[structure tool home]

(1) First Order Integrated Rate Expressions

(2) Second Order Integrated Rate Expressions

Differential rate expressions show how rates of reactions depend on concentrations of reagents. Converting rate expressions from differentials to integrals changes a rate expression from change in concentration with change in time (dc/dt) to concentration versus time [f(c) versus t]. Both expressions can help us understand the mechanism of a reaction. It is useful to know how to convert between the two expressions. Below are two examples of how differential rate expressions can be converted to expressions that give time dependence of concentration.

[Second Order Integrated Rate Expressions]

The reaction

is first order with respect to the concentration of A.

The negative sign here indicates that the [A] is being consumed; d[A]/dt is less than zero. If we let c stand for the concentration of A, the differential rate expression is

Infinitesimal quantities like dc and dt can be manipulated just like ordinary numbers. So the equation can be rearranged.

Now integrate both sides of this equation, choosing the appropriate limits of integration; co is the concentration at time zero (t=0) and c is the concentration at some later time (t=t).

Integrating gives:

This equation shows that for a first-order reaction, the natural log (ln) of the reactant concentration decreases linearly as time increases. A plot of In(c) versus t will give a straight line with slope -k and intercept ln(co).

• Mono-bumper detachment, a familiar example. Every day a certain number of car bumpers fall from cars onto the central artery of Boston (Interstate 93). Attached bumpers are converted to detached bumpers.

bumpera -> bumperd

We are considering a spontaneous detachment process, and are specifically excluding bi-bumper events involving cars crashing into each other. We shall consider bi-bumper processes later. We also assume that attached bumpers do not leave the central artery by pathways other than detachment, for example by exiting onto the Mass Pike. The rate of spontaneous bumper detachment is

d[bumperd]/dt

which is equal to -d[bumpera]/dt, (the negative sign indicates that the concentration of attached bumpers decreases over time).

-d[bumpera]/dt depends on

[bumpera]	the concentration of attached bumpers on the central artery, and
k	a constant that is determined by how well the average manufacturer attaches bumpers to cars, the amount of road salt used over several years, the number of gigantic potholes, etc.

So, the differential rate expression is first order with respect to concentration of attached bumpers is:

The rate of car bumper detachment increases linearly with the concentration of car bumpers. The concentration of trucks and truck bumpers is not relevant. A plot of In[bumpera] versus t will give a straight line with slope -k and intercept In[bumpera]initial.

[go to top]

[First Order Integrated Rate Expressions]

First and second order reactions have different time dependencies of reactant concentrations. Consider the reaction below, which is second order with respect to A.

Remember, c stands for the concentration of A.

So, for a second-order reaction, the reciprocal of the reactant concentration increases linearly with time. A plot of 1/c versus t will give a straight line with slope 2k and intercept 1/co.

• Bi-bumper detachment, another familiar example. Every day a certain number of car bumpers fall from cars onto the central artery of Boston (Interstate 93) after being bashed by other car bumpers. As above we assume that attached bumpers do not leave the central artery by pathways other than detachment. In the bi-bumper mechanism, attached bumpers are converted to detached bumpers, just as in the mono-bumper mechanism described above. But the differential rate expressions for the two mechanism differ. When detachment requires bashing by another bumper, two attached bumpers interact to produce one attached bumper and one detached bumper.

So the rate of detachment is second order with respect to bumper concentration. In this case the rate of spontaneous bumper detachment (-d[bumpera]/dt), depends on

[bumpera]2	the square of concentration of attached bumpers on the central artery, and
K	a constant that is determined by how well the average manufacturer attaches bumpers to cars, the amount of salt used over several years, the average speed of the cars, etc.

The rate equation is:

So, the rate of car bumper detachment increases with the square of the concentration of attached car bumpers. Again the concentration of trucks and truck bumpers is not relevant, but of course certain other elementary process involving trucks can detach bumpers from cars. The reciprocal of the attached bumper concentration increases linearly with time. A plot of 1/[bumpera] versus t will give a straight line with slope 2k and intercept 1/[bumper]initial.

In sum, we should be able to determine the mechanism of bumper detachment on the central artery by making two graphs. If 1/[bumpera] versus t is linear, then the mechanism of bumper detachment is cars bashing into each other. If ln[bumpera] versus t is linear, the bumpers are spontaneously falling off, with no requirement for interaction from a second bumper.

[structure tool home]

[go to top]