## Equilibrium Curves

You now have enough information to work out what happens when you heat a mix of ethanol and water and send the vapor through a reflux column, by combining Raoult's and Dalton's laws with either the Clausius-Clapeyron or the Antoine equations. The following demonstration uses the Clausius-Clapeyron equation.

And we know from mol fractions that X1 + X2 = 1 so |
Xi = (Ptotal - P2*) I (Pi* - P2*) |
Eqn 1 |

As Pi = XiP* and also (from Dalton) Pi = YiPtotal so |
Yi = XiP* I Ptotal |
Eqn 2 |

We can calculate P1* and P2* from Clausius-Clapeyron |
loge(Pn*IPo) = H(1ITo - 1IT) I R |
Eqn 3 |

P0 and T0 are the pressures and temperatures of the pure liquids at their boiling points, something we already know for water and ethanol, and PTOTAL is atmospheric pressure

These results show why it's important to keep the pressure constant, and also indicate what happens if you operate at higher or lower pressures. Equation 2 shows that raising the pressure lowers the yield of vapor, and that lowering the pressure (eg. vacuum distillation) increases it. We've seen many designs for stills that operate under high pressure in the hope that this will increase the yield. Not only is that practice dangerous, the equations show that it's based on ignorance.

These three equations, plotted for ethanol mol fractions ranging from 0 to 1 and temperatures from 78.5°C to 100°C, result in the following constant pressure plot. The lower curve is called the bubble line, because it shows the temperatures at which the mix begins to boil, and the upper curve is called the dew line, because it indicates the temperature at which vapor first condenses. Anything below the bubble line is liquid, and anything above the dew line is vapor. The area enclosed by the two curves represents points where liquid and vapor are in dynamic equilibrium.

This plot is extremely useful Because it lets you graphically plot the progress of a distillation. If, for example, you start with 11% ethanol in the boiler, the first vapor to come off contains 23% ethanol. If this is condensed and re-boiled, the resulting vapor would contain 40% ethanol, and so on up to the maximum possible concentration of around 96% ethanol. Ethanol has a maximum concentration lower than 100% because it forms an azeotropic mixture with water. We used charts like this in Chapter 2 to illustrate what happens with repeated distillations.

The stages in this plot are called theoretical plates. This name comes from the plates or trays inserted in large commercial distillation towers to facilitate the evaporation and condensation process. These plates serve the same purpose as the packing used in small-scale columns. On the diagram, a theoretical plate is marked out by going vertically from the bubble line to the dew line, and then horizontally back to the bubble line.

In the example starting at 11%, 7 plates are needed to reach the top concentration, and the first plate is the boiler itself. If a packed column is 100 cm long, and the temperature levels out at the boiling point of ethanol 60 cm up, then the Height Equivalent Theoretical Plate (HETP) for this packing would be 60/6 = 10 cm. If you are interested in calculating the number of theoretical plates needed in a column, you should use the McCabe-Thiele equations, and we use a simplified form of these in a moment when discussing Reflux. On the internet, Tony Ackland's website (see Appendix 8) gives an excellent description of them. Tony is a chemical engineer in New Zealand with a great interest in the hobby of distillation.

Liquid / Vapor chart

o o o o o Mole fraction Ethanol o o o o o Mole fraction Ethanol

To save you the trouble of plotting this equilibrium diagram, and to calculate how the mol fraction of ethanol relates to percentage by volume, the following tables give the results of the calculations. Please remember that the equations relate to "perfect" gases, and don't exactly match the results you'd get with a very sensitive thermometer and controlled laboratory conditions. However, you should find them very useful in all practical situations.

De-C |
XI |
Y1 |
N°/o |
DegC |
XI |
Y1 |
N°/o | |

100.2 |
0.00 |
0.00 |
0.00 |
89.5 |
0.39 |
0.59 |
67.81 | |

100.0 |
0.01 |
0.01 |
1.93 |
89.0 |
0.41 |
0.61 |
69.81 | |

99.5 |
0.02 |
0.04 |
6.60 |
88.5 |
0.44 |
0.63 |
71.76 | |

99.0 |
0.04 |
0.08 |
11.05 |
88.0 |
0.46 |
0.65 |
73.63 | |

98.5 |
0.05 |
0.11 |
15.29 |
87.5 |
0.48 |
0.67 |
75.45 | |

98.0 |
0.07 |
0.14 |
19.34 |
87.0 |
0.51 |
0.70 |
77.21 | |

97.5 |
0.08 |
0.17 |
23.21 |
86.5 |
0.53 |
0.72 |
78.91 | |

97.0 |
0.10 |
0.20 |
26.92 |
86.0 |
0.56 |
0.74 |
80.56 | |

96.5 |
0.12 |
0.23 |
30.47 |
85.5 |
0.58 |
0.76 |
82.16 | |

96.0 |
0.14 |
0.25 |
33.87 |
85.0 |
0.61 |
0.78 |
83.71 | |

95.5 |
0.15 |
0.28 |
37.13 |
84.5 |
0.64 |
0.80 |
85.21 | |

95.0 |
0.17 |
0.31 |
40.26 |
84.0 |
0.67 |
0.82 |
86.67 | |

94.5 |
0.19 |
0.34 |
43.26 |
83.5 |
0.69 |
0.83 |
88.09 | |

94.0 |
0.21 |
0.36 |
46.15 |
83.0 |
0.72 |
0.85 |
89.46 | |

93.5 |
0.23 |
0.39 |
48.93 |
82.5 |
0.75 |
0.87 |
90.80 | |

93.0 |
0.25 |
0.42 |
51.60 |
82.0 |
0.78 |
0.89 |
92.09 | |

92.5 |
0.27 |
0.44 |
54.18 |
81.5 |
0.81 |
0.91 |
93.35 | |

92.0 |
0.29 |
0.47 |
56.66 |
81.0 |
0.84 |
0.92 |
94.58 | |

91.5 |
0.31 |
0.49 |
59.05 |
80.5 |
0.87 |
0.94 |
95.77 | |

pi.o |
0.33 |
0.52 |
61.36 |
80.0 |
0.91 |
0.96 |
96.92 | |

90.5 |
0.35 |
0.54 |
63.58 |
79.5 |
0.94 |
0.97 |
98.05 | |

90.0 |
0.37 |
0.56 |
65.73 |
79.0 |
0.97 |
0.99 |
99.15 | |

89.5 |
0.39 |
0.59 |
67.81 |
78.6 |
1.00 |
1.00 |
Note too that the figures for mol fractions and percentage ethanol for temperatures below 80°C on this chart are those derived from theory. In real life, ethanol is 'difficult' as it exhibits a phenomenon called azeotropism, and the figures depart from those that simple theory would predict. So, what is azeotropism? |

## Brew Your Own Beer

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