metr3210 clausius-clapeyron

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

Fusi

on

Vapo

rizat

ion

T

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

Tdqds 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

Fusi

on

Vapo

rizat

ion

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 ACA 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 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

Fusi

on

Vapo

rizat

ion

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

Satu

ratio

n va

por p

ress

ure

A, D

B, C

Volume

T2

T1esw1

esw2

Satu

ratio

n va

por p

ress

ure

A D

B C

Isothermal processAdiabatic 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

TTT

QQQ

where: Q1 > 0 and Q2 < 0

21

NET

1

1

T-TW

TQ

Volume

T2

T1esw1

esw2

Satu

ratio

n va

por p

ress

ure

A D

B C

Isothermal process

Adiabatic process

WNET

Q1

Q2

Thermodynamics M. D. Eastin

Volume

T2

T1esw1

esw2Sa

tura

tion

vapo

r pre

ssur

eA 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-TW

TQ

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 liquiddm = total mass converted from vapor to liquid

Clausius-Clapeyron Equation

dmααdV wv

sw2sw1sw eede

21

NET

1

1

T-TW

TQ

dpdVWNET

Volume

T2

T1esw1

esw2Sa

tura

tion

vapo

r pre

ssur

eA 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-TW

TQ

dTdedmαα

TL swwvv

wv

vsw

ααTdTde

l

Temperature

T2 T1

esw1

esw2

Satu

ratio

n va

por p

ress

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

TΔdTdps l

Sublimatio

n

Fusi

on

Vapo

rizat

ion

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

TdT

Rede l

wv

vsw

ααTdTde

l

T

T 2v

ve

esw

sw

0

sw

s0 TdT

Rede 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 TdT

Rede l

T1

T1

Reeln

0v

v

s0

sw l

T(K)1

273.151

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

273.151

Rexp11.6(mb)e

v

vsw

l

)(ln09.5

)(680849.53exp11.6(mb)esw KTKT

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 KTKT

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

ααTdTde

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 Kmb36.21TTee

refrefsw T

eT

36.21esw

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

Fusi

on

Vapo

rizat

ion

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

Fusi

on

Vapo

rizat

ion

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

TdTdes l

iv

ssi

ααTdTde

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

273.151

Rexp11.6(mb)e

v

ssi

l

)(ln555.0

)(629316.26exp11.6(mb)esi KTKT

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

ααTdTdp

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.