Vapor Pressure, The Clapeyron Equation, and Single Pure Chemical Species Phase Equilibrium

1. Vapor Pressure, The Clapeyron Equation, and Single Pure Chemical Species Phase Equilibrium Chapter 5

2. Vapor Liquid Measurement of Vapor Pressure Thermometer Pressure Gauge Filling and emptying line

3. Cox Chart Logarithmic scale What kind of scale is this?

4. Normal Boiling Point • Temperature at which the Pure Component Vapor Pressure is one atm • 212 F for water • Ranges from -44 F for propane to 675 F for mercury • Increase regularly with increasing Molecular weight for similar compounds

5. The Clapeyron Equation If two phases of a pure substance are in equilibrium, then they have the same Gibbs free energy, per mole Now raise the temperature from T1 to T1 + dT, and establish a new equilibrium

6. And since… We know…

7. Which leads to… Now remember that the definition of g is h-ts

9. Clapeyron Equation • Rigorously correct for any phase change of a single pure chemical species P Phase 1 Phase 2 dP/dT T

10. Try an example • Find dP/dT for the steam-water equilibrium at 212 F Use the steam tables to find:Dhvap = 970.3 Btu/lbmT = 212 + 459.6 = 671.6 RDvvap = 26.78 ft3/lbm With some units conversions

11. Check your results • Use the saturation vapor pressure and temperature data at points close to 212 F

12. The Clausius-Clapeyron Equation • The Clapeyron equation is rigorous and exact • Applies to any two phase equilibrium of a pure species • Gas-liquid • Liquid-solid • Gas-solid • Solid-solid

13. Clausius-Clapeyron is a good approximation for equilibria involving gases • Assume Dh is constant • Assume vgas is large compared to: • vsolid • vliquid • Thus Dv ≈ vgas =RT/P

14. Rearrange and integrate

15. Or This works well for low pressure gas-liquid and gas-solid equilibria

16. As the pressure increases, what happens to our assumptions • Dv and vgas are no longer approximately equal • Gas deviates from the ideal gas law • Dh is not constant Fortunately, these errors tend to cancel each other out!! The Clausius-Clapeyron equation works well over a broad range of pressure and temperature – even though it is not rigorously correct.

17. Form

18. If the theory of corresponding states was exactly right, then…

20. All these lines should fall one on top of each other, if the simple corresponding states theorem is correct (But not necessarily straight). • All of the lines should also be straight if equation 5.12 is correct Eq 5.12

21. But they are not all straight and congruent • They are close to straight, but with different slopes • This suggests that B might be a good third parameter for corresponding states (Pr and Tr are the first two) • The accentric factor referred to in chapter 2 is derived from B

22. The plots of lnP vs 1/T are not exactly straight – they have a curve to them We could fit them better if we added another parameter All the constants are empirically determined The Antoine equation T is usually expressed in degrees Celsius. If C=273, the Antoine equation becomes the Clausius-Clapeyron equation

23. Antoine Equation • There is no theoretical basis for this equation • It is strictly a data fitting exercise • The more constants you add, the better you can fit the data • See page 95

25. Applying the Clapeyron Equation to other kinds of phase changes • The Clausius-Clapeyron equation makes some simplifications that are fairly accurate if one phase is a gas • For other kinds of phase changes it is not accurate to say that Dv is equal to the volume of one of the phases, because the specific volume of liquids and solids is fairly close.

26. Let’s try some other approximations Assume the ratio of enthalpy change to volume change is constant This doesn’t give very good results – our assumption is faulty We’ll need to find Dh and Dv as functions of T to do a better job

27. Extrapolating Vapor Pressure Curves • Extends from the triple point to the critical point Triple Point Critical Point ln P True Equilibrium vapor pressure Unstable region 1/T

28. Subcooled water, below the triple point • If the subcooled water is exposed to a “seed” crystal, it will solidify.

29. Vapor Pressure of Mixtures • This whole chapter is about vapor pressure and related phenomena of single pure species. We’ll get to mixtures later.

30. Summary 1. Vapor pressure is fairly easy to measure • Lots of data is available 2.The Clapeyron equation follows from the statement that if any two phases of a pure substance are in equilibrium, their molar or specific Gibbs Free energies are the same • Fundamental, rigorous relationship

31. Summary Cont 3. If one of the phases is a low pressure gas, we can use the Clausius Clapeyron equation • Very accurate approximation for gas-liquid and gas-solid equilibria 4. Clausius Clapeyron is also accurate at high pressures, because of competing inaccuracies that cancel each other out

32. Summary Continued 5. Most complex vapor pressure equations are empirical • Antoine equation 6. Vapor pressure data are often needed in equilibrium calculations • We’ll use this material again in later chapters