Phase of Water and Latent Heats
We must begin to account for the thermodynamics of water
Our atmosphere contains dry air and water vapor
Clouds contain dry air, water vapor, liquid water, and ice
Thermodynamics
M. D. Eastin
Phase of Water and Latent Heats
Outline:
 Review of Systems
 Thermodynamic Properties of Water
 Multiple phases
 Water in Equilibrium
 Equilibrium Phase Changes
 Amagat-Andrew Diagrams
 Latent Heats for Equilibrium Phase Changes
Thermodynamics
M. D. Eastin
Review of Systems
Homogeneous Systems:
• Comprised of a single component
• Oxygen gas
• Dry air
• Water vapor
• Each state variable (p, T, V, m) has
the same value at all locations
within the system
Thermodynamics
M. D. Eastin
Review of Systems
• Thus far we have worked exclusively
a homogeneous (dry air only) closed
system (no mass exchange, but some
energy exchange)
• So far, our versions of the Ideal gas law
and the first and second laws are only
applicable to dry air
• What about water vapor?
• What about the combination
of dry air and water vapor?
• What about the combination
of dry air, water vapor, and
liquid/ice water?
Thermodynamics
Dry Air
Closed
System
p, T, V, m, Rd
pα  R d T
dq  c v dT  pd 
ds 
dq rev
T
M. D. Eastin
Review of Systems
Heterogeneous Systems:
• Comprised of a single component
in multiple phases or multiple
components in multiple phases
• Water (vapor, liquid, ice)
• Each component or phase
must be defined by its own
set of state variables
Water Vapor
Pv, Tv, Vv, mv
Ice Water
Pi, Ti, Vi, mi
Liquid Water
Pw, Tw, Vw, mw
Thermodynamics
M. D. Eastin
Review of Systems
• Our atmosphere is a heterogeneous
closed system consisting of multiple
sub-systems
• Very complex…we come back to it later
Dry Air
(gas)
p, T, V, md, Rd
Liquid Water
pw, Tw, Vw, mw
Open sub-system
Water Vapor
pv, Tv, Vv, mv, Rv
Open sub-system
Closed sub-system
Energy Exchange
Mass Exchange
Thermodynamics
Ice Water
pi, Ti, Vi, mi
Open sub-system
M. D. Eastin
Review of Systems
• For now, let’s focus our attention on
the one component heterogeneous
system “water” comprised of vapor
and one other phase (liquid or ice)
Dry Air
(gas)
p, T, V, md, Rd
Liquid Water
pw, Tw, Vw, mw
Open sub-system
Water Vapor
pv, Tv, Vv, mv, Rv
Open sub-system
Closed sub-system
Energy Exchange
Mass Exchange
Thermodynamics
Ice Water
pi, Ti, Vi, mi
Open sub-system
M. D. Eastin
Thermodynamic Properties of Water
Single Gas Phase (Water Vapor):
• Can be treated like an ideal gas when it
exists in the absence of liquid water or ice
(i.e. like a homogeneous closed system):
p v  ρ v R v Tv
pv = Partial pressure of water vapor
(called vapor pressure)
ρv = Density of water vapor (or vapor density)
( The mass of the H2O molecules )
( per unit volume
)
= mv/Vv
Tv = Temperature of the water vapor
Rv = Gas constant for water vapor
( Based on the mean molecular weights )
( of the constituents in water vapor
)
= 461 J / kg K
Thermodynamics
M. D. Eastin
Thermodynamic Properties of Water
Single Gas Phase (Water Vapor):
