clausius-clapeyron equation
DESCRIPTION
p ( mb ). C. 221000. Fusion. Liquid. Vaporization. Solid. 1013. 6.11. T. Sublimation. Vapor. 0. 100. 374. T ( º C). Clausius-Clapeyron Equation. Cloud drops first form when the vaporization equilibrium point is reached (i.e., the air parcel becomes saturated) - PowerPoint PPT PresentationTRANSCRIPT
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
Sublimatio
n
Fus
ion
Vap
oriz
atio
nT
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
Thermodynamics M. D. Eastin
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• Can apply the first and second laws
Solid Phase (ice):
• Does not behave like an ideal gas• Can apply the first and second laws
Review of Water Phases
pd dTcdq v
T
dqds rev
vvvv TRρp
Thermodynamics M. D. Eastin
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
Review of Water Phases
pw, Tw
pv, Tv
wv pp
wv TT Sublim
ation
Fus
ion
Vap
oriz
atio
n
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
Equilibrium States for Water(function of temperature and pressure)
Thermodynamics M. D. Eastin
Equilibrium Phase Changes:
Vapor → Liquid Water (Condensation):
• Equilibrium state (saturation)• Does not behave like an ideal gas• Isobaric• Isothermal• Volume changes
Review of Water Phases
wv pp wv TT C
V
P(mb)
Vapor
Solid
Tt = 0ºC
Liquid
Liquidand
Vapor
Solidand
Vapor
Tc =374ºC
T1
6.11
221,000
T
B AC
A B C
Thermodynamics M. D. Eastin
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
Review of Latent Heats
constantdQ L 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
Thermodynamics M. D. Eastin
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 TTT T
TRρe vv
Ideal Gas Law for Water Vapor
Thermodynamics M. D. Eastin
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
Thermodynamics M. D. Eastin
Who are these people?
Clausius-Clapeyron Equation
Benoit Paul Emile Clapeyron1799-1864
French Engineer / Physicist
Expanded on Carnot’s work
Rudolf Clausius1822-1888German
Mathematician / Physicist
“Discovered” the Second LawIntroduced the concept of entropy
Thermodynamics M. D. Eastin
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)
Clausius-Clapeyron Equation
Sublimatio
n
Fus
ion
Vap
oriz
atio
n
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
V
P(mb)
Vapor
Liquid
Liquidand
Vapor
T
esw
Sections of the P-V and P-T diagrams for which the Clausius-Clapeyron equation is derived in the following slides
Thermodynamics M. D. Eastin
Mathematical Derivation:
Assumption: Our system consists of liquid water in equilibrium with water vapor (at saturation)
• We will return to the Carnot Cycle…
Clausius-Clapeyron Equation
Temperature
T2 T1
esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A, D
B, C
Volume
T2
T1esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A D
B C
Isothermal process
Adiabatic process
Thermodynamics M. D. Eastin
Mathematical Derivation:
• Recall for the Carnot Cycle:
• If we re-arrange and substitute:
Clausius-Clapeyron Equation
21NET QQW
1
21
1
21
T
TT
Q
where: Q1 > 0 and Q2 < 0
21
NET
1
1
T-T
W
T
Q
Volume
T2
T1esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2
Thermodynamics M. D. Eastin
Volume
T2
T1esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2
Mathematical Derivation:
Recall:
• During phase changes, Q = L
• Since we are specifically working with vaporization in this example,
• Also, let:
Clausius-Clapeyron Equation
21
NET
1
1
T-T
W
T
Q
v1 LQ
TT1
dTTT 21
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
dmααdV wv
sw2sw1sw eede
21
NET
1
1
T-T
W
T
Q
dpdVWNET
Volume
T2
T1esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A D
B C
Isothermal process
Adiabatic process
WNET
Q1
Q2
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
21
NET
1
1
T-T
W
T
Q
dT
dedmαα
T
L swwvv
wv
vsw
ααTdT
de
l
Temperature
T2 T1
esw1
esw2
Sa
tura
tio
n v
ap
or
pre
ss
ure
A, D
B, C
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
TΔdT
dps l
Sublimatio
n
Fus
ion
Vap
oriz
atio
n
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
Equilibrium States for Water(function of temperature and pressure)
Thermodynamics M. D. Eastin
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:
Clausius-Clapeyron Equation
wv αα
TRαe vvsw
2v
v
sw
sw
T
dT
Re
de l
wv
vsw
ααTdT
de
l
T
T 2v
ve
esw
sw
0
sw
s0 T
dT
Re
de l
Thermodynamics M. D. Eastin
Application: Saturation vapor pressure for a given temperature
After integration we obtain:
After some algebra and substitution for es0 = 6.11 mb and T0 = 273.15 K we get:
Clausius-Clapeyron Equation
T
T 2v
ve
esw
sw
0
sw
s0 T
dT
Re
de l
T
1
T
1
Re
eln
0v
v
s0
sw l
T(K)
1
273.15
1
Rexp11.6(mb)e
v
vsw
l
Thermodynamics M. D. Eastin
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:
Clausius-Clapeyron Equation
T(K)
1
273.15
1
Rexp11.6(mb)e
v
vsw
l
)(ln09.5
)(
680849.53exp11.6(mb)esw KT
KT
Thermodynamics M. D. Eastin
Application: Saturation vapor pressure for a given temperature
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
Clausius-Clapeyron Equation
)(ln09.5
)(
680849.53exp11.6(mb)esw KT
KT
Thermodynamics M. D. Eastin
Application: Boiling Point of Water
At typical atmospheric conditions near the boiling point:
T = 100ºC = 373 Klv = 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)
Clausius-Clapeyron Equation
wv
vsw
ααTdT
de
l
1sw Kmb36.21dT
de
Thermodynamics M. D. Eastin
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 Kesw-ref = 1005 mb
• Let esw and T represent the values on Mt. Mitchell:
esw = 750 mb
T = 366.11 KT = 93ºC (boiling point temperature on Mt. Mitchell)
Clausius-Clapeyron Equation
1
ref
refswsw Kmb36.21TT
ee
refrefsw T
eT
36.21
esw
1sw Kmb36.21dT
de
Thermodynamics M. D. Eastin
Equilibrium with respect to Ice:
• We will know examine the equilibrium vapor pressure for a heterogeneous system containing vapor and ice
Clausius-Clapeyron Equation
Sublimatio
n
Fus
ion
Vap
oriz
atio
n
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
ABesi
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
Sublimatio
n
Fus
ion
Vap
oriz
atio
n
T
C
T (ºC)
p (mb)
3741000
6.11
1013
221000
Liquid
Vapor
Solid
TdT
des l
iv
ssi
ααTdT
de
l
Thermodynamics M. D. Eastin
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:
Clausius-Clapeyron Equation
T(K)
1
273.15
1
Rexp11.6(mb)e
v
ssi
l
)(ln555.0
)(
629316.26exp11.6(mb)esi KT
KT
Thermodynamics M. D. Eastin
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 Klf = 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)
Clausius-Clapeyron Equation
iw
fwi
ααTdT
dp
l
1wi Kmb135,038dT
dp
Thermodynamics M. D. Eastin
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
Clausius-Clapeyron Equation
Thermodynamics M. D. Eastin
ReferencesPetty, 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.