Separation is one of the most important and energy consuming industrial processes. Things often like to mix and dislike to demix, as we have seen in class! Imagine you need 10 gallons of gasoline to fill up your car, but you are given 20 gallons of a mixture of two kinds of hydrocarbons, of which one is the one that you need and the other is heavier and does not burn readily. (Crude oil is in fact a mixture of many grades of hydrocarbons.) We have considered in class a slow process in which we mixed two distinct, ideal gases. The reverse process is possible, too, but would require lowering the entropy of the system. At the same time, since the temperature stays constant and the gases are ideal, it means that the energy stays constant. Consequently, the lowering of the entropy would be all at the expense of your doing the work to bring the two gases back to their respective halves of the container. (Note that distillation, which is probably the most common separation technique, is not at all like the thought experiment we considered in class but does require mechanical work, and so does separation using the centrifuge, something biologists use a lot. Your own body spatially separates things all the time, by mechanically moving them across the membranes encasing cells or organelles.) Pretend that the mixing entropy of your hydrocarbons is equal to that of an ideal gas mixture. (It is actually not a bad approximation because the two substances are so similar chemically.) Now, how much work do you need to perform to separate out your 10 gallons from that mixture, at temperature 300K? Pretend that the hydrocarbon you need is octane, C₈H₁₈, the other compound is dodecane C₁₂H₂₆, and the mass density of either compound and the mixture is all the same and equal 750 kg/m³.