Physical Chemistry , 1st ed.

Example 20.3

One example of a first-order reaction is the isomerization of hydrogen iso-

cyanide to hydrogen cyanide:

HNC (g) →HCN (g)

If the rate constant at a particular temperature is 4.403 10 ^4 s^1 , what

massof HNC remains after 1.50 hr if a 1.000-gram sample of HNC was pres-

ent at the beginning of the reaction?

Solution

We can use equation 20.14 directly by recognizing that k4.403 10 ^4 1/s,

[A] 0 1.000 g, and t1.50 hr, which we must convert to units of seconds:

t5400 s. With these numbers, we get

ln 

1.

[

0

A

0

]

0

t

g

4.403 10 ^4 

1

s

(5400 s)

ln 1.

[

0

A

0

]

0

t

g2.378

Taking the inverse logarithm of both sides, we get



1.

[

0

A

0

]

0

t

g

238.5

Solving for the final amount:

[A]t0.00419 g

Just over 0.4% of the original material remains as the unreacted HNC.

There are two other ways of expressing equation 20.14 mathematically. One

way is to take the inverse logarithm of both sides, then rearrange the variables

so we get an expression for [A]tas the time varies. We get

[A]t[A] 0 ekt (20.15)

which shows that the amount at any time tfollows a negative exponential func-

tion of time. (You should note that the initial amount, [A] 0 , is a constant for

a given experiment.) Negative exponential functions have the characteristic of

having a maximum value at the variable t0, and declining monotonically

and asymptotically toward zero. Figure 20.1 shows the general trend for [A]t

over time. The speed with which the amount [A]tapproaches zero is dictated

by the rate constant k.

Another way to rewrite equation 20.14 is to separate the logarithms of the

numerator and denominator in the fractional term. We can do that and rewrite

the equation as

ln [A]tln [A] 0 kt (20.16)

This equation has the form of a straight line ymxb,where yis ln[A]t,

the slope mis k,xis t(the elapsed time), and the y-intercept bln[A] 0.

(Again, since [A] 0 is a constant, so is the logarithm of [A] 0. We usually ignore

the units on [A] 0 and [A]t—you can take a logarithm only of a pure

number, not of a unit—but we require that they be expressed in the same

20.3 Characteristics of Specific Initial Rate Laws 687

Figure 20.1 For a first-order reaction, the con-
centration at any time, labeled [A]t, decreases in
a characteristic negative exponential way. The rate
at which it approaches a zero concentration is
dictated by the value of the rate constant.
Mathematically, the plot of [A]tdoes not reach 0
until t.

[A] 0

Time

[A]

t