Question

Compare and contrast how first-order reaction kinetics and zero-order reaction kinetics differ in terms of the depletion of a reactant A in a chemical reaction. Assume an initial concentration of 420 mg of A. Assume a half-life of 30 minutes if first-order kinetics are obeyed, and a depletion rate of 60 mg/hour if zero-order kinetics are obeyed. Use a clearly drawn graph(s) to determine which rate will result in complete depletion of A over the shortest time. Assume 99% to be complete depletion of A

Answer 1

Note: First we will describe each type of the reaction kinetics one by one and then proceed to compare them graphically to determine which is faster.

Zero Order reaction

For a generic reaction

Rate of the reaction means either the rate of appearance of product or the rate of disappearance of reactants and written as

Note that there is a -ve sign in front for the rate of disappearance of A as rate of the reaction is a positive quantity which measure how fast the reaction is happening.

Now for a zero-order reaction the rate is independent of the concentration of reactant.

or simply

where k is the depletion rate of A.

Integrating the above relation we get

Hence, given that the initial amount of A present

and k = 60 mg/hr i.e 1 mg/min.

We can plot

We can see the the amount of A drop to 0 mg after 420 min. Note we can take 100% completion for this zero order reaction instead of 99% completion.

Or simply we can solve for time t for completion of the reaction by putting in the integrated rate law for zero order scheme

Now coming to the first order reaction.

Where k is the first order rate constant

Here the rate of the reaction depends on the first power of concentration of A.

Integrating the rate law we get

Taking exponential on both side

Now given that the half life of the first order reaction is 30 min, it means the amount decreases to half its initial value value after 30 min.

Hence, the plot of [A] vs t will be

Note that here for 99% completion we can take

Hence, putting it in the equation for first order reaction

We can also see in the graph that around 200 min the amount of A has decreased to almost 0 mg whereas zero order reaction took 420 min for completion.

Hence, clearly first order rate will result in faster depletion of A.