One hundred kilogram-moles per hour of a feed containing 30 mole% n-hexane and 70% n-octane is to be distilled in a column consisting of a partial re-boiler, a theoretical plate, and a partial condenser, all operating at 1 atm (101.3 kPa). The feed, a bubble-point liquid, is fed to the re-boiler, from which a liquid bottoms product is continuously withdrawn. Bubble-point reflux is returned to the plate. The vapor distillate contains 80 mole% hexane, and the reflux ratio (L/D) is 2. Assume the partial re-boiler, plate, and partial condenser each function as an equilibrium stage. 

(a) Using the McCabe-Thiele method, calculate the bottoms composition and moles of distillate produced per hour. 

(b) If the relative volatility α is assumed constant at a value of 5 over the composition range for this example (the relative volatility actually varies from 4.3 at the re-boiler to 6.0 at the condenser) calculate the bottoms composition analytically. 

(c) Solve problem graphically, assuming the feed is introduced on plate 1, rather than into the re-boiler. 

(d) Determine the minimum number of stages required to carry out the calculated separation.