Physical Chemistry Third Edition

11.2 Forward Reactions with One Reactant 495

graph. Denote the reactant by A. To test for zero order, one makes a graph of[A]t

as a function oft. To test for first order, one makes a graph of ln([A]) as a function

oft. To test for second order, one makes a graph of 1/[A]as a function oft.To

test for third order, one makes a graph of 1/(2[A]^2 ) as a function oft. Graphs of

1 /

(

(n− 1 )[A]n−^1

)

for various nonintegral values ofncan also be made. The graph

that is most nearly linear corresponds to the correct order. Figure 11.3 shows schematic

graphs for zero, first, second, and third order. Using commercially available software

packages such as Excel, MathCad, or Mathematica, one can construct the graphs and

carry out least-squares (regression) fits to the data. These software packages automat-

ically calculate thecorrelation coefficient(or its square), which is a measure of the

closeness of the fit of the function to the data. A perfect fit corresponds to a value of

the correlation coefficient equal to unity. Some packages will print out a list ofresid-

uals, which are the set of differences between the data points and the least-squares

line. The Excel spreadsheet can also print out the expected errors in the slope and

intercept.

Slope 52 kf

0

(a)

Zero order

[A]

t

Slope 5 kf

0

(c)

Second order

1/[A]

t

Slope 5 kf

0

(d)

Third order

1/2[A]

2

t

Slope 52 kf

0

(b)

First order

In

[A]

t

Figure 11.3 Linear Graphs for Zero-, First-, Second-, and Third-Order Reactions.

(a) Zero order. (b) First order. (c) Second order. (d) Third order.