• When only water vapor is present, we
can apply the first and second laws of
thermodynamics just like we did for
parcels of dry air
p v  ρ v R v Tv
dq  c v dT  pd 
ds 
dq rev
T
Thermodynamics
M. D. Eastin
Thermodynamic Properties of Water
Multiple Phases:
• Can NOT be treated like an ideal gas
when water vapor co-exists with either
liquid water, ice, or both:
p v  ρ v R v Tv
Liquid Water
pw, Tw, Vw, mw
Open sub-system
p w  ρ w R w Tw
Water Vapor
• This is because the two sub-systems
can exchange mass between each
other when an equilibrium exists
pv, Tv, Vv, mv, Rv
Open sub-system
• This violates the Ideal Gas Law
Thermodynamics
M. D. Eastin
Water in Equilibrium
Multiple Phases:
• When an equilibrium exists, the thermodynamic properties
of each phase are equal:
Vapor and Liquid
Vapor and Ice
pv, Tv
pv, Tv
pw, Tw
pi, Ti
Thermodynamics
pv  pw
pv  pi
Tv  Tw
T v  Ti
M. D. Eastin
Water in Equilibrium
An Example: Saturation
• Assume we have a parcel of dry air
located above liquid water
• Closed system
• Air is initially “unsaturated”
• System is not at equilibrium
Dry Air
(no water)
Liquid Water
Thermodynamics
M. D. Eastin
Water in Equilibrium
An Example: Saturation
• After a short time…
• Molecules in the liquid are in constant
motion (have kinetic energy)
• The motions are “random”, so some
molecules are colliding with each other
• Some molecules near the surface gain
velocity (or kinetic energy) through
collisions
• Fast moving parcels (with a lot of
kinetic energy) leave the liquid water
at the top surface → vaporization
Thermodynamics
M. D. Eastin
Water in Equilibrium
An Example: Saturation
• Soon there are a lot of water molecules
in the air (in vapor form)…
• The water molecules in the air make
collisions as well
• Some collisions result in slower moving
(or lower kinetic energy) molecules
• The slower water molecules return to
the water surface → condensation
Thermodynamics
M. D. Eastin
Water in Equilibrium
An Example: Saturation
 Eventually, the rate of condensation
equals the rate of evaporation
Rate of
Condensation
=
Rate of
Evaporation
 We have reached “Equilibrium”
Thermodynamics
M. D. Eastin
Water in Equilibrium
Three Standard Equilibrium States:
Vaporization:
Gas ↔ Liquid
Fusion:
Liquid ↔ Ice
Sublimation:
Gas ↔ Solid
p (mb)
C
221000
Liquid
Solid
• Each of these equilibrium states
occur at certain temperatures
and pressures
• Thus we can construct an
equilibrium phase change
graph for water
Thermodynamics
1013
6.11
T
Vapor
0
100
374
T (ºC)
M. D. Eastin
Water in Equilibrium
One Unique Equilibrium State:
• It is possible for all three phases
to co-exist in an equilibrium at a
single temperature and pressure
p (mb)
C
221000
• Called the Triple Point
Liquid
pv  pw  pi
T v  T w  Ti
Solid
1013
6.11
T
Vapor
p  6.11 mb
T  273.16 K
0
Thermodynamics
100
374
T (ºC)
M. D. Eastin
Water in Equilibrium
Critical Point:
• Thermodynamic state in which
liquid and gas phases can
co-exist in equilibrium at the
highest possible temperature
p (mb)
C
221000
Tc  374 C

