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43.25. Tritium ( \(^{3} \mathrm{H} )\) undergoes \(\beta^{-}\) decay with a half- life of 12.3 years. It is also highly toxic to living things. (a) What nucleus is produced in the \(\beta^{-}\) decay of tritium? (b) Suppose some tritium gas is released into the atmosphere in a nuclear power plant accident. How long will it take for 90.0\(\%\) of the tritium to become nonradioactive?

Short Answer

Expert verified
(a) The product is helium-3 (\(^{3}\mathrm{He}\)). (b) It takes about 40.61 years for 90% to decay.

Step by step solution

01

Understanding Beta Decay

Beta decay, specifically beta-minus decay (\(\beta^{-}\) decay), involves a neutron in an atomic nucleus transforming into a proton, while emitting a beta particle (an electron) and an anti-neutrino. In this process, the atomic number increases by one because a new proton is formed, while the atomic mass number remains unchanged.
02

Identifying the New Nucleus

Tritium, denoted as \(^{3}\mathrm{H}\), undergoes \(\beta^{-}\) decay. In this process, one of its neutrons turns into a proton. This turns tritium into helium-3, which is denoted as \(^{3}\mathrm{He}\), as the atomic number increases from 1 (hydrogen) to 2 (helium), but the mass number remains at 3.
03

Understanding Half-life

The half-life of a radioactive isotope is the time it takes for half of the isotope to decay. Tritium has a half-life of 12.3 years, meaning that every 12.3 years, the remaining amount of tritium is halved.
04

Calculating the Decay Using Half-life

We need to determine how long it takes for 90% of tritium to decay, which leaves 10% remaining. We'll use the decay formula: \[ N = N_0 \times \left( \frac{1}{2} \right)^{t/T_{1/2}} \] where \( N \) is the remaining amount, \( N_0 \) is the initial amount, \( t \) is the time in years, and \( T_{1/2} \) is the half-life. We need \( N/N_0 = 0.1 \).
05

Solving the Equation for Time

Rearrange the decay formula to solve for \( t \): \[ \left( \frac{1}{2} \right)^{t/T_{1/2}} = 0.1 \] \[ t/T_{1/2} \cdot \log_{10}(0.5) = \log_{10}(0.1) \] \[ t/T_{1/2} = \frac{\log_{10}(0.1)}{\log_{10}(0.5)} \] Substitute \( T_{1/2} = 12.3 \) years: \[ t = 12.3 \times \frac{\log_{10}(0.1)}{\log_{10}(0.5)} \approx 40.61 \text{ years} \]
06

