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One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a}){ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca},\) (b) \(^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) (c) lead-206 and thorium-230. Explain your choice in each case.

Short Answer

Expert verified
In each pair, the radioactive nuclides are (a) \( _{20}^{45} \mathrm{Ca}\), (b) \(^{14} \mathrm{C}\), (c) Thorium-230.

Step by step solution

01

Understanding Radioactive vs. Stable Nuclides

Nuclides are characterized by their number of protons and neutrons. Radioactive nuclides typically have an imbalanced ratio of protons to neutrons or an atomic number higher than lead. Stable nuclides have a balanced proton-neutron ratio or are among specific known stable isotopes.
02

Analyzing Calcium Isotopes

Calcium isotopes: - \( _{20}^{40}\text{Ca} \) and \( _{20}^{45}\text{Ca} \).- \( _{20}^{40}\text{Ca} \) has a balanced ratio, it is stable.- \( _{20}^{45}\text{Ca} \) has 5 additional neutrons, causing imbalance, making it radioactive.
03

Comparing Carbon Isotopes

Carbon isotopes: - \( ^{12}\text{C} \) is a common stable isotope with equal protons and neutrons.- \( ^{14}\text{C} \) has 2 extra neutrons compared to the stable \( ^{12}\text{C} \), making it radioactive, known for use in radiocarbon dating.
04

Examining Lead and Thorium

Lead and thorium: - Lead-206 is a stable isotope, a common end-product of radioactive decay chains. - Thorium-230 has a long decay chain, part of the uranium-238 decay series and is radioactive because thorium naturally undergoes radioactive decay.

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Key Concepts

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

Nuclides
A nuclide is any distinct atom or nucleus characterized by a specific number of protons and neutrons. The nucleus is the core of an atom, where protons and neutrons are packed together. Nuclides can vary in terms of these numbers, giving them different properties. Each nuclide has a specific atomic number, which is the count of protons in its nucleus, and a mass number, the sum of protons and neutrons. Studying these differences in nuclides helps us understand why some are stable while others are radioactive.
  • Proton count determines the element's chemical identity.
  • Neutron count affects the stability of the nuclide.
Often, small variations in neutron numbers result in isotopes of an element, leading some nuclides to natural radioactivity. A radioactive nuclide will eventually decay, releasing particles and energy, until it reaches a more stable form. This transformation underlies many natural phenomena, such as radiocarbon dating and decay chains in heavier elements.
Isotopes
Isotopes are variations of the same element that have the same number of protons but differ in numbers of neutrons. This means they share the same atomic number but have different mass numbers. Every element can have multiple isotopes, and these isotopes can be either stable or radioactive.
  • Stable isotopes have a balanced proton-neutron ratio.
  • Radioactive isotopes have an imbalance in this ratio, leading them to decay over time.
For example, carbon has several isotopes, including the stable

Understanding Carbon Isotopes

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Proton-neutron ratio
The proton-neutron ratio is a crucial factor in determining the stability of a nuclide.

Importance of the Ratio

A balanced ratio tends to make a nuclide stable, as it ensures a good interaction between the forces within the nucleus. Instability often arises when this balance is lost, leading to various forms of decay to achieve a more stable state.
  • Balanced proton-neutron ratios are often found in lighter elements, typically around 1:1.
  • Heavier elements tend to have more neutrons than protons to maintain stability, often exceeding a ratio of 1:1.

Case Studies: Calcium and Carbon

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Most popular questions from this chapter

Complete and balance the nuclear equations for the following fission reactions: (a) \({ }_{94}^{239} \mathrm{Pu}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{52}^{137} \mathrm{Te}+{ }_{42}^{100} \mathrm{Mo}+\) (b) \({ }_{100}^{256} \mathrm{Fm}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{46}^{113} \mathrm{Pd}+{ }_{-}+4{ }_{0}^{1} \mathrm{n}\)

Indicate the number of protons and neutrons in the following nuclei: \((\mathbf{a}){ }_{94}^{239} \mathrm{Pu},(\mathbf{b}){ }^{142} \mathrm{Ba},(\mathbf{c})\) potassium- 41 .

The Sun radiates energy into space at the rate of \(3.9 \times 10^{26} \mathrm{~J} / \mathrm{s} .\) (a) Calculate the rate of mass loss from the Sun in \(\mathrm{kg} /\) s. (b) How does this mass loss arise? (c) It is esti- mated that the Sun contains \(9 \times 10^{56}\) free protons. How many protons per second are consumed in nuclear reactions in the Sun?

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) \(\frac{90}{38} \mathrm{Sr},(\mathbf{b})_{38}^{85} \mathrm{Sr}\) (c) potassium-40, (d) sulfur-30.

A \(65-\mathrm{kg}\) person is accidentally exposed for \(240 \mathrm{~s}\) to a 15-mCi source of beta radiation coming from a sample of \({ }^{90}\) Sr. (a) What is the activity of the radiation source in disintegrations per second? In becquerels? (b) Each beta particle has an energy of \(8.75 \times 10^{-14} \mathrm{~J}\). and \(7.5 \%\) of the radiation is absorbed by the person. Assuming that the absorbed radiation is spread over the person's entire body, calculate the absorbed dose in rads and in grays. (c) If the RBE of the beta particles is 1.0 , what is the effective dose in mrem and in sieverts? (d) Is the radiation dose equal to, greater than, or less than that for a typical mammogram \((3 \mathrm{mSv}) ?\)

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