Two chemicals A and B are combined to form a chemical C. the reaction rate is pr

Question

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 created consumes 1 gram 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

Let amount of C at time t = x

Initial amount of A = 60 g

Present amount of A = 60 - 3x

Initial amount of B = 40 g

Present amount of B = 40 - x

a)

Reaction rate is decrease in A, B or increase in C = dx/dt = k*(40-x)*(60-3x)

b)

dx/((40-x)*(60-3x)) = kdt

Solve by partial fractionand integrate

(x(t) = (20 (e^{20 c_1+60k t}-2))/(e^{20 c_1+60k t}-1))

At time t=0, x=0

((e^{20c_1}-2)/(e^{20c_1}-1) = 0, e^{20c_1}=2)

This gives C1 = (ln2)/20 = 0.035

(x = 40(e^{60kt}-1)/(2e^{60kt}-1))

At time = 5*60 = 300 s, x = 10 gm

10 =x = 40(e^{60*k*300}-1)/(2e^{60*k*300}-1)

Solve for k

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.