Clausius-Clapeyron Equation

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


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



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


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


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)


Equilibrium Phase Changes:

Vapor → Liquid Water (Condensation):

• Equilibrium state (saturation)• Does not behave like an ideal gas• Isobaric• Isothermal• Volume changes


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


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


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


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


Who are these people?


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


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)


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


Mathematical Derivation:

Assumption: Our system consists of liquid water in equilibrium with water vapor (at saturation)

• We will return to the Carnot Cycle…


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



• Recall for the Carnot Cycle:

• If we re-arrange and substitute:


21NET QQW

1

21

1

21

T

TT

Q

QQ

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


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


Recall:

• During phase changes, Q = L

• Since we are specifically working with vaporization in this example,

• Also, let:


21

NET

1

1

T-T

W

T

Q

v1 LQ

TT1

dTTT 21



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


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



• 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


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


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


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)


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:


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



After integration we obtain:

After some algebra and substitution for es0 = 6.11 mb and T0 = 273.15 K we get:


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



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:


T(K)

1

273.15

1

Rexp11.6(mb)e

v

vsw

l

)(ln09.5

)(

680849.53exp11.6(mb)esw KT

KT



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


)(ln09.5

)(

680849.53exp11.6(mb)esw KT

KT


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)


wv

vsw

ααTdT

de

l

1sw Kmb36.21dT

de


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)


1

ref

refswsw Kmb36.21TT

ee

refrefsw T

eT

36.21

esw

1sw Kmb36.21dT

de


Equilibrium with respect to Ice:

• We will know examine the equilibrium vapor pressure for a heterogeneous system containing vapor and ice


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


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


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


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:


T(K)

1

273.15

1

Rexp11.6(mb)e

v

ssi

l

)(ln555.0

)(

629316.26exp11.6(mb)esi KT

KT


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)


iw

fwi

ααTdT

dp

l

1wi Kmb135,038dT

dp


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



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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Clausius-Clapeyron Equation