Solve it as a Bernoulli differential equation using the correct substitution Two

Question

Solve it as a Bernoulli differential equation using the correct substitution Two chemicals A and B are combined to form a chemical C. the reaction rate is proportional to the product of the amounts of A and B not yet converted to chemical C. Initially, the re are 60 grams of A and 40 grams of B. Each gram of C crated consumes 1 grams of B and 3 grams of A it is observed that 10 grams of C is formed in 5 minutes. Write an autonomous differential equation which gives the amount X of chemical C created in t minutes. Solve the differential equation As time increases, how much chemical C is created? Which and how much of chemical A and B is left?

Explanation / Answer

1
A + B = C

reaction decreases as time passes and the reactants get used up.
r = -kAB

during reaction A converts 2x while B converts x to form 3x of C
let's y = C

y = 3x

amount of reactants not converted yet ,
for A: 40 - 2x
for B: 50 - x


variation of x as time passes
dx/dt = -k(40 - 2x)(50 - x)

integrate x wrt to t
1/[(40 - 2x)(50 - x)] dx = -k dt

partial fractions
2/ 60(40 - 2x) - 1/ 60(50 - x) = -k dt
?2/ 60(40 - 2x) - 1/ 60(50 - x) dx = ?-k dt

ln[(50 - x) /(40 - 2x) ] = -60kt + 60D

(50 - x) /(40 - 2x) = e^(-60kt) e^(60D)

initial conditions: t = 0, x = 0
(50 - 0) /(40 - 0) = e^(0) e^(60D)
e^(60D) = 5/4 = 1.25

(50 - x) /(40 - 2x) = 1.25 e^(-60kt)
(50 - x) = (50 - 2.5x)e^(-60kt)

x[ 2 - 5e^(-60kt) ] = 100[ 1 - e^(-60kt) ]
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