Presentation is loading. Please wait.

Presentation is loading. Please wait.

Clausius-Clapeyron Equation

Published byJunior Taylor Modified over 6 years ago

Similar presentations

Presentation on theme: "Clausius-Clapeyron Equation"— Presentation transcript:

1 Clausius-Clapeyron Equation
And you thought you were already saturated!

2 Reading Hess Tsonis Wallace & Hobbs Bohren & Albrecht pp 39 - 42

3 Objectives Be able to describe the physical processes of net evaporation, net condensation and equilibrium Be able to describe the changes that occur to water substance as pressure and volume changes on a phase diagram

4 Objective Be able to state the only factor that determines the rate of evaporation from a water surface Be able to state the factor that determines the rate of condensation of water molecules from the air

5 Objective Be able to determine if net evaporation, net condensation or equilibrium exists given a temperature and pressure on a pressure vs. temperature diagram for water substance

6 Objective Be able to perform calculations using the Clausius Clapyeron Equation, including the determination of saturation vapor pressure. Be able to describe the change in boiling point temperature with pressure

7 Objective Be able to provide the definition of dew point and frost point from memory Be able to provide the definition of relative humidity from memory

8 Objective Be able to calculate relative humidity
Be able to distinguish the difference between WMO and traditional defintions of relative humidity

9 Water at Equilibrium Twater Rate of evaporation Constant
Function of water temperature Twater

10 Water at Equilibrium Rate of Condensation Variable
Function of water vapor mass in air

11 Water at Equilibrium At Equilibrium Rate of evaporation condensation =

12 Water at Equilibrium Twater At Equilibrium Evaporation = f(T)
Rate of evaporation is a function of temperature Evaporation = f(T) Twater Rate of evaporation Rate of condensation =

13 Water at Equilibrium Tair Twater At Equilibrium Condensation = f(T)
Rate of condensation also a function of temperature. Twater Tair Condensation = f(T) Rate of evaporation condensation =

14 Water at Equilibrium Tair Tair = Twater Twater At Equilibrium Rate of
evaporation condensation =

15 Water at Equilibrium Water Vapor Partial Pressure
Function of mass of water vapor

16 Rate of Condendation = Rate of Evaporation
Water at Equilibrium Equilibrium Curve Rate of Condendation = Rate of Evaporation es Equilibrium es = water vapor pressure at equilibrium (saturation) Pressure Temperature

17 Condensation Water Vapor Pressure > Equilibrium es e > es e
Temperature

18 Condensation Water Vapor Pressure > Equilibrium Net Condensation es
e > es e Pressure Temperature

19 Condensation = Evaporation
Water Vapor Pressure > Equilibrium Condensation = Evaporation es e = es Pressure e Temperature

20 Evaporation Water Vapor Pressure < Equilibrium es Net Evaporation
e < es e Temperature

21 Evaporation Water Vapor Pressure < Equilibrium es Net Evaporation
e < es e Temperature

22 Condensation = Evaporation
Water Vapor Pressure < Equilibrium Condensation = Evaporation es e = es Pressure e Temperature

23 Water at Equilibrium Equilibrium Curve Values es Pressure Temperature
1013 mb Pressure 6 mb 0oC 100oC Temperature

24 Water at Equilibrium Equilibrium Curve
Where do these numbers come from? es 1013 mb Pressure 6 mb 0oC 100oC Temperature

25 Famous Clip Art Scientists!
Water at Equilibrium Equilibrium Curve Where do these numbers come from? Clausius-Clapeyron Equation Jaimie Clapeyron Thelma Clausius Famous Clip Art Scientists!

