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Chapter 30: Problem 31

The radioactive hydrogen isotope \({ }^{3} \mathrm{H}\) is called tritium. It decays by beta-minus decay with a half-life of 12.3 years. a. What is the daughter nucleus of tritium? b. A watch uses the decay of tritium to energize its glowing dial. What fraction of the tritium remains 20 years after the Watch was created?

Short Answer

Expert verified
a. The daughter nucleus of tritium is helium-3. b. About 66.22% of the tritium remains in the watch after 20 years.

Step by step solution

01

Identify Daughter Nucleus

In beta-minus decay, a neutron in the nucleus is converted into a proton and an electron (the beta particle), with the electron being emitted. This means the atomic number increases by 1, but the atomic mass remains the same. Hydrogen-3 (tritium) has an atomic number of 1 and a mass number of 3. After beta-minus decay, the atomic number will increase by 1 to become 2, while the atomic mass remains the same - 3. An atom with atomic number 2 and mass number 3 is helium-3. Therefore, the daughter nucleus of tritium is helium-3.
02

Calculate Fraction of Tritium Remaining

The half-life of a substance is the time taken for half of the substance to decay. In this case, tritium's half-life is 12.3 years. Two half-lives would therefore be 24.6 years. However, the watch has only been present for 20 years, which is less than this. This suggests that less than half of the tritium has decayed. To identify exactly how much remains, calculate the fraction of a half-life that 20 years represents (20 years / 12.3 years per half-life) and then use this number as a power of 0.5 (representing halving) to calculate the remaining fraction of tritium: (0.5^(20 / 12.3)).

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Key Concepts

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

Tritium Half-Life
Understanding the half-life of an isotope like tritium is crucial for comprehending how it decays over time. The half-life is the period required for half of the isotope's atoms to undergo radioactive decay.

In the case of tritium (tritium{}), which has a half-life of 12.3 years, we can calculate how much of this substance will remain after a given number of years. This is essential not only for scientific calculations but also for practical applications, such as the longevity of glow-in-the-dark watch dials that rely on tritium's decay to provide luminescence over time.

Over one half-life, the quantity of tritium would be halved. Consequently, after two half-lives, a quarter of the original amount would remain. This geometric progression continues as more half-lives pass. The concept of half-life helps us predict how long a radioactive material like tritium will continue to be active, which is fundamental for safety protocols in industries and healthcare where radioactive materials are used.
Daughter Nucleus Identification
When a radioactive element undergoes decay, it transforms into a different element; this new element is known as the 'daughter nucleus'. Beta-minus decay is a type of decay in which a neutron is transformed into a proton while emitting an electron (beta particle) and an antineutrino.

For tritium (tritium{}), this decay process increases its atomic number by 1 while the atomic mass remains unchanged. Tritium, with an atomic number of 1, transforms into helium-3 ({}^{3}He), with an atomic number of 2, which is the daughter nucleus in this scenario. Identifying the daughter nucleus is a fundamental aspect of understanding radioactive decay processes and helps with tracing the decay chains of various isotopes, providing insights into their eventual stability or further transformations.
Radioactive Decay Calculations
Radioactive decay calculations involve determining the remaining quantity of a radioactive isotope after a certain period, relying on the half-life concept. The mathematical approach to such calculations involves using the exponential decay formula.

In practice, let's consider the tritium in the watch's dial with a half-life of 12.3 years. To determine the fraction remaining after 20 years, you divide the elapsed time (20 years) by the half-life (12.3 years) to find the number of passed half-lives. Using the formula (0.5^{20 / 12.3}), we find the fraction of tritium that remains. Knowing how to calculate this decay is essential for predicting the behavior of radioactive substances over time, which has applications in fields ranging from geochronology to medicine, where precise dosages of radioactive materials are critical for treatments.

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Most popular questions from this chapter

The decay chain of uranium includes radon, a noble gas. When uranium in the soil decays to radon, it may seep into houses; this can be a significant source of radiation exposure. Most of the exposure comes from the decay products of radon, but some comes from alpha decay of the radon itself. If radon in the air in your home is at the maximum permissible level, the gas in your lungs will have an activity of about \(0.22 \mathrm{Bq} .\) Each decay generates an alpha particle with \(5.5 \mathrm{MeV}\) of energy, and essentially all that energy is deposited in lung tissue. Over the course of 1 year, what will be the dose equivalent in Sv to the approximately \(0.90 \mathrm{kg}\) mass of your lungs?

Calculate (in MeV) the total binding energy and the binding energy per nucleon (a) for \({ }^{40} \mathrm{Ar}\) and (b) for \({ }^{40} \mathrm{K}\).

The technique known as potassiumargon dating is used to date volcanic rock and ash, and thus establish dates for nearby fossils, like this 1.8 -millionyear- old hominid skull. The potassium isotope \({ }^{40} \mathrm{K}\) decays with a 1.28 -billionyear half-life and is naturally present at very low levels. The most common decay mode is beta-minus decay into the stable isotope \({ }^{40}\) Ca, but \(10.9 \%\) of decays result in the stable isotope \({ }^{40}\) Ar. The high temperatures in volcanoes drive argon out of solidifying rock and ash, so there is no argon in newly formed material. After formation, argon produced in the decay of \({ }^{40} \mathrm{K}\) is trapped, so \({ }^{40}\) Ar builds up steadily over time. Accurate dating is possible by measuring the ratio of the number of atoms of \({ }^{40} \mathrm{Ar}\) and \({ }^{40} \mathrm{K} .1 .8\) million years after its formation, a. What fraction of the \({ }^{40} \mathrm{K}\) initially present in a sample has decayed? b. What is the \({ }^{40} \mathrm{Ar} /{ }^{40} \mathrm{K}\) ratio of the sample?

Medical gamma imaging is generally done with the technetium isotope \({ }^{99} \mathrm{Tc}^{*},\) which decays by emitting a gamma-ray photon with energy 140 keV. What is the mass loss of the nucleus, in u, upon emission of this gamma ray?

\(^{90} \mathrm{Sr}\) decays with the emission of a \(2.8 \mathrm{MeV}\) beta particle. Strontium is chemically similar to calcium and is taken up by bone. A 75 kg person exposed to waste from a nuclear accident absorbs \({ }^{90}\) Sr with an activity of 370,000 Bq. Assume that all of this \({ }^{90} \mathrm{Sr}\) ends up in the skeleton. The skeleton forms \(17 \%\) of the person's body mass. If \(50 \%\) of the decay energy is absorbed by the skeleton, what dose equivalent in Sv will be received by the person's skeleton in the first month?
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