Clausius-Clapeyron Equation
p (mb)
C
221000
Liquid
Solid
1013
6.11
T
Vapor
0
100
374
T (ºC)
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
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
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
Thermodynamics
M. D. Eastin
Review of Water Phases
Homogeneous Systems (single phase):
Gas Phase (water vapor):
• Behaves like an ideal gas
• Can apply the first and second laws
p v  ρ v R vTv
Liquid Phase (liquid water):
• Does not behave like an ideal gas
• Can apply the first and second laws
Solid Phase (ice):
• Does not behave like an ideal gas
• Can apply the first and second laws
dq  cvdT  pd
dq rev
ds 
T
Thermodynamics
M. D. Eastin
Review of Water Phases
Heterogeneous Systems (multiple phases):
Liquid Water and Vapor:
• Equilibrium state
• Saturation
• Vaporization / Condensation
• Does not behave like an ideal gas
• Can apply the first and second laws
Equilibrium States for Water
(function of temperature and pressure)
p (mb)
C
221000
Liquid
Solid
pv, Tv
1013
6.11
T
pv  pw
Vapor
Tv  Tw
pw, Tw
Thermodynamics
0
100
374
T (ºC)
M. D. Eastin
Review of Water Phases
Equilibrium Phase Changes:
Vapor → Liquid Water (Condensation):
• Equilibrium state (saturation)
• Does not behave like an ideal gas
• Isobaric
• Isothermal
• Volume changes
pv  pw
P
(mb)
Liquid
Tv  Tw
C
A
B
C
221,000
C
Solid
6.11
B
Liquid
and
Vapor
Solid
and
Vapor
Tc =
374ºC
A
Vapor
T1
T
Tt =
0ºC
V
Thermodynamics
M. D. Eastin
Review of Latent Heats
Equilibrium Phase Changes:
• Heat absorbed (or given away)
during an isobaric and isothermal
phase change
L  dQ  constant
P
(mb)
Liquid
C
221,000
Tc =
374ºC
L
• From the forming or breaking of
molecular bonds that hold water
molecules together in its different
phases
• Latent heats are weak function of
temperature
Vapor
L
Solid
T1
T
6.11
L
Tt =
0ºC
V
Values for lv, lf, and ls are given
in Table A.3 of the Appendix
Thermodynamics
M. D. Eastin
Changes to Notation
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:
Ideal Gas Law for Water Vapor
p v  ρ v R vTv
e  ρ v R vT
• We will drop all subscripts to water and
dry air temperatures since we will assume
the heterogeneous system is always in
equilibrium
T  Tv  Tw  Ti
Thermodynamics
M. D. Eastin
Changes to Notation
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
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Who are these people?
Rudolf Clausius
1822-1888
German
Mathematician / Physicist
Benoit Paul Emile Clapeyron
1799-1864
French
Engineer / Physicist
“Discovered” the Second Law
Introduced the concept of entropy
Expanded on Carnot’s work
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Basic Idea:
p (mb)
• 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)
C
221000
Liquid
Solid
1013
6.11
T
Vapor
P
(mb)
Vapor
esw
100
374
T (ºC)
T
Liquid
Liquid
and
Vapor
Thermodynamics
0
Sections of the P-V and P-T diagrams for
which the Clausius-Clapeyron equation
is derived in the following slides
V
M. D. Eastin
Clausius-Clapeyron Equation
Mathematical Derivation:
Assumption:
Our system consists of liquid water in equilibrium with
water vapor (at saturation)
Isothermal process
Adiabatic process
B
C
esw1
esw2
T1
A
Volume
D
T2
Saturation vapor pressure
Saturation vapor pressure
• We will return to the Carnot Cycle…
B, C
esw1
A, D
esw2
T2
T1
Temperature
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Mathematical Derivation:
• Recall for the Carnot Cycle:
Q1  Q2 T1  T2

Q1
T1
where: Q1 > 0 and Q2 < 0
• If we re-arrange and substitute:
Saturation vapor pressure
WNET  Q1  Q2
Isothermal process
Adiabatic process
Q1
B
C
esw1
T1
WNET
esw2
A
Q2
D
T2
Volume
Q1 WNET

T1 T1 - T2
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Mathematical Derivation:
Recall:
Q1 WNET

T1 T1 - T2
• During phase changes, Q = L
• Since we are specifically working
with vaporization in this example,
Q1  Lv
T1  T
T1  T2  dT
Saturation vapor pressure
• Also, let:
Isothermal process
Adiabatic process
Q1
B
C
esw1
T1
WNET
esw2
A
Q2
D
T2
Volume
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Mathematical Derivation:
Recall:
Q1 WNET

