Chemical reactions are often described using a three statemodel:
Reactants→Transition State→ProductsReactants→TransitionState→Products
In most cases the energy of the transition state is much higherthan the energy of the reactant state. This means that the reactioncannot proceed until there is a random thermal fluctuation largeenough to `kick' the reactant molecule(s) up to the transitionstate energy. Say we have a reaction in which the transition stateis 9.0×10−20 J above the reactant state.
a.) Calculate the ratio of the probability the system is in thetransition state to the probability that it is in the reactantstate (PTS/PR) at 37 C.
PTS/PR = (7.3 x 10-10 iscorrect)
b.) Use your answer above to estimate the time it takes for thereaction to occur spontaneously at 37 C. Assume that the systemsamples a new microstate every nanosecond (10−9 s).
[Hint: If you throw a 6-sided die, what is the probability youwill get a 3? How many times would you expect to have to throw thedie to get a 3? Using similar logic, use your answer in the firstpart to tell you how many times the system has to try before itgets 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. Ifthe temperature is increased to 1000 K, how long does it take forthe reaction to occur?
time: (.009325 s is incorrect)
d.) In biological systems it is not feasible to acceleratereactions by heating them to 1000 K. Instead, the reactions rely onenzymes that catalyze reactions by lowering the energy of thetransition state. If catalyzed reaction has a transition stateenergy 2.0×10−20 J, what is the approximate reactiontime? Use a temperature 37 C.
time: (6.52 x 10-9 s is incorrect)
e.) Finally, estimate the uncatalyzed reaction time if thetransition state is not a unique microstate, but actually anensemble of 100 microstates. Use a temperature 37 C.
time: (4.675 x 10-9 s is incorrect)
