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.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.