To design a distillation column to separate a mixture of benzene and toluene, we need to determine the reflux ratio and the number of theoretical trays required to meet the purity specification. We will use the Fenske-Underwood-Gilliland FUG method to estimate these parameters.1. Determine the relative volatility of benzene to toluene at 1 atm:At 1 atm, the boiling points of benzene and toluene are 80.1C and 110.6C, respectively. Using the Clausius-Clapeyron equation, we can estimate the vapor pressures of benzene Pb and toluene Pt at their boiling points. Then, we can calculate the relative volatility as = Pb/Pt. For this problem, we will assume 2.3.2. Calculate the minimum reflux ratio Rmin using the Underwood equation:Rmin = - 1 / - xF , where xF is the mole fraction of benzene in the feed.Rmin = 2.3 - 1 / 2.3 - 0.4 0.763. Choose an operating reflux ratio R above the minimum reflux ratio:A common practice is to use 1.2 times the minimum reflux ratio to ensure adequate separation.R = 1.2 * Rmin 1.2 * 0.76 0.914. Calculate the number of theoretical trays N using the Fenske equation:N = ln[ xD * 1 - xB * 1 - xF / xB * 1 - xD * xF ] / ln , where xD is the mole fraction of benzene in the distillate and xB is the mole fraction of benzene in the bottoms.For this problem, xD = 0.9 90% purity of benzene in the distillate and xB = 0.1 assuming 10% benzene in the bottoms to ensure adequate separation .N ln[ 0.9 * 0.9 * 0.6 / 0.1 * 0.1 * 0.4 ] / ln 2.3 7.5Since the number of theoretical trays must be an integer, we will round up to 8 trays to ensure adequate separation.5. Design the distillation column:The distillation column should be designed with 8 theoretical trays and an operating reflux ratio of 0.91. The column should be able to process a feed rate of 500 kg/h of the benzene-toluene mixture at 1 atm and constant temperature.