1550078481-Ordinary_Differential_Equations__Roberts_

30 Ordinary Differential Equations

and

(5)

Evaluating equation (3) at t = t 0 and imposing the condition (4), we see

the constant P must be chosen to satisfy Q(t 0 ) = Qo = Pekto. Hence,

P = Q 0 e-kto. Substituting this expression into equation (3), we find the
solution of the DE (1) which satisfies condit ion (4) has the form

(6) Q(t) = Qoek(t-to).

Evaluating equation (6) at t =ti and imposing the condition (5), we see the

constant k must satisfy

(7)

Solving equation (7) for k , we find the constant of proportionality is

k = ln Qi - ln Qo.

ti - to

(8)

In the radioactive decay process, the amount of substance present decreases

with increasing time. Since we have assumed ti > to, it follows that Qi < Qo.

Hence, ti - to > 0, and since ln Qi < ln Qo, we h ave ln Qi - ln Qo < 0. It

then follows from equation (8) that k < 0.

The rate of decay of a radioactive substance is often expressed in terms of

half-life- that is , the time required for a ny given quantity of the substance

to be reduced by a factor of one-half. If in equation (8) we let Qi = ~Qo,

then the half-life T =ti - to satisfies the equation

(9) kT = -ln2.

Consequently, if either k or Tis known or can be determined experimentally,

then the other variable can be determined from equation (9).

The half-life of uranium 238 is 4.5 billion years, the half-life of potassium

40 is 1.4 billion years, and the h alf-life of rubidium 87 is 60 billion years.

By checking the ratio of elements such as these to the elements into which

they decay radioactively, geologists and archaeologists can reliably estimate

dates of significant events that occurred millions and even billions of years

ago. However, since they decay so slowly, radioactive elements with h alf-lives

of millions or billions of years are not suitable for dating events which took

place relatively recently.

In the late 1940s and early 1950s, the American chemist Willard F. Libby

(1908-1980) developed the technique of radiocarbon dating, which can be

used to estimate the dates of events that occurred up to 50,000 years ago.

In 1960, Libby was awarded the Nobel Prize in Chemistry for this achieve-

ment. Libby's technique is based on a phenomenon which involves the radioac-

tive isotope carbon 14 , i^4 c, which is called radiocarbon and has a half-life of