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Chapter 42: Problem 9

Generally, more massive nuclides tend to be more unstable to alpha decay. For example, the most stable isotope of uranium, \({ }^{238} \mathrm{U}\), has an alpha decay half-life of \(4.5 \times 10^{9} \mathrm{y} .\) The most stable isotope of plutonium is \({ }^{244} \mathrm{Pu}\) with an \(8.0 \times 10^{7} \mathrm{y}\) half-life, and for curium we have \({ }^{248} \mathrm{Cm}\) and \(3.4 \times 10^{5} \mathrm{y}\). When half of an original sample of \({ }^{238} \mathrm{U}\) has decayed, what fraction of the original sample of (a) plutonium and (b) curium is left?

Short Answer

Expert verified
After half of the uranium has decayed, approximately 0 plutonium and 0 curium remain.

Step by step solution

01

Understanding the half-life concept

The half-life of a substance is the time required for half of the original quantity of a radioactive substance to decay. This concept will be used to determine how much of the plutonium and curium are left when half of the uranium has decayed.
02

Calculate time for uranium half-life decay

Start with knowing the half-life of uranium: \( t_{1/2}^{238\text{U}} = 4.5 \times 10^{9} \text{ years} \). This is the time we set as a reference when half of the original \(^{238}\text{U}\) sample has decayed.
03

Determine remaining fraction of plutonium

The half-life of plutonium \(^{244}\text{Pu}\) is \(8.0 \times 10^{7} \text{ years} \). The number of half-lives that occur in \(4.5 \times 10^{9}\) years is \( \frac{4.5 \times 10^{9}}{8.0 \times 10^{7}} = 56.25 \). Thus, the remaining fraction of plutonium is \( \left( \frac{1}{2} \right)^{56.25} \).
04

Determine remaining fraction of curium

The half-life of curium \(^{248}\text{Cm}\) is \(3.4 \times 10^{5} \text{ years} \). The number of half-lives that occur in \(4.5 \times 10^{9}\) years is \( \frac{4.5 \times 10^{9}}{3.4 \times 10^{5}} = 13235.29 \). Thus, the remaining fraction of curium is \( \left( \frac{1}{2} \right)^{13235.29} \).
05

Evaluate the fractions

Calculate the exponential decay formulas: \( \left( \frac{1}{2} \right)^{56.25} \) for plutonium and \( \left( \frac{1}{2} \right)^{13235.29} \) for curium. These practically result in extremely small numbers, close to zero.

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Key Concepts

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

Half-life
Every radioactive substance has a special term associated with its decay process鈥攃alled the half-life. The half-life of a substance is the period during which half of the initial amount of radioactive atoms in a sample have transformed due to decay. This concept is crucial because it allows scientists and researchers to estimate how long a radioactive isotope will remain active.
  • Think of the half-life as a predictable timer for radioactive decay. After each half-life period, only 50% of the substance remains, regardless of the starting amount.
  • This process is exponential鈥攎eaning after the first half-life, half remains, after the second half-life, half of that half remains, and so on.
The concept is universally applied in radioactivity studies, environmental science, medicine, and even archaeology through carbon dating.
Alpha Decay
During alpha decay, an unstable nucleus ejects an alpha particle. An alpha particle consists of two protons and two neutrons鈥攅ssentially a helium nucleus. This ejection reduces both the atomic number and mass number of the original nuclide.
  • Alpha decay is a common mode of decay for heavy nuclides, such as uranium and radium.
  • This process results in the transformation of the original nuclide into a new element.
For example, when uranium-238 undergoes alpha decay, it transforms into thorium-234, reducing its atomic number by 2 and mass number by 4. The energy released during alpha decay contributes to the heat produced within the Earth's core.
Isotopes
Isotopes are variants of a particular chemical element that have the same number of protons but a different number of neutrons in their nuclei. This means isotopes share an atomic number but differ in atomic mass.
  • Isotopes can be stable or unstable (radioactive).
  • Radioactive isotopes, like uranium-238, eventually decay into other elements through the emission of particles.
Understanding isotopes is crucial in applications such as carbon dating, medical diagnostics, and nuclear energy production. By studying isotopes, scientists gain insights into processes like radioactive decay, energy release, and elemental transformation.
Nuclear Stability
Nuclear stability refers to the tendency of a nucleus to remain unchanged over time or transform into a new nucleus. Stability is determined by the ratio of neutrons to protons.
  • Nuclei with equal or nearly equal numbers of protons and neutrons (like carbon-12) tend to be stable.
  • The larger the nucleus gets, the more neutrons are needed for stability due to increased proton-proton repulsion.
Nuclides that have an imbalance in this ratio often undergo radioactive decay, such as alpha decay, to reach a more stable state. Heavier elements, like uranium and plutonium, typically have more neutrons to counterbalance the repulsion among their many protons but still might be unstable, leading them to decay.

