ThermodynamicsM. D. Eastin Phase of Water and Latent Heats We must begin to account for the...

Thermodynamics M. D. Eastin

Phase of Water and Latent Heats

We must begin to account for the thermodynamics of water

Our atmosphere contains dry air and water vaporClouds contain dry air, water vapor, liquid water, and ice


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



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

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?

Review of Systems

Dry Air

ClosedSystem

p, T, V, m, Rd

TRpα d

pd dTcdq v

T

dqds rev


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

Review of Systems

Water VaporPv, Tv, Vv, mv

Liquid WaterPw, Tw, Vw, mw

Ice WaterPi, Ti, Vi, mi


• Our atmosphere is a heterogeneous closed system consisting of multiple sub-systems

• Very complex…we come back to it later

Review of Systems

Water Vapor

pv, Tv, Vv, mv, Rv

Open sub-system

Ice Waterpi, Ti, Vi, mi

Open sub-system

Dry Air(gas)

p, T, V, md, Rd

Closed sub-system

Liquid Waterpw, Tw, Vw, mw

Open sub-system

Energy Exchange

Mass Exchange


• For now, let’s focus our attention on the one component heterogeneous system “water” comprised of vapor and one other phase (liquid or ice)

Review of Systems

Water Vapor

pv, Tv, Vv, mv, Rv

Open sub-system

Ice Waterpi, Ti, Vi, mi

Open sub-system

Dry Air(gas)

p, T, V, md, Rd

Closed sub-system


Open sub-system

Energy Exchange

Mass Exchange


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):

Thermodynamic Properties of Water

vvvv TRρp

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


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


pd dTcdq v

T

dqds rev

vvvv TRρp


Multiple Phases:

• Can NOT be treated like an ideal gas when water vapor co-exists with either liquid water, ice, or both:

• This is because the two sub-systems can exchange mass between each other when an equilibrium exists

• This violates the Ideal Gas Law


Water Vapor

pv, Tv, Vv, mv, Rv

Open sub-system


Open sub-system

vvvv TRρp wwww TRρp


Multiple Phases:

• When an equilibrium exists, the thermodynamic properties of each phase are equal:

Water in Equilibrium

pw, Tw

pv, Tv

Vapor and Liquid Vapor and Ice

pv, Tv

pi, Ti

wv pp

wv TT iv TT iv pp


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



• 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




• 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




Eventually, the rate of condensation equals the rate of evaporation

Rate of Rate of Condensation = Evaporation

We have reached “Equilibrium”



Three Standard Equilibrium States:

Vaporization: Gas ↔ Liquid

Fusion: Liquid ↔ Ice

Sublimation: Gas ↔ Solid

• Each of these equilibrium states occur at certain temperatures and pressures

• Thus we can construct an equilibrium phase change graph for water


Sublimatio

n

Fus

ion

Vap

oriz

atio

n

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid


One Unique Equilibrium State:

• It is possible for all three phases to co-exist in an equilibrium at a single temperature and pressure

• Called the Triple Point


iwv ppp

iwv TTT

mb 6.11p

K273.16T

Sublimatio

n

Fus

ion

Vap

oriz

atio

n

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid


Critical Point:

• Thermodynamic state in which liquid and gas phases can co-exist in equilibrium at the highest possible temperature

• Above this temperature, water can NOT exist in the liquid phase

Other Atmospheric Gases:


mb 221,000pc

C374Tc

Sublimatio

n

Fus

ion

Vap

oriz

atio

n

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

C119TO c2

C147TN c2


Equilibrium Phase Changes on P-V Diagrams:

Amagat-Andrews Diagram

T

C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc = 374ºC

T1

6.11

221,000



Vapor Phase (A → B)

• Behaves like an ideal gas

• Decrease in volume• Increase in pressure• Heat Removed


C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc =374ºC

T1

6.11

221,000

T

A

B

vvvv TRρp



Liquid and Vapor Phase (B → B’)

• Small change in volume causes condensation

• Some liquid water begins to form

• No longer behaves like an ideal gas


C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc =374ºC

T1

6.11

221,000

T

B’ B



Liquid and Vapor Phase (B’ → B”)

• Condensation occurs due to a decrease in volume

• Constant temperature• Constant pressure

• Water vapor pressure is at equilibrium


C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc =374ºC

T1

6.11

221,000

T

B’B”



Liquid and Vapor Phase (B” → C)

• All the vapor has condensed into liquid water


C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc =374ºC

T1

6.11

221,000

T

C B”



Liquid Phase (C → D)

• Small changes in volume produce large increases in pressure

• Liquid water is virtually incompressible


C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Liquidand

Vapor

Solidand

Vapor

Tc =374ºC

T1

6.11

221,000

T

C

D



C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Tc =374ºC

T1

6.11

221,000

T

Equilibrium Phase-Change Range:

• The range of volumes for which equilibrium occurs decreases with increasing temperature


Critical Point:

• Maximum temperature at which condensation (or vaporization) can occur

• Water vapor obeys the Ideal Gas Law at higher temperatures


C374Tc

mb 221,000pc

C

V

P(mb)

Vapor

Solid

Tt = 0ºC

Liquid

Tc =374ºC

T1

6.11

221,000

T

Solidand

Vapor

Liquidand

Vapor


Homogeneous System:

• Vapor only• Behaves like Ideal Gas

Isobaric Process

• Heat (dQ) added or removed from the system• Temperature changes• Volume changes

Latent Heats during Phase Changes

Vdp dTmcdQ p

dTmcdQ p

vvvv TRρp

p

V

273K 373K

dQ


Heterogeneous System:

• Liquid and Vapor

Isobaric Process

• Heat (dQ) added or removed from the system• Temperature constant• Volume changes


C

V

P(mb)

Vapor

Solid

Tt

Liquid

Tc

T1

T

dQ

dQ

dQ


Definition of Latent Heat (L):

• Heat absorbed (or given away) during an isobaric and isothermal phase change

• The heat is needed to form (or (results from the breaking of) the molecular bonds that hold water molecules together


C

V

P(mb)

Vapor

Solid

Tt

Liquid

Tc

T1

T

dQ

dQ

dQ

dQ L


Definition of Latent Heat (L):

• Heat absorbed (or given away) during an isobaric and isothermal phase change

• Magnitude varies with temperature

• However, the range of variation is very small for the range of pressures and temperatures observed in the troposphere

• Assumed constant in practice


C

V

P(mb)

Vapor

Solid

Tt

Liquid

Tc

T1

T

L

L

L

constantdQ L


The Different Latent Heats:


FusionFusion(L(Lf f or lor lff))

SublimationSublimation(L(Ls s or lor lss))

VaporizationVaporizationCondensation Condensation

(L(Lv v or lor lvv))

SolidSolidLiquidLiquid

GasGas

Values for lv, lf, and ls are given in Table A.3 of the Appendix as a function of temperature


Heat is Absorbed (dQ > 0):







GasGas


Heat is Released (dQ < 0):







GasGas


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



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.

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