Physical Chemistry , 1st ed.

Here, we can see that k 1 is much larger than k 2 (that is, 10^6 is two orders

of magnitude higher than 10^8 ), so we might expect that there would be a

“momentary” buildup of polonium. Then, as time proceeds, the amount of

polonium would decrease as it decays to the stable lead isotope. Figure 20.14

shows a plot of what ultimately happens to 1.00 gram of^210 Bi.

The complete 4n2 series has 14 nuclear reactions between^238 U and

(^206) Pb, with a total of 15 nuclear species. Can you imagine the 15 mathemat-
ical expressions that give their concentrations over time?

20.6 Temperature Dependence

The rates of chemical reactions are strongly affected by temperature. This is

one reason why most declarations of rate constants include a temperature at

which that constant is valid. Common temperatures are 25°C (a common stan-

dard temperature) and 37°C (“normal” human body temperature). Because

temperature is an obvious thermodynamic variable, this section considers an-

other relationship between thermodynamics and kinetics.

Perhaps the most straightforward relationship between temperature and

rate constants was suggested by Svante Arrhenius (Figure 20.15) in 1889. He

used a thermodynamic approach in the form of an analogy. According to an

expression known as the van’t Hoff equation (notthe van’t Hoff equation from

osmotic pressure considerations), the temperature variation in the equilibrium

constant of a process is







(

(

l

1

n

/T

K

)

)



r

R

xnH (20.48)

where rxnHis the change in enthalpy of the reaction and Ris the ideal gas

law constant. Arrhenius proposed an analogous equation by suggesting an

“equilibrium” between reactant molecules and some transition species that is

higher in energy (that is, less stable) than the reactants. The energy difference

is called the energy of activationor, more simply, the activation energyof the

reaction. The “equilibrium constant” of this so-called equilibrium is the rate

702 CHAPTER 20 Kinetics

1.0

0

0

Time (s)

6  107

(^210) Po
(^206) Pb
(^210) Bi
Fraction present
0.8
0.6
0.4
0.2
1  107 2  107 3  107 4  107 5  107
Figure 20.14 See Example 20.7. This plot shows the concentrations of^210 Bi,^210 Po, and^206 Pb
over time. Note the “temporary” buildup of^210 Po, which does start at 0. Note that the x-axis is
in units of seconds, but in this example the right side of the plot is equivalent to a time of
1.9 years.
Figure 20.15 Svante Arrhenius (1859–1927),
a Swedish chemist who—among other things—
came up with a simple relationship between the
rate constant and the absolute temperature.
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