26 Clausius-Clapeyron Equation
Rudolf Clausius 1822 – 1888 German Mathematical Physicist Emile Clapeyron French Engineer

27 Clausius-Clapeyron Equation
The amount of moisture in the air (at equilibrium) depends on Temperature of Air/Water Water Vapor Pressure of Air Volume (or Specific Volume) of Air Equilibrium es Pressure Constant Volume Temperature P-T Diagram

28 Clausius-Clapeyron Equation
Our discussion assumed constant volume Temperature Pressure es Equilibrium P-T Diagram Constant Volume

29 Clausius-Clapeyron Equation
The volume of water substance changes during phase changes Gas Solid Liquid

30 Clausius-Clapeyron Equation
Pressure-Volume Diagram Ideal Gas Pressure Volume

31 Clausius-Clapeyron Equation
Pressure-Volume Diagram Ideal Gas Pressure Volume

32 Clausius-Clapeyron Equation
Pressure-Volume Diagram Vapor Water T3 Pressure (e) O2 … -119oC N2 … -147oC H oC Water & Vapor Vapor T2 T1 Volume (V)

33 Pressure-Volume Diagram
Also known as Phase Diagram Water Vapor Pressure (e) Water & Vapor T2 T1 Volume (V)

34 Pressure-Volume Diagram
Isotherms T2>T1 Water Vapor Pressure (e) Water & Vapor T2 T1 Volume (V)

35 Pressure-Volume Diagram
Vapor Phase Ideal Gas Law Decrease Volume Increase Pressure Water Vapor Pressure (e) Water & Vapor T2 B T1 A Volume (V)

36 Pressure-Volume Diagram
Water & Vapor Phase (B) Slight Change in Volume Causes Condensation Water Vapor Pressure (e) Water & Vapor T2 B T1 Volume (V)

37 Pressure-Volume Diagram
Water & Vapor Phase (B to C) Condensation Volume Decreasing Constant Pressure at Constant Temp. Water Vapor Pressure (e) Water & Vapor T2 C B T1 Volume (V) Condensation

38 Pressure-Volume Diagram
Water & Vapor Phase (B to C) Condensation Water Vapor Pressure is at Equilibrium (es) Water Vapor Pressure (e) Water & Vapor T2 C B T1 Volume (V)

39 Pressure-Volume Diagram
Water Phase (C) All Water Vapor Has Condensed Water Vapor Pressure (e) Water & Vapor T2 C T1 Volume (V)

40 Pressure-Volume Diagram
Water Phase (C to D) Volume Decreases Little Virtually Incompressible D Water Vapor Pressure (e) Water & Vapor T2 C T1 Volume (V)

41 Clausius-Clapeyron Equation
We’ve also ignored the heat required for phase change + =

42 Clausius-Clapeyron Equation
Let’s investigate all these variables using the Carnot Cycle! Vapor Water & Volume (V) Pressure (e) T1 T2

43 Clausius-Clapeyron Equation
Reversible cycle Water B C Vapor Pressure (e) Water & Vapor D T2 A T1 Specific Volume (a)

44 Clausius-Clapeyron Equation
At Point A Water Vapor es = Pressure T1 = Temperature aw = Volume C B Water & Vapor T2 Pressure (e) A D es T1 es,T1 , aw Specific Volume (a)

45 Clausius-Clapeyron Equation
At Point B es+Des T1+DT aw +Daw Water Vapor Vapor Pressure es+Des Temperature T2 = T1+DT Volume aw +Daw es +Des C B Water & Vapor T2 Pressure (e) A D es T1 es,T1 , aw Specific Volume (a)

46 Clausius-Clapeyron Equation
At Point C es+Des T1+DT av +Dav Water Vapor Pressure es+Des Temperature T2 = T1+DT Volume av +Dav es +Des C B Vapor Water & Vapor T2 Pressure (e) A D es T1 es,T1 , av Specific Volume (a)

47 Clausius-Clapeyron Equation
At Point D es+Des T1+DT av +Dav Water Vapor es = Pressure T1 = Temperature av = Volume es +Des C B Vapor Water & Vapor T2 Pressure (e) A D es T1 es,T1 , av Specific Volume (a)

