Clausius-Clapeyron Equation - Dronacharyaggn.dronacharya.info/MEDept/Downloads/QuestionBank/IIIsem/th_21.pdf · Assumption: Our system consists of liquid water in equilibrium with

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

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

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 anyequilibrium state of water as a functionof temperature and pressure.

• Accounts for phase changes at eachequilibrium state (each temperature)


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 forwhich the Clausius-Clapeyron equation is derived in the following slides

Mathematical Derivation:

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

• We will return to the Carnot Cycle…


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


• Recall for the Carnot Cycle:

• If we re-arrange and substitute:


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


WNET

Q1

Q2

Volume

T2

T1esw1

esw2Sa

tura

tion

vapo

r pre

ssur

eA D

B C


WNET

Q1

Q2


Recall:

• During phase changes, Q = L

• Since we are specifically workingwith vaporization in this example,

• Also, let:


21

NET

1

1

T-TW

TQ

v1 LQ

TT1

dTTT 21


Recall:

• The net work is equivalent to thearea enclosed by the cycle:

• The change in pressure is:

• The change in volume of our system ateach temperature (T1 and T2) is:

where: αv = specific volume of vaporαw = specific volume of liquiddm = total mass converted from

vapor to liquid


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


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 ofvaporization (lv = Lv /dm) to obtain:

Clausius-Clapeyron Equation for the equilibrium vapor pressurewith respect to liquid water


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

General Form:

• Relates the equilibrium pressure between two phases to the temperatureof the heterogeneous system

where: T = Temperature of the system l = Latent heat for given phase change

dps = Change in system pressure at saturationdT = Change in system temperatureΔα = Change in specific volumes between

the two phases


TΔdTdps l

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

TdT

Rede l

wv

vsw

ααTdTde

l

T

T 2v

ve

esw

sw

0

sw

s0 TdT

Rede 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 TdT

Rede l

T1

T1

Reeln

0v

v

s0

sw l

T(K)1

273.151

Rexp11.6(mb)e

v

vsw

l


A more accurate form of the above equation can be obtained when we do notassume lv is constant (recall lv is a function of temperature). See your book forthe derivation of this more accurate form:


T(K)1

273.151

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 functionof atmospheric pressure when the saturated with respect to water (esw)


wv

vsw

ααTdTde

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 pointat ~1005 mb is 100ºC

• Let this be our reference point:

Tref = 100ºC = 373.15 Kesw-ref = 1005 mb

• Let esw and T represent thevalues on Mt. Mitchell:

esw = 750 mb

T = 366.11 KT = 93ºC (boiling point temperature on Mt. Mitchell)


1

ref

refswsw Kmb36.21TTee

refrefsw T

eT

36.21esw

1sw Kmb36.21dT

de

Equilibrium with respect to Ice:

• We will know examine the equilibriumvapor pressure for a heterogeneoussystem containing vapor and ice


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 theClausius-Clapeyron equation

• Make the appropriate substitution forthe two phases (vapor and ice)

Clausius-Clapeyron Equation for the equilibrium vaporpressure with respect to ice


T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

TdTdes l

iv

ssi

ααTdTde

l

Application: Saturation vapor pressure of ice for a given temperature

Following the same logic as before, we can derive the following equation forsaturation with respect to ice

A more accurate form of the above equation can be obtained when we do notassume ls is constant (recall ls is a function of temperature). See your book forthe derivation of this more accurate form:


T(K)1

273.151

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 theappropriate substitutions for our two phases (liquid water and ice)

At typical atmospheric conditions near the melting point:

T = 0ºC = 273 Kls = 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 functionof pressure when liquid water is saturated with respect to ice (pwi)


iw

fwi

ααTdTdp

l

1wi Kmb135,038dT

dp

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Clausius-Clapeyron Equation - Dronacharyaggn.dronacharya.info/MEDept/Downloads/QuestionBank/IIIsem/th_21.pdf · Assumption: Our system consists of liquid water in equilibrium with