Clausius-Clapeyron Equation · 2020-02-19 · Clausius-Clapeyron equation •Make the appropriate substitution for the two phases (vapor and ice) Clausius-Clapeyron Equation for the

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

T

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


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


Solid Phase (ice):




pd dTcdq v

T

dqds rev

vvvv TRρp


Heterogeneous Systems (multiple phases):

Liquid Water and Vapor:

• Equilibrium state

• Saturation

• Vaporization / Condensation




pw, Tw

pv, Tv

wv pp

wv TT

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)


• Isobaric

• Isothermal

• Volume changes


wv pp wv TT C

V

P

(mb)

Vapor

Solid

Tt =

0ºC

Liquid

Liquid

and

Vapor

Solid

and

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 a 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 Clapeyron

1799-1864

French

Engineer / Physicist

Expanded on Carnot’s work

Rudolf Clausius

1822-1888

German

Mathematician / Physicist

“Discovered” the Second Law

Introduced 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)


T

C

T (ºC)

es (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

V

es

(mb)

Vapor

Liquid

Liquid

and

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:


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

ααT

l

dT

de

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

lx = Latent heat for given phase change

desx = Change in system pressure at saturation

dT = Change in system temperature

Δα = Change in specific volumes between

the two phases


TΔdT

desx xl

T

C

T (ºC)

es (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

2

v

v

sw

sw

T

dT

R

l

e

de

wv

vsw

ααT

l

dT

de

T

T 2

v

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 2

v

ve

esw

sw

0

sw

s0 T

dT

R

l

e

de

T

1

T

1

R

l

e

eln

0v

v

s0

sw

T(K)

1

273.15

1

R

l(mb)e

v

vsw exp11.6



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

R

l(mb)e

v

vsw exp11.6

)(ln09.5

)(

680849.53exp11.6 KT

KT(mb)esw



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 temperature

(near sea level)

esw = 1005 mb


T(K)

1

273.15

1

R

l(mb)e

v

vsw exp11.6


Application: Boiling Point of Water

At typical atmospheric conditions near the boiling point:

T = 100ºC = 373 K

lv = 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

ααT

l

dT

de

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 K

esw-ref = 1005 mb

• Let esw and T represent the

values on Mt. Mitchell:

esw = 750 mb

T = 366.11 K

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


1

ref

refswswKmb36.21

TT

ee

ref

refswswT

36.21

eeT

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


T

C

T (ºC)

es (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

C

V

es

(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)


for the equilibrium vapor

pressure with respect to ice


T

C

T (ºC)

es (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

TdT

desx xl

iv

ssi

ααT

l

dT

de


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

R

l(mb)e

v

ssi exp11.6

)(ln555.0

)(

629316.26exp11.6 KT

KT(mb)esi


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 K

lf = 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 (ewi)


iw

fwi

ααT

l

dT

de

1wi Kmb135,038dT

de


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



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.

Documents

Clausius-Clapeyron Equation · 2020-02-19 · Clausius-Clapeyron equation •Make the appropriate substitution for the two phases (vapor and ice) Clausius-Clapeyron Equation for the