Conclusion

(a) The nucleus produced in the \(\beta^{-}\) decay of tritium is helium-3, \(^{3}\mathrm{He}\). (b) It will take approximately 40.61 years for 90% of tritium to decay and become non-radioactive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tritium
Tritium, or hydrogen-3, is a rare isotope of hydrogen. Like other isotopes, it has the same number of protons as hydrogen, but differs in the number of neutrons.
While regular hydrogen has no neutrons, tritium contains two neutrons in its nucleus, giving it a greater atomic mass.
Tritium is naturally occurring but can also be produced artificially. It is used in scientific applications and is a byproduct in nuclear reactors. Despite its utility, tritium is highly radioactive and poses a threat to living organisms due to its ability to replace regular hydrogen atoms in molecular structures.
When considering its applications and dangers, it is important to handle tritium with care and respect due to its radioactive nature and potential health impacts.
Beta Decay
Beta decay is a type of radioactive process where an unstable atomic nucleus transforms to achieve stability by emitting particles. In beta-minus decay ( \( \beta^- \) decay), a neutron in the nucleus turns into a proton.
This transformation increases the atomic number by one, converting the element into a new element.
Beta-minus decay involves the emission of a beta particle, which is essentially an electron. An anti-neutrino is also released during this decay. The emitted beta particle is key to distinguishing beta decay from other decay types.
Understanding beta decay allows us to predict changes in elements and aids in calculations pertaining to nuclear reactions or decay processes. In the case of tritium, during its beta decay, it transitions into helium-3.
Half-life
The concept of half-life is fundamental to understanding radioactive decay. It refers to the time required for half the atoms in a radioactive sample to decay. For tritium, this period is 12.3 years.
This means that after 12.3 years, only half of any given sample of tritium would remain unchanged.
The half-life is crucial in predicting how long a radioactive isotope remains active. It is calculated using decay formulas that mathematicians and scientists utilize to model decay over time.
Understanding half-life helps us measure the rate of decay and estimate timelines for the decay process. This concept is not only important in nuclear chemistry but also in gauging the longevity of materials in various fields.
Helium-3
Helium-3 is a stable isotope of helium, composed of two protons and one neutron. It is the product of tritium's beta decay.
This isotope is non-radioactive and less hazardous to living beings compared to its predecessor, tritium.
Helium-3 has unique properties that make it valuable for different applications. It is used in cryogenics, thermonuclear weapons, and nuclear fusion research. Particularly in fusion, helium-3 is considered a potential fuel due to its large energy yield in reactions. Because of its scarcity on Earth, helium-3 is often touted as a valuable resource for future space exploration and colonization efforts, where it might be mined from lunar surfaces or other celestial bodies.

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Most popular questions from this chapter

43.31. The radioactive nuclide "Pt has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of \(7.56 \times 10^{11} \mathrm{Bq}\) . (a) How many 199 \(\mathrm{Pt}\) nuclei are initially present in the sample? (b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a time 92.4 minutes after the sample is first prepared.

43.36. To Sean or Not to Scan? It has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans using \(x\) rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the radiation they impart. Typically, one such scan gives a dose of 12 \(\mathrm{mSv}\) , applied to the whole body. By contrast, a chest \(\mathbf{x}\) ray typically administers 0.20 \(\mathrm{mSv}\) to only 5.0 \(\mathrm{kg}\) of tissue. How many chest \(\mathrm{x}\) rays would deliver the same total amount of energy to the body of \(75-\mathrm{kg}\) person as one whole-body scan?

43.44. The United States uses \(1.0 \times 10^{19} \mathrm{J}\) of electrical energy per year. If all this energy came from the fission of \(^{235} \mathrm{U}\) , which releases 200 MeV per fission event, (a) how many kilograms of 235 \(\mathrm{U}\) would be used per year and (b) how many kilograms of uranium would have to be mined per year to provide that much 235 \(\mathrm{U} ?\) (Recall that only 0.70\(\%\) of naturally occurring uranium is \(^{235} \mathrm{U.}\) )

43.72. An Oceanographic Tracer. Nuclear weapons tests in the 1950 s and 1960 s released significant amounts of radioactive tritium \((3 \mathrm{H}, \text { half-life } 12.3 \text { years) into the atmosphere. The tritium }\) atoms were quickly bound into water molecules and rained out of the air, most of them ending up in the ocean. For any of this tritium-tagged water that sinks below the surface, the amount of time during which it has been isolated from the surface can be calculated by measuring the ratio of the decay product, \(\frac{3}{2} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \(\frac{3}{1}\) He to \(\frac{3}{1} \mathrm{H}\) in a sample of water is \(1 : 1,\) the water has been below the surface for one half-life, or approximately 12 years. This method has provided oceanographers with a convenient way to trace the movements of subsurface currents in parts of the ocean. Suppose that in a particular sample of water, the ratio of \(\frac{3}{2} \mathrm{He}\) to \(\frac{3}{1} \mathrm{H}\) is 4.3 to 1.0 . How many years ago did this water sink below the surface?

43.27. Measurements on a certain isotope tell you that the decay rate decreases from 8318 decays/min to 3091 decays/min in 4.00 days. What is the half-life of this isotope?

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