(Solved): 1. ( 32 points) A mixture of benzene and toluene is to be distilled in a simple batch still such as ...

1. ( 32 points) A mixture of benzene and toluene is to be distilled in a simple batch still such as the one shown in the figure. The feed charge to the still pot, i.e., i.e. the initial feed is \( \mathrm{F}=10 \mathrm{~kg} \) mole with \( \mathrm{x}_{\mathrm{F}}=0.45 \) mole fraction benzene. The relative volatility is \( \alpha_{\text {bensene- }} \) solucee \( =2.0 \). Assume that the still pot functions as a single equilibrium stage. a. When the distillation was completed, the mole fraction of benzene remaining in the still pot was found to be \( X_{5 i n}=0.10 \). Find the composition of benzene in the collection pot, \( x_{\text {Dang }} \) b. In a different experiment where \( \mathrm{F}=10 \mathrm{~kg} \) mole and \( \mathrm{x}_{\mathrm{F}}=0.45 \), a distillation is completed and the composition in the collection pot is measured to be \( \mathrm{x}_{\mathrm{D}, \mathrm{ng}}=0.850 .55 \). However, we do not know \( \mathrm{X} \) in nor the final total number of moles in the still pot, \( W_{\text {final }} \). Find \( \mathbf{x}_{\text {fin }} \) and \( \mathbf{W}_{\text {final. }} \). Hints:(1) Since \( \alpha_{\text {bensene-toluene }} \) provides an analytical relation between vapor and liquid compositions, consider using the integrated form of the Rayleigh equation that uses \( \alpha \) to relate \( \mathrm{W}_{\text {final }} \) and \( \mathrm{x}_{\text {fin- }} \) (2) The solution for part (b) requires a trial-and-error approach to solve the nonlinear equation, i.e., we guess a value of \( \mathrm{X}_{\text {fin }} \) and calculate a value of \( \mathrm{W}_{\text {final. }} \). How will we know when the correct value of \( \mathrm{W}_{\text {final }} \) has been found? Consider how \( \mathrm{W}_{\text {final }} \) is related to \( \mathrm{x}_{\mathrm{D} \text { avg }} \) through a component balance. Show the details of the trial-and-error calculations for each guessed value of \( \mathrm{x}_{\text {fia }} \). Tabulate your results in a table with 2 columns showing the \( \mathrm{x}_{\mathrm{fin}} \) values you guessed and the resulting \( \mathrm{x}_{\mathrm{Davg}} \) values. If you find two successive values of \( \mathrm{x}_{\text {fin }} \). \( 0.55<\mathrm{XD} \).ngit1, then you can use linear interpolation to estimate a "final" value of \( \mathrm{X}_{\text {in }} \) that results in the specified XDavg \( =0.850 .55 \).