which the Clausius-Clapeyron equation
is derived in the following slides

Mathematical Derivation:
Assumption: Our system consists of liquid water in equilibrium with
water vapor (at saturation)
• We will return to the Carnot Cycle…
Temperature
T2 T1
esw1
esw2
Saturationvaporpressure
A, D
B, C
Volume
T2
T1esw1
esw2
A D
B C
Isothermal process
Adiabatic process

• Recall for the Carnot Cycle:
• If we re-arrange and substitute:
21NET QQW +=
1
21
1
21
T
TT
Q
QQ −
=
+
where: Q1 > 0 and Q2 < 0
21
NET
1
1
T-T
W
T
Q
=
Volume
T2
T1esw1
esw2
A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2

Volume
T2
T1esw1
esw2
Saturationvaporpressure A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2
Recall:
• During phase changes, Q = L
• Since we are specifically working
with vaporization in this example,
• Also, let:
21
NET
1
1
T-T
W
T
Q
=
v1 LQ =
TT1 =
dTTT 21 =−

Recall:
• The net work is equivalent to the
area enclosed by the cycle:
• The change in pressure is:
• The change in volume of our system at
each temperature (T1 and T2) is:
where: αv = specific volume of vapor
αw = specific volume of liquid
dm = total mass converted from
vapor to liquid
( )dmααdV wv −=
sw2sw1sw eede −=
21
NET
1
1
T-T
W
T
Q
=
dpdVWNET ×=
Volume
T2
T1esw1
esw2
Saturationvaporpressure A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2

• We then make all the substitutions into our Carnot Cycle equation:
• We can re-arrange and use the
definition of specific latent heat of
vaporization (lv = Lv /dm) to obtain:
for the equilibrium vapor pressure
with respect to liquid water
21
NET
1
1
T-T
W
T
Q
=
( )
dT
dedmαα
T
L swwvv −
=
( )wv
vsw
ααTdT
de
−
=
l
Temperature
T2 T1
esw1
esw2
A, D
B, C

General Form:
• Relates the equilibrium pressure
between two phases to the temperature
of the heterogeneous system
where: T = Temperature of the system
l = Latent heat for given phase change
dps= Change in system pressure at
saturation
dT = Change in system temperature
Δα = Change in specific volumes
between
the two phases
αTΔdT
dps l
=
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
Equilibrium States for Water
(function of temperature and pressure)

Application: Saturation vapor pressure for a given temperature
Starting with:
Assume: [valid in the atmosphere]
and using: [Ideal gas law for the water vapor]
We get:
If we integrate this from some reference point (e.g. the triple point: es0, T0) to some
arbitrary point (esw, T) along the curve assuming lv is constant:
wv αα >>
TRαe vvsw =
2
v
v
sw
sw
T
dT
Re
de l
=
( )wv
vsw
ααTdT
de
−
=
l
∫∫ =
T
T 2
v
v
e
e
sw
sw
0
sw
s0 T
dT
Re
de l

After integration we obtain:
After some algebra and substitution for es0 = 6.11 mb and T0 = 273.15 K we get:
∫∫ =
T
T 2
v
v
e
e
sw
sw
0
sw
s0 T
dT
Re
de l






−=
T
1
T
1
Re
e
ln
0v
v
s0
sw l












−=
T(K)
1
273.15
1
R
exp11.6(mb)e
v
v
sw
l

A more accurate form of the above equation can be obtained when we do not
assume lv is constant (recall lv is a function of temperature). See your book for
the derivation of this more accurate form:












−=
T(K)
1
273.15
1
R
exp11.6(mb)e
v
v
sw
l
[ ]





−−= )(ln09.5
)(
6808
49.53exp11.6(mb)esw KT
KT

 What is the saturation vapor pressure with respect to water at 25ºC?
T = 298.15 K
esw = 32 mb
 What is the saturation vapor pressure with respect to water at 100ºC?
T = 373.15 K Boiling point
esw = 1005 mb
[ ]





−−= )(ln09.5
)(
6808
49.53exp11.6(mb)esw KT
KT

Application: Boiling Point of Water
 At typical atmospheric conditions near the boiling point:
T = 100ºC = 373 K
lv = 2.26 ×106
J kg-1
αv = 1.673 m3
kg-1
αw = 0.00104 m3
kg-1
 This equation describes the change in boiling point temperature (T) as a function
of atmospheric pressure when the saturated with respect to water (esw)
( )wv
vsw
ααTdT
de
−
=
l
1sw
Kmb36.21
dT
de −
=

Application: Boiling Point of Water
 What would the boiling point temperature be on the top of Mount Mitchell
if the air pressure was 750mb?
• From the previous slide
we know the boiling point
at ~1005 mb is 100ºC
• Let this be our reference point:
Tref = 100ºC = 373.15 K
esw-ref = 1005 mb
• Let esw and T represent the
values on Mt. Mitchell:
esw = 750 mb
T = 366.11 K
T = 93ºC (boiling point temperature on Mt. Mitchell)
1
ref
refswsw
Kmb36.21
TT
ee −−
=
−
−
ref
refsw
T
e
T +
−
=
−
36.21
esw
1sw
Kmb36.21
dT
de −
=

Equilibrium with respect to Ice:
• We will know examine the equilibrium
vapor pressure for a heterogeneous
system containing vapor and ice
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
C
V
P
(mb)
Vapor
Solid
Liquid
T
6.11 T
AB
esi

Equilibrium with respect to Ice:
• Return to our “general form” of the
Clausius-Clapeyron equation
• Make the appropriate substitution for
the two phases (vapor and ice)
for the equilibrium vapor
pressure with respect to ice
Sublim
ation
Fusion
Vaporization
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solidα∆
=
TdT
des l
( )iv
ssi
ααTdT
de
−
=
l

Application: Saturation vapor pressure of ice for a given temperature
Following the same logic as before, we can derive the following equation for
saturation with respect to ice
A more accurate form of the above equation can be obtained when we do not
assume ls is constant (recall ls is a function of temperature). See your book for
the derivation of this more accurate form:












−=
T(K)
1
273.15
1
R
exp11.6(mb)e
v
s
si
l
[ ]





−−= )(ln555.0
)(
6293
16.26exp11.6(mb)esi KT
KT

Application: Melting Point of Water
• Return to the “general form” of the Clausius-Clapeyron equation and make the
appropriate substitutions for our two phases (liquid water and ice)
 At typical atmospheric conditions near the melting point:
T = 0ºC = 273 K
lf = 0.334 ×106
J kg-1
αw = 1.00013 × 10-3
m3
kg-1
αi = 1.0907 × 10-3
m3
kg-1
 This equation describes the change in melting point temperature (T) as a function
of pressure when liquid water is saturated with respect to ice (pwi)
( )iw
fwi
ααTdT
dp
−
=
l
1wi
Kmb135,038
dT
dp −
−=

Summary:
• Review of Water Phases
• Review of Latent Heats
• Changes to our Notation
• Clausius-Clapeyron Equation
• Basic Idea
• Derivation
• Applications
• Equilibrium with respect to Ice
• Applications

References
Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp.
Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467
pp.

Metr3210 clausius-clapeyron

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