Clausius-Clapeyron Equation Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid 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 Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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 Thermodynamics M. D. Eastin

Review of Water Phases 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): Thermodynamics M. D. Eastin

Equilibrium States for Water (function of temperature and pressure) Review of Water Phases 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 Equilibrium States for Water (function of temperature and pressure) Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid pv, Tv pw, Tw Thermodynamics M. D. Eastin

Review of Water Phases Equilibrium Phase Changes: Vapor → Liquid Water (Condensation): Equilibrium state (saturation) Does not behave like an ideal gas Isobaric Isothermal Volume changes C V P (mb) Vapor Solid Tt = 0ºC Liquid and Tc = 374ºC T1 6.11 221,000 T B A A B C Thermodynamics M. D. Eastin

Review of Latent Heats 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 C V P (mb) Vapor Solid Tt = 0ºC Liquid Tc = 374ºC T1 6.11 221,000 T L Values for lv, lf, and ls are given in Table A.3 of the Appendix Thermodynamics M. D. Eastin

Ideal Gas Law for Water Vapor Changes to Notation 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 Ideal Gas Law for Water Vapor Thermodynamics M. D. Eastin

Changes to Notation 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 Thermodynamics M. D. Eastin

Benoit Paul Emile Clapeyron Clausius-Clapeyron Equation Who are these people? Rudolf Clausius 1822-1888 German Mathematician / Physicist “Discovered” the Second Law Introduced the concept of entropy Benoit Paul Emile Clapeyron 1799-1864 French Engineer / Physicist Expanded on Carnot’s work Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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) Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid P (mb) Vapor esw T Liquid Sections of the P-V and P-T diagrams for which the Clausius-Clapeyron equation is derived in the following slides Liquid and Vapor V Thermodynamics M. D. Eastin

Saturation vapor pressure Saturation vapor pressure Clausius-Clapeyron Equation Mathematical Derivation: Assumption: Our system consists of liquid water in equilibrium with water vapor (at saturation) We will return to the Carnot Cycle… Volume T2 T1 esw1 esw2 Saturation vapor pressure A D B C Isothermal process Adiabatic process Temperature T2 T1 esw1 esw2 Saturation vapor pressure A, D B, C Thermodynamics M. D. Eastin

Saturation vapor pressure Clausius-Clapeyron Equation Mathematical Derivation: Recall for the Carnot Cycle: If we re-arrange and substitute: Volume T2 T1 esw1 esw2 Saturation vapor pressure A D B C Isothermal process Adiabatic process WNET Q1 Q2 where: Q1 > 0 and Q2 < 0 Thermodynamics M. D. Eastin

Saturation vapor pressure Clausius-Clapeyron Equation Mathematical Derivation: Recall: During phase changes, Q = L Since we are specifically working with vaporization in this example, Also, let: Volume T2 T1 esw1 esw2 Saturation vapor pressure A D B C Isothermal process Adiabatic process WNET Q1 Q2 Thermodynamics M. D. Eastin

Saturation vapor pressure Clausius-Clapeyron Equation Mathematical Derivation: 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 Volume T2 T1 esw1 esw2 Saturation vapor pressure A D B C Isothermal process Adiabatic process WNET Q1 Q2 Thermodynamics M. D. Eastin

Saturation vapor pressure Clausius-Clapeyron Equation Mathematical Derivation: 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 Temperature T2 T1 esw1 esw2 Saturation vapor pressure A, D B, C Thermodynamics M. D. Eastin

Equilibrium States for Water (function of temperature and pressure) Clausius-Clapeyron Equation 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 Equilibrium States for Water (function of temperature and pressure) Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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: Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation Application: Saturation vapor pressure for a given temperature After integration we obtain: After some algebra and substitution for es0 = 6.11 mb and T0 = 273.15 K we get: Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation Application: Saturation vapor pressure for a given temperature 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: Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation Application: Saturation vapor pressure for a given temperature 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 Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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) Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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) Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation Equilibrium with respect to Ice: We will know examine the equilibrium vapor pressure for a heterogeneous system containing vapor and ice Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid C V P (mb) Vapor Solid Liquid T 6.11 A B esi Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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 Sublimation Fusion Vaporization T C T (ºC) p (mb) 374 100 6.11 1013 221000 Liquid Vapor Solid Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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: Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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 (pwi) Thermodynamics M. D. Eastin

Clausius-Clapeyron Equation 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 Thermodynamics M. D. Eastin

References Thermodynamics M. D. Eastin 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. Thermodynamics M. D. Eastin