Page 294 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 19] NUCLEAR REACTIONS 283

Examples 19.4 and 19.5. How much remains is determined from the half-life; how much decomposes is determined
from the original quantity and how much remains.

Half-life problems may be solved with the equation

N 0 0.693
ln = t
N t 1/2
where N is the number of radioactive atoms of the original isotope remaining at time t, N 0 is the number of
those atoms at the beginning of the process, t 1/2 is the half-life, and ln is the natural logarithm. The electronic
calculator gives natural logarithms and antilogarithms at the touch of a key.

EXAMPLE 19.6. Calculate the time required for a radioactive sample to lose one-fourth of the atoms of its parent isotope.
The half-life is 32.0 s.
Ans. If one-fourth of the atoms are lost, three-fourths remain. Therefore,

1 0.693
ln = t
3/4 32.0s
(32.0s)(0.288)
t = = 13.3s
0.693
In 13.3 s, one-fourth of the atoms will disintegrate. Note that this interval of time is less than the half-life, in which
one-half the atoms would disintegrate, which is reasonable.

19.4. RADIOACTIVE SERIES
When an isotope disintegrates spontaneously, the products include one of the small particles from Table 19-1
and a large nucleus. The new nucleus is not necessarily stable; it might itself decompose spontaneously, yielding
another small particle and still another nucleus. A disintegrating nucleus is called a parent nucleus, and the
product is called the daughter nucleus. Such disintegrations can continue until a stable nucleus is produced. It
turns out that four series of nuclei are generated from the disintegration of four different naturally occurring
isotopes with high mass numbers. Since each nucleus can produce only an alpha particle, a beta particle, or a
gamma particle, it turns out that the mass number of the daughter nucleus can be different from that of the parent
nucleus by 4 units or 0 unit. If an alpha particle is emitted, the daughter nucleus will be 4 units smaller; if a
beta or gamma particle is produced, the daughter nucleus will have the same mass number as the parent nucleus.
Thus, the mass numbers of all the members of a given series can differ from each other by some multiple of
4 units.

EXAMPLE 19.7. When a 238 U nucleus disintegrates, the following series of alpha and beta particles is emitted: alpha,
beta, beta, alpha, alpha, alpha, alpha, alpha, beta, alpha, beta, beta, beta, alpha. (Since emission of gamma particles ac-
companies practically every disintegration and since gamma particles do not change the atomic number or mass number
of an isotope, they are not listed.) Show that each isotope produced has a mass number that differs from 238 by some
multiple of 4.
Ans. Each alpha particle loss lowers the mass number of the product nucleus by 4. Each beta particle loss lowers the mass
number by 0. The mass numbers thus go down as follows:

238 −→ 234 −→ 234 −→ 234 −→ 230 −→ 226 −→ 222 −→ 218
α β β α α α α
α
↓
206 ←− 210 ←− 210 ←− 210 ←− 210 ←− 214 ←− 214
α β β β α β

There are four such series of naturally occurring isotopes. The series in which all mass numbers are evenly divisible
by 4 is called the 4n series. The series with mass numbers 1 more than the corresponding 4n series members is called
the 4n + 1 series. Similarly, there are a 4n + 2 series and a 4n + 3 series. Since the mass numbers change by 4 or 0,
no member of any series can produce a product in a different series.