Partial Molal Properties

1. Partial Molal Properties Chapter 6

2. Partial Molal Gibbs Free Energy • Hard to visualize • No direct way to measure it • Let’s look at other partial molal properties first, then come back when we understand them better

3. The derivative is taken of an extensive property, with respect to the number of moles. The result is an intensive property, because it is “per mole”. Basic Definition of the partial molal property in phase 1

4. Partial Molal Properties exist only for extensive properties!!! It wouldn’t make sense to have K/mole for example in the case of temperature It wouldn’t make sense to have Btu/lbm/mole in the case of specific enthalpy

5. A simple example • Consider a graduated cylinder with 1000 g of pure water at 200C • Add ethanol, a little bit at a time, at constant T and P • After each addition, stir and measure the resulting volume

6. Volume vs Molality The slope is the partial molal volume

7. Partial Molal Volume Eq. 6.1 How does the total volume change as you add compound i, while keeping the temperature, Pressure and moles of all the other components constant?

8. Method of Tangent Slopes This looks like a straight line, but it actually curves slightly

9. Example 6.1 The volume of 1000 g of water, plus m moles of ethanol can be modeled as… m is the molality

10. Find the partial molal volume • At m = 0, =54.7 cm3/mol • At m = 1, =53.8 cm3/mol What does it mean? It’s the amount the solution volume will change if you add one mole of ethanol. It is not the same as the solution volume of one mole of pure ethanol!!

11. An aside… • What would happen if you repeated the experiment with pure water? • The partial molal volume would be the same as the pure species molar volume, v0 • The partial molal volume would be a constant

12. This is the total volume before mixing Just be careful of units!! Example 6.2 • Estimate the volume change on mixing for 1 mol of ethanol with 1kg water at 200C =1.0603 L

13. 1.0603 L 1.05615 L Use the equation for V to find the mixture volume This is the total volume after mixing V = 1.05615 L The volume change on mixing is…

14. Here’s another way to think of it Function of the number of moles

15. Here’s another way to think of it Equation 6.5, page 109

16. Example 6.3 • Estimate the volume change on mixing, using equation 6.5 • To do this you need to know the average value of the partial molal volume, over the concentration range of interest

18. For a pure substance, any partial molal property is the same as the molar property. • In the volume example, if we add water to water, there is no volume change with mixing • In mixtures with multiple species, the partial molal property is different from the molar property

19. This time let’s measure the heat added or subtracted from the system to keep the temperature constant An energy balance on this control volume gives us: Let’s repeat our experiment

20. The system pressure is constant, so the only work is the work of driving back the surroundings Substitute and rearrange

21. If we divide by dnin we get the partial molal enthalpy Equation 6.10 Heat of mixing

22. The Partial Molal Equation • Consider some extensive property Y • Y could be V, S, U, H, A, G etc • Remember that for a pure substance, the state is defined if you know 2 independent state properties • For a mixture, you also need to know the composition – knowing how many moles defines that

23. We did this for Gibbs Free Energy in Chapter 4

24. The partial molal properties are functions of composition only, since T and P are constant

25. Start with an empty vessel Add species a and b at a constant rate Keep T and P constant Mix while pouring Now consider adding two species at the same time to a mixer

26. The composition is constant at all times • T and P are constant • Therefore the partial molal properties (in this case partial molal volume) are constant Constants

27. Now integrate Don’t forget you started with zero moles of each component, and therefore zero initial volume

28. We used a trick to allow us to do the integrations • We defined a special path • However, V is a state function • It doesn’t matter what path you follow, you always arrive at the same final position • This relation is true, no matter what path we follow!!

29. For any extensive property, it follows that…

30. Partial Molal equations

31. Tangent Intercepts Molality is a satisfactory unit of concentration for dilute solutions but does not work well over the entire concentration range. Molality goes to infinity for pure solute

32. v0ethanol v0water A better plotting scheme Molar propertyIn this case molar volume of the solution, cm3/mole solution This plot looks straight, but actually has a slight “S” shape Mole fraction

33. Let’s use an exaggerated figure Pure b Pure a Molar volume at the tangent Tangent Line Molar volume, v v0a Composition at the tangent point v0b Mole fraction of component a, xa

34. Let’s use an exaggerated figure Pure b Pure a a Molar volume, v v0a b v0b Mole fraction of component a, xa

35. b Let’s use an exaggerated figure Pure b Pure a a Molar volume, v v0a v0b Mole fraction of component a, xa

36. b Let’s use an exaggerated figure Pure b Pure a a Molar volume, v v0a Remember that a and b depend on the value of xa v0b Mole fraction of component a, xa

37. Relate this equation to partial molal volumes Equation 6.17 page 113 Equation 6.16 page 112 These equations are identical, suggesting that a and b are equal to the partial molal volumes!!!

38. The method of tangent intercepts • This result is general for all partial molal quatities • It is not restricted to partial molal volume

39. Plots like Figure 6.2 are not widely available Thus this technique is not widely used to find partial molal quantities The idea of tangent slopes is intuitively pleasing Figure 6.2

40. It is more widely used because figures like 6.5 and 6.6 are more readily available It is much more common to use mole fraction, xa, than to use molality The technique of tangent intercepts is less intuitive

41. The two equations for partial molal properties • If you have experimental data, you can construct either type of plot (using molality or mole fraction) • Depending on the graph you draw, use either • Tangent slopes • Tangent intercepts • It would be nice if we could calculate the partial molal properties, instead of using geometry to find them If your plot is vs molality If your plot is vs mole fraction

42. The two Tangent intercepts calculating equations Fit the data to an equation Look at Figure 6.6 to prove this to yourself, if you don’t believe it

43. Pure b Pure a a Molar volume, v v0a b v0b Mole fraction of component a, xa v + slope*(1-xa)= a= 1-xa

44. Using figure 6.5, estimate the partial molal volumes of EtOH and of water in a solution that is 1.00 molal in ethanol, by equation 6.17 Example 6.4 pg 113

45. First… convert from 1 molal to mole fraction

46. Second – fit the data to an empirical equation for specific volume This equation applies only to x from 0 to 0.04 At x=0.01768, from the above equation v=0.018675 L/mol and dv/dxEtOH = 0.035806 L/mol

47. Third - Plug into equation 6.19

48. Example 6.5 pg 115 • Repeat the previous example, for ethanol only, using the method of tangent slopes

49. Convert the equation to a mole basis Multiply both sides of the equation by nT

50. Substitute into definition for partial molal properties