Page 293 - Theory and Problems of BEGINNING CHEMISTRY

282 NUCLEAR REACTIONS [CHAP. 19

19.3. HALF-LIFE
Not all the nuclei of a given sample of a radioactive isotope disintegrate at the same time, but do so over a
period of time. The number of radioactive disintegrations per unit time that occur in a given sample of a naturally
radioactive isotope is directly proportional to the quantity of that isotope present. The more nuclei present, the
more will disintegrate per second (or per year, etc.).

EXAMPLE 19.3. Sample A of 235 U has a mass of 0.500 kg; sample B of the same isotope has a mass of 1.00 kg. Compare
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the rate of decay (the number of disintegrations per second) in the two samples.
Ans. There will be twice the number of disintegrations per second in sample B (the 1.00-kg sample) as in sample A,
because there were twice as many 235 U atoms there to start. The number of disintegrations per gram per second is a
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constant, because both samples are the same isotope— 235 U.
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After a certain period of time, sample B of Example 19.3 will have disintegrated so much that there will
be only 0.500 kg of 235 U left. (Products of its decay will also be present.) How would the decay rate of this
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0.500 kg of 235 U compare with that of 0.500 kg of 235 U originally present in sample A of Example 19.3? The
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rates should be the same, since each contains 0.500 kg of 235 U. That means that sample B is disintegrating at a
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slower rate as its number of 235 U atoms decreases. Half of this sample will disintegrate in a time equal to the time
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it took one-half the original sample B to disintegrate. The period in which half a naturally radioactive sample
disintegrates is called its half-life because that is the time required for half of any given sample of the isotope to
disintegrate (Fig. 19-1). The half-lives of several isotopes are given in Table 19-2.

Table 19-2 Half-Lives of Some Nuclei
If 0.500 kg takes 1 half-life to undergo the process shown
for sample A, it also should take the same 1 half-life to Isotope Half-Life Radiation ∗
undergo the process shown for sample B, after sample B
gets down to 0.500 kg.
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238 U 4.5 × 10 years Alpha
8
235 U 7.1 × 10 years Alpha
237 Np 2.2 × 10 years Alpha
6
14 C 5730 years Beta
90 Sr 19.9 years Beta
3 H 12.3 years Beta
140 Ba 12.5 days Beta
Sample A 0.500 kg 0.250 kg 131 I 8.0 days Beta
15 O 118 s Beta
94 Kr 1.4 s Beta
Sample B 1.00 kg 0.500 kg 0.250 kg

Fig. 19-1. A half-life example ∗ In most of these processes, gamma radiation is also emitted.

EXAMPLE 19.4. A certain isotope has a half-life of 3.00 years. How much of a 8.00-g sample of this isotope will remain
after 12.0 years?
Ans. After the first 3.00 years, half the 8.00-g sample will still be the same isotope. After 3.00 years more, half of the
4.00 g remaining will still be the original isotope. That is, 2.00 g will remain. After the third 3.00-year period, half
of this 2.00 g will remain, or 1.00 g, of the original isotope will have its original identity, and after the fourth 3.00
years, half of that sample, or 0.500 g, will remain.

EXAMPLE 19.5. A certain isotope has a half-life of 3.00 years. How much of an 8.00-g sample of this isotope will have
decomposed after 12.0 years?
Ans. We calculated in Example 19.4 that 0.500 g would remain. That means that 8.00 g − 0.500 g = 7.50 g of the isotope
will have disintegrated. (Most of the 7.50 g will produce some other isotope.) Notice the difference in the wording of