T1 T1 - T2
• The net work is equivalent to the
area enclosed by the cycle:
WNET  dV  dp
• The change in pressure is:
• The change in volume of our system at
each temperature (T1 and T2) is:
dV  α v  α w dm
where:
αv = specific volume of vapor
αw = specific volume of liquid
Saturation vapor pressure
desw  esw1  esw2
Isothermal process
Adiabatic process
Q1
B
C
esw1
T1
WNET
esw2
A
Q2
D
T2
Volume
dm = total mass converted from
vapor to liquid
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Mathematical Derivation:
• We then make all the substitutions into our Carnot Cycle equation:
L v α v  α w  dm de sw

T
dT
• We can re-arrange and use the
definition of specific latent heat of
vaporization (lv = Lv /dm) to obtain:
de sw
lv

dT Tα v  α w 
Clausius-Clapeyron Equation
for the equilibrium vapor pressure
with respect to liquid water
Thermodynamics
Saturation vapor pressure
Q1 WNET

T1 T1 - T2
B, C
esw1
A, D
esw2
T2
T1
Temperature
M. D. Eastin
Clausius-Clapeyron Equation
General Form:
• Relates the equilibrium pressure
between two phases to the temperature
of the heterogeneous system
Equilibrium States for Water
(function of temperature and pressure)
p (mb)
dp s
l

dT TΔ
where:
T =
l =
dps =
dT =
Δα =
Temperature of the system
Latent heat for given phase change
Change in system pressure at saturation
Change in system temperature
Change in specific volumes between
the two phases
C
221000
Liquid
Solid
1013
6.11
T
Vapor
0
Thermodynamics
100
374
T (ºC)
M. D. Eastin
Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature
Starting with:
de sw
lv

dT Tα v  α w 
Assume:
α v  α w
[valid in the atmosphere]
and using:
esw α v  R v T
[Ideal gas law for the water vapor]
We get:
de sw
lv dT

esw
R v T2
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:

esw
es0
Thermodynamics
de sw
lv

esw
Rv
dT
T0 T 2
T
M. D. Eastin
Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature

esw
es0
de sw
l
 v
esw
Rv
dT
T0 T 2
T
After integration we obtain:
esw
lv  1 1 
  
ln

es0 R v  T0 T 
After some algebra and substitution for es0 = 6.11 mb and T0 = 273.15 K we get:
 lv  1
1 

esw (mb)  6.11 exp 

 R v  273.15 T (K)
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature
 lv  1
1 

esw (mb)  6.11 exp 

 R v  273.15 T (K)
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:


6808
esw (mb)  6.11 exp53.49 
 5.09lnT ( K )
T (K )


Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature


6808
esw (mb)  6.11 exp53.49 
 5.09lnT ( K )
T (K )


 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
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
Application: Boiling Point of Water
de sw
lv

dT Tα v  α w 
 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
de sw
 36.21 mb K 1
dT
 This equation describes the change in boiling point temperature (T) as a function
of atmospheric pressure when the saturated with respect to water (esw)
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
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
de sw
 36.21 mb K 1
dT
• Let this be our reference point:
esw  esw ref
 36.21 mb K 1
T  Tref
Tref = 100ºC = 373.15 K
esw-ref = 1005 mb
• Let esw and T represent the
values on Mt. Mitchell:
T
esw  esw  ref
36.21
 Tref
esw = 750 mb
T = 366.11 K
T = 93ºC
Thermodynamics
(boiling point temperature on Mt. Mitchell)
M. D. Eastin
Clausius-Clapeyron Equation
Equilibrium with respect to Ice:
• We will know examine the equilibrium
vapor pressure for a heterogeneous
system containing vapor and ice
p (mb)
C
221000
Liquid
P
(mb)
Solid
1013
6.11
Liquid C
T
Vapor
Vapor
Solid
0
Thermodynamics
374
T (ºC)
T
6.11
esi
100
B
A
T
V
M. D. Eastin
Clausius-Clapeyron Equation
Equilibrium with respect to Ice:
• Return to our “general form” of the
Clausius-Clapeyron equation
p (mb)
de s
l

dT T
• Make the appropriate substitution for
the two phases (vapor and ice)
de si
ls

dT Tα v  α i 
C
221000
Liquid
Solid
1013
6.11
T
Vapor
0
100
374
T (ºC)
Clausius-Clapeyron Equation
for the equilibrium vapor
pressure with respect to ice
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
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
 ls  1
1 

esi (mb)  6.11 exp 

 R v  273.15 T (K)
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:


6293
esi (mb)  6.11 exp26.16 
 0.555lnT ( K )
T (K )


Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
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)
dp wi
lf

dT Tα w  αi 
 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
dp wi
 135,038 mb K 1
dT
 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)
Thermodynamics
M. D. Eastin
Clausius-Clapeyron Equation
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
Thermodynamics
M. D. Eastin
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.
Thermodynamics
M. D. Eastin