Show and explain all steps. Ethyl acetate is made from acetic acid and ethyl alc
ID: 1032065 • Letter: S
Question
Show and explain all steps.
Ethyl acetate is made from acetic acid and ethyl alcohol. Let x pounds of acetic acid input, y pounds of ethyl alcohol input, and z pounds of ethyl acetate output. The relationship of output to input is z2 (1.47x z)(1.91y -z) (a) Determine the output of ethyl acetate per pound of acetic acid, where the ratio of acetic acid of ethyl alcohol is 2.0,1.0, and 0.67, and graph the result. (b) Graph the cost of material per pound of ethyl acetate for each of the ratios given and determine the ratio for which the material cost per pound of ethyl acetate is a minimum if acetic acid costs $0.80 per pound and ethyl alcohol costs $0.92 per pound.Explanation / Answer
a)
Ratio of acetic acid and ethyl alcohol = x/y = 2.0 (given) , so y/x = 1/2 = 0.5
we need to determine z/x = k
Rewrite the equation
z2 = 3.9*[(1.47x - z)(1.91y - z)]
Divide the whole equation by x2
(z/x)2 = 3.9*[(1.47 - z/x)(1.91y/x - z/x)]
k2 = 3.9*[(1.47 - k)(1.91*0.5 - k)]
2.9k2 - 9.4575k + 5.475 = 0
k = (9.4575 +/- ?((-9.4575)2 - 4*2.9*5.475))/(2*2.9) = 2.51 or 0.753