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1.9.31 Formulate first order differential equations (A-level only)

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Question 25

A scientist is tracking the concentration of two specific catalysts in a chemical reaction chamber.

The concentration of the first catalyst, C1C_1C1​ (in ppm), is modelled by the equation

C1=Aekt,t≥0 C_1 = A e^{kt}, \quad t \ge 0 C1​=Aekt,t≥0

where A A\,A and k k\,k are positive constants and t t\,t is the time in hours from the start of the reaction.

Given that:

  • the concentration of the first catalyst was 500 ppm at the start of the reaction
  • the concentration was 2500 ppm after 4 hours
a.

Find the exact value of A A\,A and the value of k k\,k to 4 significant figures.

[4]
b.

The concentration of the second catalyst, C2C_2C2​ (in ppm), is modelled by the equation

C2=50000e−0.6t,t≥0 C_2 = 50000 e^{-0.6t}, \quad t \ge 0 C2​=50000e−0.6t,t≥0

where t t\,t is the time in hours from the start of the reaction.

Find the rate of decrease of the concentration of this second catalyst exactly 5 hours from the start. Give your answer to 3 significant figures.

[3]
c.

At time t=Tt = Tt=T, the concentrations of the two catalysts are equal.

Find the value of TTT, giving your answer to 3 significant figures.

(Solutions relying entirely on calculator technology are not acceptable.)

[4]

1.9.31 Formulate first order differential equations (A-level only) Questions

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  3. /1.9.31 Formulate first order differential equations (A-level only)