Liquid
p c  221,000 mb
Solid
• Above this temperature, water
can NOT exist in the liquid phase
1013
6.11
T
Vapor
Other Atmospheric Gases:
O2

N2

Thermodynamics
Tc   119 C

0
100
374
T (ºC)
Tc   147 C

M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
P
(mb)
Liquid
C
221,000
Tc = 374ºC
Vapor
Liquid
and
Vapor
T1
Solid
6.11
T
Solid
and
Vapor
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
Vapor Phase (A → B)
• Behaves like an ideal gas
P
(mb)
p v  ρ v R v Tv
Liquid
C
221,000
Tc =
374ºC
• Decrease in volume
• Increase in pressure
• Heat Removed
B
Solid
6.11
Liquid
and
Vapor
Solid
and
Vapor
Vapor
A
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
Liquid and Vapor Phase (B → B’)
• Small change in
volume causes
condensation
• Some liquid water
begins to form
P
(mb)
Liquid
C
221,000
• No longer behaves like
an ideal gas
Tc =
374ºC
B’ B
Solid
6.11
Liquid
and
Vapor
Solid
and
Vapor
Vapor
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
Liquid and Vapor Phase (B’ → B”)
• Condensation occurs due
to a decrease in volume
• Constant temperature
• Constant pressure
P
(mb)
Liquid
C
221,000
• Water vapor pressure
is at equilibrium
Tc =
374ºC
B” B’
Solid
6.11
Liquid
and
Vapor
Solid
and
Vapor
Vapor
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
Liquid and Vapor Phase (B” → C)
• All the vapor has condensed
into liquid water
P
(mb)
Liquid
C
221,000
C
Solid
6.11
Tc =
374ºC
B”
Liquid
and
Vapor
Solid
and
Vapor
Vapor
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase Changes on P-V Diagrams:
Liquid Phase (C → D)
• Small changes in volume
produce large increases
in pressure
P
(mb)
Liquid
• Liquid water is virtually
incompressible
C
221,000
D
Tc =
374ºC
Vapor
C
Solid
6.11
Liquid
and
Vapor
Solid
and
Vapor
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Equilibrium Phase-Change Range:
• The range of volumes for
which equilibrium occurs
decreases with increasing
temperature
P
(mb)
Liquid
C
221,000
Tc =
374ºC
Vapor
Solid
T1
T
6.11
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Amagat-Andrews Diagram
Critical Point:
• Maximum temperature
at which condensation
(or vaporization) can
occur
• Water vapor obeys the
Ideal Gas Law at
higher temperatures
P
(mb)
Liquid
221,000
Tc =
374ºC
Vapor
Tc  374 C

p c  221,000 mb
C
Solid
6.11
Liquid
and
Vapor
Solid
and
Vapor
T1
T
Tt = 0ºC
V
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
Homogeneous System:
• Vapor only
• Behaves like Ideal Gas
273K 373K
Isobaric Process
• Heat (dQ) added or removed
from the system
• Temperature changes
• Volume changes
p
dQ
dQ  mc p dT  Vdp
V
dQ  mc p dT
p v  ρ v R v Tv
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
Heterogeneous System:
• Liquid and Vapor
Isobaric Process
• Heat (dQ) added or removed
from the system
• Temperature constant
• Volume changes
P
(mb)
Liquid
C
Tc
dQ
Vapor
dQ
Solid
T1
T
dQ
Tt
V
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
Definition of Latent Heat (L):
• Heat absorbed (or given away)
during an isobaric and isothermal
phase change
L  dQ
• The heat is needed to form (or
(results from the breaking of)
the molecular bonds that hold
water molecules together
P
(mb)
Liquid
C
Tc
dQ
Vapor
dQ
Solid
T1
T
dQ
Tt
V
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
Definition of Latent Heat (L):
• Heat absorbed (or given away)
during an isobaric and isothermal
phase change
• Magnitude varies with temperature
P
(mb)
Liquid
• However, the range of variation
is very small for the range of
pressures and temperatures
observed in the troposphere
C
Tc
L
Vapor
L
Solid
T1
T
• Assumed constant in practice
L
Tt
L  dQ  constant
V
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
The Different Latent Heats:
Vaporization
Condensation
(Lv or lv)
Gas
Liquid
Solid
Fusion
(Lf or lf)
Thermodynamics
Sublimation
(Ls or ls)
Values for lv, lf, and ls
are given in Table A.3
of the Appendix as a
function of temperature
M. D. Eastin
Latent Heats during Phase Changes
Heat is Absorbed (dQ > 0):
Vaporization
Condensation
(Lv or lv)
Gas
Liquid
Sublimation
(Ls or ls)
Solid
Fusion
(Lf or lf)
Thermodynamics
M. D. Eastin
Latent Heats during Phase Changes
Heat is Released (dQ < 0):
Vaporization
Condensation
(Lv or lv)
Gas
Liquid
Sublimation
(Ls or ls)
Solid
Fusion
(Lf or lf)
Thermodynamics
M. D. Eastin
Phase of Water and Latent Heats
Summary:
• Review of Systems
• Thermodynamic Properties of Water
• Multiple phases
• Water in Equilibrium
• Equilibrium Phase Changes
• Amagat-Andrew Diagrams
• Latent Heats for Equilibrium Phase Changes
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