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Most popular questions from this chapter

The radionuclide \({ }^{56} \mathrm{Mn}\) has a half-life of \(2.58 \mathrm{~h}\) and is produced in a cyclotron by bombarding a manganese target with deuterons. The target contains only the stable manganese isotope \({ }^{55} \mathrm{Mn}\), and the manganese-deuteron reaction that produces \({ }^{56} \mathrm{Mn}\) is $$ { }^{55} \mathrm{Mn}+\mathrm{d} \rightarrow{ }^{56} \mathrm{Mn}+\mathrm{p} $$ If the bombardment lasts much longer than the half-life of \({ }^{56} \mathrm{Mn}\), the activity of the \({ }^{56} \mathrm{Mn}\) produced in the target reaches a final value of \(8.88 \times 10^{10} \mathrm{~Bq}\). (a) At what rate is \({ }^{56} \mathrm{Mn}\) being produced? (b) How many \({ }^{56} \mathrm{Mn}\) nuclei are then in the target? (c) What is their total mass?

An organic sample of mass \(4.00 \mathrm{~kg}\) absorbs \(2.00 \mathrm{~mJ}\) via slow neutron radiation \((\mathrm{RBE}=5)\). What is the dose equivalent \((\mathrm{mSv})\) ?

A radiation detector records 9500 counts in \(1.00 \mathrm{~min}\). Assuming that the detector records all decays, what is the activity of the radiation source in (a) becquerels and (b) curies?

Because the neutron has no charge, its mass must be found in some way other than by using a mass spectrometer. When a neutron and a proton meet (assume both to be almost stationary), they combine and form a deuteron, emitting a gamma ray whose energy is \(2.2233 \mathrm{MeV}\). The masses of the proton and the deuteron are \(1.007276467 \mathrm{u}\) and \(2.013553212 \mathrm{u}\), respectively. Find the mass of the neutron from these data.

Plutonium isotope \({ }^{239} \mathrm{Pu}\) decays by alpha decay with a half- life of \(24100 \mathrm{y}\). How many milligrams of helium are produced by an initially pure \(10.0 \mathrm{~g}\) sample of \({ }^{239} \mathrm{Pu}\) at the end of \(20000 \mathrm{y}\) ? is emitted by a nucleus, along with a neutrino. The emitted particles share the available disintegration energy. The electrons and positrons emitted in beta decay have a continuous spectrum of energies from near zero up to a limit \(K_{\max }\left(=Q=-\Delta m c^{2}\right)\). Radioactive Dating Naturally occurring radioactive nuclides provide a means for estimating the dates of historic and prehistoric events. For example, the ages of organic materials can often be found by measuring their \({ }^{14} \mathrm{C}\) content; rock samples can be dated using the radioactive isotope \({ }^{40} \mathrm{~K}\). Radiation Dosage Three units are used to describe exposure to ionizing radiation. The becquerel ( \(1 \mathrm{~Bq}=1\) decay per second) measures the activity of a source. The amount of energy actually absorbed is measured in grays, with 1 Gy corresponding to \(1 \mathrm{~J} / \mathrm{kg}\). The estimated biological effect of the absorbed energy is measured in sieverts; a dose equivalent of 1 Sv causes the same biological effect regardless of the radiation type by which it was acquired. Nuclear Models The collective model of nuclear structure assumes that nucleons collide constantly with one another and that relatively long-lived compound nuclei are formed when a projectile is captured. The formation and eventual decay of a compound nucleus are totally independent events. The independent particle model of nuclear structure assumes that each nucleon moves, essentially without collisions, in a quantized state within the nucleus. The model predicts nucleon levels and magic nucleon numbers \((2,8,20,28,50,82\), and 126) associated with closed shells of nucleons; nuclides with any of these numbers of neutrons or protons are particularly stable. The combined model, in which extra nucleons occupy quantized states outside a central core of closed shells, is highly successful in predicting many nuclear properties.
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