An industrial chemist is monitoring the degradation of a catalyst in a chemical reactor. The amount of catalyst, A A\,A units, remaining in the system t t\,t hours after it is introduced is modelled by the differential equation
dAdt=−λA \frac{dA}{dt} = -\lambda A dtdA=−λAwhere λ \lambda\,λ is a positive constant. Given that the solution to this differential equation is of the form A=A0e−λtA = A_0 e^{-\lambda t}A=A0e−λt, where A0 A_0\,A0 is the initial amount of catalyst:
On average, the catalyst has a half-life of 8.4 hours. At 8:00 am, 500 units of the catalyst are added to the reactor. Use the model to calculate the amount of catalyst remaining at 6:30 pm on the same day.
To maintain the reaction safely, the chemist must ensure that the total amount of catalyst in the reactor never exceeds 600 units. The chemist intends to add a second batch of 500 units of catalyst. Determine the earliest time at which this second batch can be added. Give your answer to the nearest minute.
Suggest one limitation of using this model for the degradation of the catalyst in a real industrial environment.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