48 Clausius-Clapeyron Equation
First Law of Thermodynamics Second Law of Thermodynamics

49 Clausius-Clapeyron Equation
Combine the equations Integrate over the closed path

50 Clausius-Clapeyron Equation
For a cyclic process So …..

51 Clausius-Clapeyron Equation
es+Des T1+DT aw +Daw es+Des T1+DT av +Dav Water Vapor Apply to the Phase Diagram es +Des C B Vapor Water & Vapor Des T2 Pressure (e) A es D av - aw T1 es,T1 , aw es,T1 , av es,T1 , av es = Pressure Specific Volume (a)

52 Clausius-Clapeyron Equation
The area ABCD is similar to … So … Water Vapor es +Des C B Vapor Water & Vapor Des T2 Pressure (e) A es D av - aw T1 Specific Volume (a)

53 Clausius-Clapeyron Equation
Let’s evaluate the left hand side But the exact differential

54 Clausius-Clapeyron Equation
and So …

55 Clausius-Clapeyron Equation
From A to B esA+Des T1+DT aw +Daw es+Des T1+DT av +Dav Water Vapor C B Vapor Water & Vapor T2 sw = entropy of water Pressure (e) A D T1 es,T1 , aw es,T1 , av es,T1 , av Specific Volume (a)

56 Clausius-Clapeyron Equation
From B to C Isothermal esA+Des T1+DT aw +Daw es+Des T1+DT av +Dav Water Vapor C B Vapor Water & Vapor T2 Pressure (e) A D T1 es,T1 , aw es,T1 , av es,T1 , av Specific Volume (a)

57 Clausius-Clapeyron Equation
From C to D esA+Des T1+DT aw +Daw es+Des T1+DT av +Dav Water Vapor C B Vapor Water & Vapor T2 Pressure (e) sv = entropy of vapor A D T1 es,T1 , aw es,T1 , av es,T1 , av Specific Volume (a)

58 Clausius-Clapeyron Equation
From D to A Isothermal esA+Des T1+DT aw +Daw es+Des T1+DT av +Dav Water Vapor C B Vapor Water & Vapor T2 Pressure (e) A D T1 es,T1 , aw es,T1 , av es,T1 , av Specific Volume (a)

59 Clausius-Clapeyron Equation
After substitution, the left hand side is

60 Clausius-Clapeyron Equation
Rearrange terms and take the limit

61 Clausius-Clapeyron Equation
Water Vapor C B Vapor Water & Vapor T2 During the isothermal process Pressure (e) sw sV A D dq = LV T1 Specific Volume (a) Lv = Latent Heat of Vaporization/Condensation

62 Clausius-Clapeyron Equation
Substitute Water Vapor C B Vapor Water & Vapor T2 Pressure (e) sw sV A D dq = LV T1 Specific Volume (a)

63 Clausius-Clapeyron Equation
Specific Volume of Liquid Water (aw ) Rearrange terms If av > aw, then ... Specific Volume of Liquid Water (av )

64 Clausius-Clapeyron Equation
es Describes the change of the equilibrium water vapor pressure curve versus temperature Pressure Constant Volume Temperature

65 Clausius-Clapeyron Equation
Ideal Gas Law Substitute

66 Clausius-Clapeyron Equation
Rearrange terms Integrate with reference es = 273K

67 Clausius-Clapeyron Equation
Integrate Lv = Latent Heat of Vaporization = 2.5x106 J kg-1 Rv = Gas Constant for Water Vapor = 461 J K-1 kg-1

68 Clausius-Clapeyron Equation
Substitute for Lv and Rv Rearrange Terms

69 Clausius-Clapeyron Equation
Simplify Simplify more ....

70 Clausius-Clapeyron Equation
Raise to e es = equilibrium vapor pressure (in mb) T = temperature (in K) Good approximation for low values of temperature and pressure

71 Magnus Formula A better approximation
es = equilibrium vapor pressure (in mb) T = temperature (in K) Accounts for variation in Latent Heat

