A liquid mixture contains N components (N may be any number from 2 to 10) at pressure P(mm Hg). The mole fraction of the ith component is x_i (i = 1. 2...., N), and the vapor pressure of that component is given by the Antoine equation (see Table B.4) with constants A,, B,, and C_i. Raoults law may be applied to each component.

(a) Write the equations you would use to calculate the bubble-point temperature of the mixture, ending with an equation of the form f (T) = 0. (The value of T that satisfies this equation is the bubble-point temperature.) Then write the equations for the component mole fractions (y₁, y₂,... , y_N) in the first bubble chat forms, assuming that the temperature is now known.

(b) Prepare a spreadsheet to perform the calculations of part (a). The spreadsheet should include a title tine and two tables: the first table should contain the Antoine-equation constants and the total pressure, and the second should contain columns for the liquid-phase mole fractions, guessed values of the bubble-point temperature, any intermediate quantities generated in the bubble-point calculation (such as vapor pressures at the guessed temperatures), the function f(T), and the values of the vapor-phase mole fractions. Enter the values of A_i, B_i, C_i, P, and x, for each species in the mixture, assume a value of T. and enter formulas for the other variables in the spreadsheet including f. Then determine the bubble-point temperature by using the goal seek tool (or simple trial and error) to find the value of T for which f = 0. Test your program by calculating the bubble-point temperatures and vapor compositions for liquids at 760 mm Hg containing (i) 22.6 mole% benzene, 44.3 mole% ethylbenzene, and the balance toluene; (ii) 44.3 mole% benzene, 22.6% ethylbenzene, and the balance toluene: and (iii) 22.6 mole% benzene, 22.6% ethylbenzene, and the balance toluene Briefly explain why the variations in bubble-point temperature for these three cases make sense.