Question

Chemical reactions are often described using a three state model:

Reactants→Transition State→ProductsReactants→Transition State→Products

In most cases the energy of the transition state is much higher than the energy of the reactant state. This means that the reaction cannot proceed until there is a random thermal fluctuation large enough to `kick' the reactant molecule(s) up to the transition state energy. Say we have a reaction in which the transition state is 9.0×10⁻²⁰ J above the reactant state.

a.) Calculate the ratio of the probability the system is in the transition state to the probability that it is in the reactant state (P_TS/P_R) at 37 C.

P_TS/P_R = (7.3 x 10^-10 is correct)

b.) Use your answer above to estimate the time it takes for the reaction to occur spontaneously at 37 C. Assume that the system samples a new microstate every nanosecond (10⁻⁹ s).

[Hint: If you throw a 6-sided die, what is the probability you will get a 3? How many times would you expect to have to throw the die to get a 3? Using similar logic, use your answer in the first part to tell you how many times the system has to try before it gets a fluctuation that is big enough.]

time: (2.1 x 10^-8 s is incorrect)

c.) One way to speed up the reaction is to heat the system. If the temperature is increased to 1000 K, how long does it take for the reaction to occur?

time: (.009325 s is incorrect)

d.) In biological systems it is not feasible to accelerate reactions by heating them to 1000 K. Instead, the reactions rely on enzymes that catalyze reactions by lowering the energy of the transition state. If catalyzed reaction has a transition state energy 2.0×10⁻²⁰ J, what is the approximate reaction time? Use a temperature 37 C.

time: (6.52 x 10^-9 s is incorrect)

e.) Finally, estimate the uncatalyzed reaction time if the transition state is not a unique microstate, but actually an ensemble of 100 microstates. Use a temperature 37 C.

time: (4.675 x 10^-9 s is incorrect)

Answer 1