72 Goff-Gratch Formula Yet another approximation
Used in Smithsonian Meteorological Tables

73 Equilibrium with Ice Equilibrium Curve Water Vapor vs. Liquid Water es
Temperature Pressure Equilibrium es

74 Equilibrium with Ice What about water vapor vs. ice?

75 Equilibrium with Ice Phase Diagram Pressure (e) Volume (V) Water Water
& Vapor Ice Vapor 0oC Ice & Vapor T Volume (V)

76 Equilibrium with Ice Clausius-Clapeyron Equation
Use the Latent Heat of Sublimation Ls = Latent Heat of Sublimation = x106 J kg-1 Rv = Gas Constant for Water Vapor = 461 J K-1 kg-1

77 Equilibrium with Ice Equilibrium Curve for Ice esw Pressure esi 0.01oC
Liquid Water esw Pressure Equilibrium with Ice esi 0.01oC Temperature

78 Equilibrium with Ice Supercooled Liquid Water (SLW) Absence of ice esw
Temperature Pressure Equilibrium with Liquid Water esw with Ice 0.01oC esi SLW

79 Boiling Point Point at which vapor pressure in the liquid is equal to the atmospheric pressure on the liquid surface

80 Boiling Point Heat Energizes Water Molecules Bonds Broken Vapor Phase

81 Boiling Point Temperature of Boiling Point Varies with Atmospheric Pressure Equilibrium with Liquid Water Pressure es 1000 mb Boiling Point 100oC Temperature

82 Boiling Point Temperature of Boiling Point Varies with Atmospheric Pressure Equilibrium with Liquid Water Pressure es 1000 mb 750 mb Boiling Point 95oC 100oC Temperature

83 Boiling Point Change in Boiling Points Can Be Calculated Using Clausius-Clapeyron Equation Equilibrium with Liquid Water Pressure es 1000 mb 750 mb Boiling Point 95oC 100oC Temperature

84 Boiling Point dT = change in boiling point
des = change in atmospheric pressure av = specific volume of vapor at boiling aw = specific volume of water at boiling

85 Moisture Variables We now have equilibrium curves for liquid water and ice Equilibrium with Liquid Water esw Pressure Equilibrium with Ice esi 0.01oC Temperature

86 Moisture Variables However, the atmosphere is often not in equilibrium (or saturated). es Pressure ‘Saturated’ eequilibrium ‘Less Than Saturated’ eatmosphere Temperature

87 Moisture Variables How close it the air to equilibrium? es Pressure
‘Saturated’ eequilibrium ‘Less Than Saturated’ eatmosphere Temperature

88 Moisture Variables Relative Humidity (RH) (Traditional)
ratio of the actual water vapor pressure (e) to the saturation (or equilibrium) vapor pressure (es)

89 Moisture Variables Relative Humidity (RH) (WMO)
ratio of the actual mixing ratio (w) to the saturation (or equilibrium) mixing ratio (ws)

90 Moisture Variables Relative Humidity (RH) Problem
They are not the same Slight difference

91 Moisture Variables Relative Humidity es Pressure RH = 100% esaturation
eatmosphere Temperature

92 Moisture Variables Dew Point (Td)
Temperature to which air must be cooled in order for it to become saturated with respect to liquid water at the initial pressure and mixing ratio Td

93 Moisture Variables Dew Point (Td)
Water vapor pressure and atmospheric pressure are constant Temperature Pressure es Tatmosphere esaturation RH = 100% Td Isobaric Cooling

94 Moisture Variables Frost Point (Tf)
Temperature to which air must be cooled in order for it to become saturated with respect to ice at the initial pressure and mixing ratio Temperatures less than 0oC Tf

95 Moisture Variables Frost Point (Tf)
Water vapor pressure and atmospheric pressure are constant Temperature Pressure es Tatmosphere esaturation RH = 100% Tf Isobaric Cooling esi

96

Download ppt "Clausius-Clapeyron Equation"

Similar presentations

Water in the Atmosphere

Water in the Atmosphere
Chapter 5 Atmospheric Moisture. The process whereby molecules break free of liquid water is known as evaporation. The opposite process is condensation,

Chapter 5 Atmospheric Moisture. The process whereby molecules break free of liquid water is known as evaporation. The opposite process is condensation,
Section 04 Thermodynamics Adiabatic Processes Lesson 10/11.

