1. Clausius-Clapeyron Equation
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid
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

2. 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
Thermodynamics
M. D. Eastin

3. Review of Water Phases 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):
Thermodynamics
M. D. Eastin

4. Equilibrium States for Water (function of temperature and pressure)
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)
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid
pv, Tv
pw, Tw
Thermodynamics
M. D. Eastin

5. 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 C V P
(mb)
Vapor
Solid
Tt =
0ºC
Liquid
and
Tc =
374ºC T1
6.11
221,000 T B A A B C
Thermodynamics
M. D. Eastin

6. Review of Latent Heats 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 a weak function of
temperature C V P
(mb)
Vapor
Solid
Tt =
0ºC
Liquid
Tc =
374ºC T1
6.11
221,000 T L
Values for lv, lf, and ls are given
in Table A.3 of the Appendix
Thermodynamics
M. D. Eastin

7. Ideal Gas Law for Water Vapor
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:
We will drop all subscripts to water and
dry air temperatures since we will assume
the heterogeneous system is always in
equilibrium
Ideal Gas Law for Water Vapor
Thermodynamics
M. D. Eastin

8. 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

9. Benoit Paul Emile Clapeyron
Clausius-Clapeyron Equation
Who are these people?
Rudolf Clausius
German
Mathematician / Physicist
“Discovered” the Second Law
Introduced the concept of entropy
Benoit Paul Emile Clapeyron
French
Engineer / Physicist
Expanded on Carnot’s work
Thermodynamics
M. D. Eastin

10. Clausius-Clapeyron Equation
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)
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid P
(mb)
Vapor
esw T
Liquid
Sections of the P-V and P-T diagrams for
which the Clausius-Clapeyron equation
is derived in the following slides
Liquid
and
Vapor V
Thermodynamics
M. D. Eastin

11. Saturation vapor pressure Saturation vapor pressure
Clausius-Clapeyron Equation
Mathematical Derivation:
Assumption: Our system consists of liquid water in equilibrium with
water vapor (at saturation)
We will return to the Carnot Cycle…
Volume T2 T1
esw1
esw2
Saturation vapor pressure A D B C
Isothermal process
Adiabatic process
Temperature T2 T1
esw1
esw2
Saturation vapor pressure
A, D
B, C
Thermodynamics
M. D. Eastin

12. Saturation vapor pressure
Clausius-Clapeyron Equation
Mathematical Derivation:
Recall for the Carnot Cycle:
If we re-arrange and substitute:
Volume T2 T1
esw1
esw2
Saturation vapor pressure A D B C
Isothermal process
Adiabatic process
WNET Q1 Q2
where: Q1 > 0 and Q2 < 0
Thermodynamics
M. D. Eastin

13. Saturation vapor pressure
Clausius-Clapeyron Equation
Mathematical Derivation:
Recall:
During phase changes, Q = L
Since we are specifically working
with vaporization in this example,
Also, let:
Volume T2 T1
esw1
esw2
Saturation vapor pressure A D B C
Isothermal process
Adiabatic process
WNET Q1 Q2
Thermodynamics
M. D. Eastin

14. Saturation vapor pressure
Clausius-Clapeyron Equation
Mathematical Derivation:
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
Volume T2 T1
esw1
esw2
Saturation vapor pressure A D B C
Isothermal process
Adiabatic process
WNET Q1 Q2
Thermodynamics
M. D. Eastin

15. Saturation vapor pressure
Clausius-Clapeyron Equation
Mathematical Derivation:
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:
Clausius-Clapeyron Equation
for the equilibrium vapor pressure
with respect to liquid water
Temperature T2 T1
esw1
esw2
Saturation vapor pressure
A, D
B, C
Thermodynamics
M. D. Eastin

16. Equilibrium States for Water (function of temperature and pressure)
Clausius-Clapeyron Equation
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
Equilibrium States for Water
(function of temperature and pressure)
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid
Thermodynamics
M. D. Eastin

17. Clausius-Clapeyron Equation
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:
Thermodynamics
M. D. Eastin

18. Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature
After integration we obtain:
After some algebra and substitution for es0 = 6.11 mb and T0 = K we get:
Thermodynamics
M. D. Eastin

19. Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature
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:
Thermodynamics
M. D. Eastin

20. Clausius-Clapeyron Equation
Application: Saturation vapor pressure for a given temperature
What is the saturation vapor pressure with respect to water at 25ºC?
T = K
esw = 32 mb
What is the saturation vapor pressure with respect to water at 100ºC?
T = K Boiling point
esw = 1005 mb
Thermodynamics
M. D. Eastin

21. Clausius-Clapeyron Equation
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 = m3 kg-1
αw = 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)
Thermodynamics
M. D. Eastin

22. 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
Let this be our reference point:
Tref = 100ºC = K
esw-ref = 1005 mb
Let esw and T represent the
values on Mt. Mitchell:
esw = 750 mb
T = K
T = 93ºC (boiling point temperature on Mt. Mitchell)
Thermodynamics
M. D. Eastin

23. Clausius-Clapeyron Equation
Equilibrium with respect to Ice:
We will know examine the equilibrium
vapor pressure for a heterogeneous
system containing vapor and ice
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid C V P
(mb)
Vapor
Solid
Liquid T
6.11 A B
esi
Thermodynamics
M. D. Eastin

24. Clausius-Clapeyron Equation
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)
Clausius-Clapeyron Equation
for the equilibrium vapor
pressure with respect to ice
Sublimation
Fusion
Vaporization T C
T (ºC)
p (mb)
374
100
6.11
1013
221000
Liquid
Vapor
Solid
Thermodynamics
M. D. Eastin

25. 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
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:
Thermodynamics
M. D. Eastin

26. 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)
At typical atmospheric conditions near the melting point:
T = 0ºC = 273 K
lf = ×106 J kg-1
αw = × 10-3 m3 kg-1
αi = × 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)
Thermodynamics
M. D. Eastin

27. 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
Thermodynamics
M. D. Eastin

28. References Thermodynamics M. D. Eastin
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