Page 462 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 36] NUCLEAR PHYSICS 447

HALF-LIFE
A nucleus subject to radioactive decay always has a certain definite probability of decay during any time interval.
The half-life of a radioactive isotope is the time required for half of any initial quantity to decay. If an isotope
has a half-life of, say 5 h, and we start with 100 g of it, 50 g will be left undecayed after 5 h; 25 g will be left
undecayed after 10 h; 12.5 g will be left undecayed after 15 h; and so on.

SOLVED PROBLEM 36.12
3
Tritium is the hydrogen isotope H whose nucleus contains two neutrons and a proton. Tritium is beta-
1
radioactive and emits an electron. (a) What does tritium become after beta decay? (b) The half-life of
tritium is 12.5 yr. How much of a 1-g sample will remain undecayed after 25 yr?
(a) In the beta decay of a nucleus, one of its neutrons becomes a proton. Since the atomic number 2 corresponds
3
to helium, the beta decay of Hisgiven by
1
3
3 H → He + e −
1 2
3
and the new atom is He.
2
(b) Twenty-five years is two half-lives of tritium, and so 1 × 1 × 1g = 1 g of tritium remains undecayed.
2 2 4
SOLVED PROBLEM 36.13
24
The half-life of the sodium isotope Na against beta decay is 15 h. How long does it take for seven-eighths
11
of a sample of this isotope to decay?
After seven-eighths has decayed, one-eighth is left, and 1 = 1 × 1 × 1 which is three half-lives. Hence the
8 2 2 2
answer is (3)(15) h = 45 h.

SOLVED PROBLEM 36.14
14
The carbon isotope C (called radiocarbon) is beta-radioactive with a half-life of 5600 yr. Radiocarbon
6
is produced in the earth’s atmosphere by the action of cosmic rays on nitrogen atoms, and the carbon
dioxide of the atmosphere contains a small proportion of radiocarbon as a result. All plants and animals
12
therefore contain a certain amount of radiocarbon along with the stable isotope C. When a living thing
6
dies, it stops taking in radiocarbon and the radiocarbon it already contains decays steadily. By measuring
the ratio between the 14 C and 12 C contents of the remains of an animal or plant and comparing it with
6 6
the ratio of these isotopes in living organisms, the time that has passed since the death of the animal or
plant can be found. (a) How old is a piece of wood from an ancient dwelling if its relative radiocarbon
content is one-fourth that of a modern specimen? (b) If it is one-sixteenth that of a modern specimen?
1
(a) Since 1 = 1 × , the specimen is two half-lives old, which is 11,200 yr old.
4 2 2
1
(b) Since 1 = 1 × 1 × 1 × , the specimen is four half-lives old, which is 22,400 yr old.
16 2 2 2 2
SOLVED PROBLEM 36.15
The rate at which a sample of radioactive substance decays is called its activity. A unit of activity is the
curie (Ci), where 1 curie = 3.7 × 10 10 decays/s. If a luminous watch dial contains 5 µCi of the radium
isotope 226 Ra, how many decays per second occur in it? (This isotope emits alpha particles which cause
88
flashes of light when they strike a special material the isotope is mixed with.)
Since 1 µCi is 10 −6 Ci, the activity of the watch dial is

decays/s
5
3.7 × 10 10 (5 × 10 −6 Ci) = 1.85 × 10 decays/s
Ci