Concise Physical Chemistry

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

SECOND-ORDER REACTIONS 147

ln X

ln X 0

t

slope = –k

FIGURE 10.2 Logarithmic decay of a radioactive element.

measurement. From the form of the equation, lnXmay be taken as the dependent

variable whiletmay be as independent, whereupon the slope is equivalent to minus

the rate constant. The constant of integration can be evaluated by settingt=0,

C=−lnX 0

enabling us to write

−lnX=kt−lnX 0

which is equivalent to the forms obtained by integration between limits:

ln

X

X 0

=−kt

10.2 SECOND-ORDER REACTIONS

From the simple physical picture of a reacting mixture of reagents of two different

kinds, say A and B, in which molecules collide to produce one or more products, it is

logical to suppose that if either A or B is in short supply, the reaction must go slowly

because fewer A–B collisions take place even though the total number of collisions

may still be large. In the form of a rate law, the rate is proportional to both A and B

concentrations, hence it is proportional to theABproduct^2 :

Rate=−

dA

dt

=−

dB

dt

=kAB

(^2) The identity of the molecules is given in regular type AB, but the concentration variable referring to the
different species is italicAB.