Concise Physical Chemistry

c10 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

OTHER REACTION ORDERS 149

10.3 OTHER REACTION ORDERS

In general, the order of a chemical reaction is the sum of the exponents in the rate

equationa+b+c+···:

−

dA

dt

=kAaBbCc...

The order may be fractional or even decimal, but it is not a large number. Zero- and

third-order reactions exist, but they are not common. A zero-order reaction is one in

which the rate does not vary with time:

dA

dt

=k

This rate law might be followed by a reaction depending on some constant like the

surface area of a catalyst on which the reaction is taking place.

There are systematic mathematical methods for determining the order of a reaction

from a data set of concentration vs. time even when the order is not simple.

10.3.1 Mathematical Interlude: The Laplace Transform

TheLaplace transformis used to simplify differential equations. For the function

F(t) wheretis the time andF(t) is zero att<0, the Laplace transformf(s)ofF(t)

is defined as

f(s)=L[F(t)]=

∫∞

0

e−stF(t)dt

The inverse Laplace transform off(s)isF(t). The Laplace transformL[F(t)]is a

function of a function; it is called afunctional. Extensive tables of Laplace transforms

and inverse Laplace transforms are available (CRC Handbook of Chemistry and

Physics 2008–2009, 89th ed.).

First we shall convert the derivative of the functionF(t) to a simple algebraic

form. By definition,

L

[

dF(t)

dt

]

=

∫∞

0

e−st

dF(t)

dt

dt

Integration by parts gives

L

[

dF(t)

dt

]

=F(t)e−st

∣

∣∞

0 −

∫∞

0

F(t)d(e−st)

=−F(t=0)+s

∫∞

0

F(t)e−stdt

=−F(t=0)+sf(s)