(1 point) Two chemicals A and B are combined to form a chemical C. The rate of t
ID: 2920940 • Letter: #
Question
(1 point) Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B noft converted to chemical C. Initially there are 58 grams of A and 85 grams of B, and for each gram of B, 0.7 grams of A is used. It has been observed that 35.75 grams of C is formed in 10 minutes. How much is formed in 40 minutes? What is the limiting amount of C after a long time? 49 grams of C are formed in 40 minutes grams is the limiting amount of C after a long timeExplanation / Answer
Solution:
A + B = C
reaction decreases as time passes and the reactants get used up.
r = - k A B
during reaction A converts 0.7x while B converts x to form 1.7x of C.
let's y = C
y = 1.7x
amount of reactants not converted yet,
for A: 58 - 0.7x
for B: 85 - x
variation of x as time passes; dx/dt = -k(58 - 0.7x)(85 - x)
integrate x w.r.t. to t; 1/[(58 - 0.7x)(85 - x)] dx = -k dt
partial fractions
0.7/ 1.5(58 - 0.7x) - 1/ 1.5(85 - x) = -k dt
?0.7/ 1.5(58 - 0.7x) - 1/ 1.5(85 - x) = ?-k dt
0.7((-1/0.7)*ln(58 - 0.7x) - (-ln(85 - x)) = -1.5kt + 1.5D
ln[(85 - x)/(58 - 0.7x)] = -1.5kt + 1.5D
(85 - x)/(58 - 0.7x) = e-1.5kt e1.5D
initial conditions: t = 0, x = 0
(85 - 0)/(58 - 0.7*0) = e-1.5k*0 e1.5D
e1.5D = 85/58 = 1.465
(85 - x)/(58 - 0.7x) = 1.465*e-1.5kt
(85 - x) = (85 - 1.0255x)e-1.5kt
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after t = 10mins , C = 35.75g = 1.7x
=> 1.7x = 35.75
=> x = 21.029
(85 - 21.029) = (85 - 1.0255*21.029)e-1.5k*10
63.971 = 63.435*e-15k
e-15k = 1.008
-15k = ln(1.008)
k = -ln(1) / 15
-----------------------
e-1.5kt = e-1.5t*(-ln(1.008) / 15) = e1/10 * t ln(1.008)
(85 - x) = (85 - 1.0255x)e-1.5kt
(85 - x) = (85*e-1.5kt - 1.0255x*e-1.5kt)
85(1 - e-1.5kt) = x(1 - 1.0255*e-1.5kt)
x = 85(1 - e-1.5kt) / (1 - 1.0255*e-1.5kt)
formation of C
but y = 1.7x
y = 1.7*85(1 - e-1.5kt) / (1 - 1.0255*e-1.5kt)
y = 144.5(1 - (1.008)t/10 / (1 - 1.0255*(1.008)t/10)
formed in 40 mins;
x(40) = 144.5(1 - (1.008)40/10 / (1 - 1.0255*(1.008)40/10)= 79.707g
limiting of C
y = 144.5[ - (1.008)40/10 / - 1.0255*(1.008)40/10 ] = 140.91g
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