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Clausius-Clapeyron Equation

p ( mb ). C. 221000. Fusion. Liquid. Vaporization. Solid. 1013. 6.11. T. Sublimation. Vapor. 0. 100. 374. T ( º C). Clausius-Clapeyron Equation. Cloud drops first form when the vaporization equilibrium point is reached (i.e., the air parcel becomes saturated)

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Clausius-Clapeyron Equation

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  1. p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) 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 M. D. Eastin

  2. 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 • Applications M. D. Eastin

  3. 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 • Can apply the first and second laws • Solid Phase (ice): • Does not behave like an ideal gas • Can apply the first and second laws M. D. Eastin

  4. p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) 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) pv, Tv pw, Tw M. D. Eastin

  5. P (mb) C Liquid 221,000 Tc = 374ºC C B A Vapor Liquid and Vapor T1 A B C Solid T 6.11 Solid and Vapor Tt = 0ºC V 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 M. D. Eastin

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

  7. 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 M. D. Eastin

  8. 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 M. D. Eastin

  9. 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 M. D. Eastin

  10. p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) 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) 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 M. D. Eastin

  11. Isothermal process Adiabatic process B, C esw1 B C esw1 T1 A, D Saturation vapor pressure Saturation vapor pressure esw2 esw2 T2 A D T2 T1 Volume Temperature 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… M. D. Eastin

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

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

  14. Isothermal process Adiabatic process Q1 B C esw1 T1 Saturation vapor pressure WNET esw2 T2 A D Q2 Volume 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 M. D. Eastin

  15. B, C esw1 A, D Saturation vapor pressure esw2 T2 T1 Temperature 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 M. D. Eastin

  16. p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) 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) M. D. Eastin

  17. 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: M. D. Eastin

  18. 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: M. D. Eastin

  19. 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: M. D. Eastin

  20. 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 M. D. Eastin

  21. 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) M. D. Eastin

  22. 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) M. D. Eastin

  23. p (mb) C 221000 P (mb) Fusion Liquid Vaporization Solid Liquid C 1013 6.11 T Vapor Sublimation Vapor Solid T 6.11 esi B A T 0 100 374 T (ºC) V Clausius-Clapeyron Equation • Equilibrium with respect to Ice: • We will know examine the equilibrium • vapor pressure for a heterogeneous • system containing vapor and ice M. D. Eastin

  24. p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) 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 M. D. Eastin

  25. 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: M. D. Eastin

  26. 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 × 10-3m3 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) M. D. Eastin

  27. 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 • Applications M. D. Eastin

  28. 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. M. D. Eastin

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