Section 04 Thermodynamics Adiabatic Processes Lesson 10/11.
Weather Water in the Atmosphere May 14, 2008. Precipitation Precipitation is any form of water that falls from a cloud. Water vapor is the source of all.

Weather Water in the Atmosphere May 14, Precipitation Precipitation is any form of water that falls from a cloud. Water vapor is the source of all.
Lecture 7 Water Vapor.

Lecture 7 Water Vapor.
Humidity First let us consider five humidity variables

Humidity First let us consider five humidity variables
MET 61 1 MET 61 Introduction to Meteorology MET 61 Introduction to Meteorology - Lecture 4 “Heat in the atmosphere” Dr. Eugene Cordero San Jose State University.

MET 61 1 MET 61 Introduction to Meteorology MET 61 Introduction to Meteorology - Lecture 4 “Heat in the atmosphere” Dr. Eugene Cordero San Jose State University.
* Reading Assignments: All sections of Chapter 5.

* Reading Assignments: All sections of Chapter 5.
Humidity. Water Vapor Can make up as little as 1/10 th of 1% to 4% of the atmosphere. Scientists agree that it is the most important atmospheric gas when.

Humidity. Water Vapor Can make up as little as 1/10 th of 1% to 4% of the atmosphere. Scientists agree that it is the most important atmospheric gas when.
Water Vapor and Humidity in the Atmosphere. Vapor Pressure The vapor pressure (e) is the pressure exerted by the water vapor molecules in the air. As.

Water Vapor and Humidity in the Atmosphere. Vapor Pressure The vapor pressure (e) is the pressure exerted by the water vapor molecules in the air. As.
Water in the Atmosphere Section 1 Changing Forms of Water Water in the atmosphere exists in three states, or phases. One phase is known as a gas called.

Water in the Atmosphere Section 1 Changing Forms of Water Water in the atmosphere exists in three states, or phases. One phase is known as a gas called.
THERMODYNAMIC PROPERTY RELATIONS

THERMODYNAMIC PROPERTY RELATIONS
+ Atmospheric Moisture Chapter 4. + Learning Targets LT #1: I can describe the movement of water through the hydrologic cycle. LT #2: I can summarize.

+ Atmospheric Moisture Chapter 4. + Learning Targets LT #1: I can describe the movement of water through the hydrologic cycle. LT #2: I can summarize.
Moisture Variables on Skew T Log P Diagram AOS 330 LAB 8.

Moisture Variables on Skew T Log P Diagram AOS 330 LAB 8.
Prof. Fred Remer University of North Dakota Phase Changes and Latent Heat Where’s the heat? Solid Liquid Gas.

Prof. Fred Remer University of North Dakota Phase Changes and Latent Heat Where’s the heat? Solid Liquid Gas.
Atmospheric Moisture Vapor pressure (e, Pa) The partial pressure exerted by the molecules of vapor in the air. Saturation vapor pressure (e s, Pa ) The.

Atmospheric Moisture Vapor pressure (e, Pa) The partial pressure exerted by the molecules of vapor in the air. Saturation vapor pressure (e s, Pa ) The.
ThermodynamicsM. D. Eastin Water Vapor in the Air How do we compute the dewpoint temperature, the relative humidity, or the temperature of a rising air.

ThermodynamicsM. D. Eastin Water Vapor in the Air How do we compute the dewpoint temperature, the relative humidity, or the temperature of a rising air.
The Second Law of Thermodynamics

The Second Law of Thermodynamics
Chapter 4: Applications of the First Law Different types of work: Configuration work: (reversible process) Dissipative work: (irreversible process) Adiabatic.

Chapter 4: Applications of the First Law Different types of work: Configuration work: (reversible process) Dissipative work: (irreversible process) Adiabatic.
HUMIDITY  Humidity is the amount of water vapor in the air. Water vapor is the gas phase of water and is invisible. Humidity indicates the likelihood.

HUMIDITY  Humidity is the amount of water vapor in the air. Water vapor is the gas phase of water and is invisible. Humidity indicates the likelihood.

Similar presentations

Ads by Google