In chemical kinetics, it is well understood how temperature influences the rates of collisions between reactants....

 

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In chemical kinetics, it is well understood how temperature influences the rates of collisions between reactants. To slow down this interaction, the ambient temperature T(t) can be reduced. The temperature in the reaction chamber (measured in Celsius) at time t (measured in minutes) is given by T(t)=4ta(t)-b(t) cos(2t) + 1. It is known a(t) ≥ 0, and that 4ta(t) is a decreasing function for t21. We also know that [b(1)| ≤ for 120, and that a(t) ≤ for t2t where the precise value of t* is unknown. (a) To preserve sufficient amounts of each reactant, we require that the temperature in the chamber be between degrees zero Celsius and three degrees Celsius before 12 minutes has passed. If a(1) = 1, a(2), and a(6) ≤ can we 20 guarantee that a sufficient amount of the reactants will be preserved? (b) Based on this model and the information provided, can we say for certain that the temperature in the chamber will eventually stabilize to a fixed value? That is, can we conclude lim T(t)- K, for some constant K? 140 In chemical kinetics, it is well understood how temperature influences the rates of collisions between reactants. To slow down this interaction, the ambient temperature T(t) can be reduced. The temperature in the reaction chamber (measured in Celsius) at time t (measured in minutes) is given by T(t)=4ta(t)-b(t) cos(2t) + 1. It is known a(t) ≥ 0, and that 4ta(t) is a decreasing function for t21. We also know that [b(1)| ≤ for 120, and that a(t) ≤ for t2t where the precise value of t* is unknown. (a) To preserve sufficient amounts of each reactant, we require that the temperature in the chamber be between degrees zero Celsius and three degrees Celsius before 12 minutes has passed. If a(1) = 1, a(2), and a(6) ≤ can we 20 guarantee that a sufficient amount of the reactants will be preserved? (b) Based on this model and the information provided, can we say for certain that the temperature in the chamber will eventually stabilize to a fixed value? That is, can we conclude lim T(t)- K, for some constant K? 140

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