Question 25.3: Using the Half-Life Concept and the Radioactive Decay Law to...
Using the Half-Life Concept and the Radioactive Decay Law to Describe the Rate of Radioactive Decay
The phosphorus isotope _{}^{32}\text{P} is used in biochemical studies to determine the pathways of phosphorus atoms in living organisms. Its presence is detected through its emission of \beta^- particles. (a) What is the decay constant for _{}^{32}\text{P}, expressed in the unit s^{-1}? (b) What is the activity of a 1.00 mg sample of _{}^{32}\text{P} (that is, how many atoms disintegrate per second)? (c) Approximately what mass of _{}^{32}\text{P} will remain in the original 1.00 mg sample after 57 days? (See Table 25.1.) (d) What will be the rate of radioactive decay after 57 days?
Analyze
To solve these types of problems, we need to determine the decay constant, λ, of the radioactive species, which is related to the concept of half-life through equation (25.13). After determining the decay constant from the half-life of the sample, we can then use it to determine the activity for part (b).
t_{1 / 2}=\frac{0.693}{\lambda} (25.13)
| TABLE 25.1 Some Representative Half-Lives | |||||
| Nuclide | \text{Half-Life}^a | Nuclide | \text{Half-Life}^a | Nuclide | \text{Half-Life}^a |
| _{1}^{3}\text{H} | 12.26 y | _{19}^{40}\text{K} | 1.25 \times 10^9 y | _{84}^{214}\text{Po} | 1.64 \times 10^{-4} s |
| _{6}^{14}\text{C} | 5730 y | _{35}^{80}\text{Br} | 17.6 min | _{86}^{222}\text{Rn} | 3.823 d |
| _{8}^{13}\text{O} | 8.7 \times 10^{-3} s | _{38}^{90}\text{Sr} | 27.7 y | _{88}^{226}\text{Ra} | 1.60 × 10³ y |
| _{12}^{28}\text{Mg} | 21 h | _{53}^{131}\text{I} | 8.040 d | _{90}^{234}\text{Th} | 24.1 d |
| _{15}^{32}\text{P} | 14.3 d | _{55}^{137}\text{Cs} | 30.23 y | _{92}^{238}\text{U} | 4.51 \times 10^9 y |
| _{16}^{35}\text{S} | 88 d | ||||
_{}^{a}\text{s}, second; min, minute; h, hour; d, day; y, year.
