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

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

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

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

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

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

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

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

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

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

Water at Equilibrium At Equilibrium Rate of evaporation condensation =

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 =

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 =

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

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

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

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

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

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

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

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

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

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

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

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!

Clausius-Clapeyron Equation Rudolf Clausius 1822 – 1888 German Mathematical Physicist Emile Clapeyron 1799 - 1864 French Engineer

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

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

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

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

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

Clausius-Clapeyron Equation Pressure-Volume Diagram Vapor Water T3 Pressure (e) O2 … -119oC N2 … -147oC H20 .. 347oC Water & Vapor Vapor T2 T1 Volume (V)

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

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

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

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

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

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)

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

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)

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

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

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

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)

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)

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)

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)

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

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

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

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)

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)

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

Clausius-Clapeyron Equation and So …

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)

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)

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)

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)

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

Clausius-Clapeyron Equation Rearrange terms and take the limit

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

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

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

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

Clausius-Clapeyron Equation Ideal Gas Law Substitute

Clausius-Clapeyron Equation Rearrange terms Integrate with reference es = 6.11mb @ 273K

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

Clausius-Clapeyron Equation Substitute for Lv and Rv Rearrange Terms

Clausius-Clapeyron Equation Simplify Simplify more ....

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

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

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

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

Equilibrium with Ice What about water vapor vs. ice?

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

Equilibrium with Ice Clausius-Clapeyron Equation Use the Latent Heat of Sublimation Ls = Latent Heat of Sublimation = 2.834 x106 J kg-1 Rv = Gas Constant for Water Vapor = 461 J K-1 kg-1

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

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

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

Boiling Point Heat Energizes Water Molecules Bonds Broken Vapor Phase

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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