Absolutely! Let's through each part step by step with explanations. --- ### 1. **Boiling Point Elevation as a Function of Weight Fraction** The boiling point elevation (\(\Delta T_b\)) for a non-volatile solute in water is given by: \[\Delta T_b = i K_b m\] Here, - \(i\) = van't Hoff factor (accounts for the number of particles the solute splits into) - \(K_b\) = ebullioscopic constant of the solvent (for water, \(K_b \approx .512 \, \mathrm{K\,kg\,mol^{-1}}\)) - \(m\) = molality of the solution (moles of solute per kg of solvent) The molality \(m\) can be related to the weight fraction \(w\) of solute (let’s call the solute B and solvent A): \[ w_B = \frac{\text{mass of B}}{\text{total mass}} = \frac{n_B M_B}{n_B M_B + n_A M_A} \] But for dilute solutions, \(m \approx \frac{w_B}{M_B(1-w_B)} \approx \frac{w_B}{M_B}\) (since \(w_B\) is small). **To find the concentration at which the boiling point increases by 2.5 K:** Set \(\Delta T_b = 2.5\,\text{K}\): \[ 2.5 = i K_b m \] \[ m = \frac{2.5}{i K_b} \] So, the required molality is \(\frac{2.5}{i K_b}\). **In words:** As you increase the weight fraction of solute, the boiling point elevation increases linearly with molality, which can be related back to weight fraction for dilute solutions. --- ### 2. **Poynting Correction Factor for High-Pressure Liquid-Vapor Equilibrium** The Poynting correction accounts for the effect of pressure on the fugacity (effective pressure) of the liquid phase. The correction factor is: \[ \exp \left[ \frac{V_L (P - P^{sat})}{RT} \right] \] where - \(V_L\) = molar volume of liquid - \(P\) = system pressure - \(P^{sat}\) = saturation pressure - \(R\) = gas constant - \(T\) = temperature The fugacity of component \(i\) in the liquid is: \[ \phi_i y_i P = x_i \gamma_i P_i^{sat} \exp \left[ \frac{V_L (P - P^{sat})}{RT} \right] \] where \(\gamma_i\) is the activity coefficient, \(x_i\) is mole fraction in liquid, \(y_i\) in vapor. **Fugacity for vapor phase:** Use the truncated Virial equation (second coefficient): \[ \ln \phi_i = \frac{P}{RT} \left( B_i + y_j B_{ij} - y_i B_{ii} \right) \] A free body diagram at the interface would show molecules in both phases, with arrows indicating evaporation and condensation events, and the net flux depending on differences in chemical potential. --- ### 3. **Concentration Profile of Gas Diffusing into Liquid** For gas \(A\) dissolving into a liquid film, the steady-state diffusion (Fick's Law) with Henry's Law at the interface: \[ P_A = H x_A \] \[ N_A = - D_{AB} \frac{dC_A}{dx} \] At steady-state and for a flat film of thickness \(\delta\), the concentration profile is linear: \[ C_A(x) = C_{A,\text{interface}} + \left( \frac{C_{A,\text{bulk}} - C_{A,\text{interface}}}{\delta} \right) x \] where \(C_{A,\text{interface}}\) is related to \(P_A\) by Henry's Law. --- ### 4. **Coordinate Where Gas Partial Pressure Equals Interfacial Equilibrium Pressure** This occurs exactly at the liquid-gas interface. At this point, the partial pressure of \(A\) in the gas phase equals \(H x_A\) (Henry's Law), which is the equilibrium pressure. --- ### 5. **Ternary System Residue Curve and Distillation Path** A residue curve tracks the composition of the remaining liquid during distillation. For simple distillation, the path moves towards the heaviest-boiling component in the composition triangle. The differential equation for the residue curve is: \[ \frac{dz_i}{dt} = z_i - y_i \] where \(z_i\) is the liquid composition and \(y_i\) is the vapor composition in equilibrium with the liquid. The path will move towards the corner representing the heaviest component (highest boiling point), because that is what remains as the more volatile components are removed. --- ### 6. **Stable Node in Composition Triangle (No Azeotropes)** For a ternary system with no azeotropes, the stable node (final composition) is at the vertex of the triangle that represents the pure component with the highest boiling point. This is where the residue curve will end up after all the more volatile components are distilled off. --- #### **Summary** Each step connects physical chemistry concepts with mathematical expressions to solve phase equilibrium and distillation problems. For boiling point elevation, you use the van’t Hoff law and relate molality to weight fraction, setting the elevation to 2.5 K to find the needed concentration. For vapor-liquid equilibrium, you derive the correction factor, show how fugacity is calculated using equations of state, and illustrate molecular interactions at interfaces. For diffusion and distillation, you describe the profiles, equilibrium points, and how residue curves behave in ternary systems, especially when no azeotropes are present—the final stable composition is the pure, least volatile component.