As an example, consider the distillation of a mixture of ethanol and water. Ethanol boils at 78.5C whilst water boils at 100C. So by gently heating the mixture, the ethanol will boil off first.
The apparatus is assembled as in the diagram. The mixture is put into the round bottomed flask along with a few anti bumping granules , and the fractionating column is fitted into the top. As the mixture boils, vapour rises up the column. The vapour condenses on the glass platforms inside the column, and runs back down into the liquid below, refluxing distillate. Only the most volatile of the vapours stays in gaseous form all the way to the top. The vapour at the top of the column will be almost pure ethanol. This then passes into the condenser, which cools it down until it liquefies. The process continues until all the ethanol boils out of the mixture. This point can be recognised by the sharp rise in temperature shown on the thermometer, from the boiling point of ethanol to the boiling point of water.
The most important industrial application of fractional distillation is the distillation of crude oil. The process is similar in principle to the laboratory method described above except for scale, continuous feed and operation, and the fact that crude oil has many different compounds mixed together. The fractionating column has outlets at regular intervals up the column which allow the different fractions to run out at different temperatures, with the highly volatile gases coming out the topmost outlet graduating to the less volatile road tar, (bitumen) coming out at the bottom.
Fractional distillation process is also used in production of liquid oxygen (LOx).
Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small.
The first term, as we see, is the first fraction; the first and second together give the second fraction, 22/7; the first, the second and the third give the third fraction 333/106, and so on with the rest; the result being that the series entire is equivalent to the original value.
Continued fractions also play a role in the study of chaos, where they tie together the Farey fractions which are seen in the Mandelbrot set with the Minkowski question mark function and the modular group Gamma.
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