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Question 2. Flory-Huggins systems. The Flory-Huggins theory applied to linear alkane theory states the following for a binary mixture comprising a smaller molecule (component 1) and a polymer (component 2): AG/RT-x+x₂+x₂ (a) Convert the above expression to mole fractions and derive an expression for the activity coefficient for the diluent and the polymer components. Write the corresponding activities in terms of the volume fraction polymer present in the mixture. (8) (b) Derive expressions for the melting point depression curves and explain how these can be used to determine the temperature and composition of the eutectic point. (4) (c) Blends of tetratricontane and tetracontane are prepared and their melting points were measured. The thermodynamic behaviour of this system follows the Flory-Huggins activity coefficient model. Assume that the interaction parameter (c) is temperature independent and estimate its value if it is found that this system features an eutectic temperature of 72.17 °C. Compute the eutectic composition and report it in terms of volume fraction and mole fraction units. (d) Assume that, at ambient temperature, the interaction parameter equals c = 0.3 for mixtures comprising alkanes. Calculate and plot the solubility of icosane, triacontane, tetracontane, pentacontane, hexacontane, heptacontane and ultrahigh molar mass high density polyethylene in decane at ambient temperature (25 °C) (6) (7) (e) Show that the logarithm of the solubility (expressed in terms of the volume fraction) varies linearly with the carbon chain length. (2) (f) Consider the alkanes below icosane and determine the one with the highest carbon number which is still 100% miscible with decane at 25 °C. (2)