Let's break down **Problem 3.3** into three main steps, with explanations after each step: --- ### **Step 1: Initial Setup and Key Equations** **Given:** - Initial moles = 30 - Benzene = 40 mol%, Toluene = 60 mol% - Vaporization rate = 15 mol/hr - Relative volatility (α) = 2.5 - Pressure = 1 atm - Differential distillation (Rayleigh distillation) #### **Key Equations:** - **Rayleigh Equation:** \[ \ln \left( \frac{L_0}{L} \right) = \int_{x_{A,0}}^{x_A} \frac{dx_A}{y_A - x_A} \] where \( L_0 \) = initial moles of liquid, \( L \) = moles of liquid remaining, \( x_A \) = mole fraction of benzene in liquid, \( y_A \) = mole fraction of benzene in vapor. - **Vapor-Liquid Equilibrium (VLE) for Binary with constant α:** \[ y_A = \frac{\alpha x_A}{1 + (\alpha - 1)x_A} \] **Explanation:** We have established the starting conditions and the fundamental equations for differential distillation using the Rayleigh equation and the VLE relation for an ideal binary mixture. --- ### **Step 2: Calculate at Each Time Step** Assume small time increments (e.g., Δt = 1 hr), and perform calculations at each increment: #### **At Each Step:** 1. **Vaporized moles:** \( \Delta V = \) vaporization rate × Δt (here, 15 mol/hr × 1 hr = 15 mol per step) 2. **Update Liquid Remaining:** \( L_{new} = L_{old} - \Delta V \) 3. **Update Composition using Rayleigh equation:** \[ \ln \left( \frac{L_0}{L} \right) = \int_{x_{A,0}}^{x_A} \frac{dx_A}{y_A - x_A} \] Or rearrange for small changes: \[ \frac{dL}{L} = \frac{dx_A}{y_A - x_A} \] Numerically, for each step, update \( x_A \) using: \[ \Delta x_A = (y_A - x_A) \frac{\Delta L}{L} \] Where \( y_A \) is calculated from the VLE equation. **Explanation:** This step outlines how to incrementally update the number of moles and composition of benzene in the still pot at each time step, making use of VLE and the Rayleigh equation. This iterative process is typically done in a spreadsheet. --- ### **Step 3: Calculate Other Quantities and Bubble-Point Temperature** **a) Average distillate composition (\( x_{AD, av} \)):** \[ x_{AD, av} = \frac{\text{total moles of benzene vaporized}}{\text{total moles vaporized}} \] **b) Bubble-point temperature:** At each step, use Raoult's law: \[ P = x_A P_A^*(T) + (1-x_A) P_B^*(T) = 1.0~\text{atm} \] Solve for T, where \( P_A^*(T) \) and \( P_B^*(T) \) are vapor pressures of benzene and toluene at temperature T. **c) Plot:** Create plots of \( x_A \), \( y_A \), \( L \), \( x_{AD, av} \), and bubble-point temperature vs. time. **Explanation:** This step shows how to calculate the distillate composition and bubble-point temperatures at each time interval, and then plot the required variables as a function of time. --- ## **Final Answer / Summary:** **To solve the problem:** 1. **Set up a spreadsheet** with time steps, starting at 30 moles and 40% benzene. 2. **At each step,** subtract 15 moles (or less for the last step), update the benzene fraction using Rayleigh and VLE equations, and calculate the new bubble-point temperature using Raoult’s law. 3. **Plot \( x_A \), \( y_A \), \( L \), \( x_{AD, av} \), and bubble-point temperature** vs. time. **These steps will allow you to track the change in composition and temperature of the liquid in the still as distillation proceeds!** If you need a sample spreadsheet setup or code, let me know!