The UNIFAC and UNIQUAC equations use surface fraction and volume fractions. This problem explores the differences.

(a) Calculate the surface area and volume for a cylinder of diameter d = 1.0 and length L = 5 where the units are arbitrary. Calculate the surface area for a sphere of the same volume. Which object has a higher surface area to volume ratio?

(b) Calculate the volume fractions and surface area fractions for an equimolar mixture of the cylinders and spheres from part (a). Use subscript s to denote spheres and subscript c to denote cylinders.

(c) For this equimolar mixture, calculate the local composition ratios x_cs /x_ss and x_sc /x_cc for the UNIQUAC equation if the energy variables τ_cs and τ_sc are unity. For the equimolar mixture, substitute the values of volume fraction and surface fraction into the expression for UNIQUAC activity coefficients, and simplify as much as possible, leaving the q’s as unknowns.

(d) Consider n-pentane and 2,2-dimethyl propane (also known as neopentane). Calculate the UNIQUAC r and q values for each molecule using group contribution methods. Compare the results with part (a).