/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Both \({ }^{14} \mathrm{O}\) and... [FREE SOLUTION] | 91影视

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Both \({ }^{14} \mathrm{O}\) and \({ }_{8}^{19} \mathrm{O}\) undergo beta decay. Which would you expect to emit a positron and which an electron? Why?

Short Answer

Expert verified
^{14}O emits a positron, and _{8}^{19}O emits an electron, due to differences in neutron-to-proton ratios.

Step by step solution

01

Understanding Beta Decay

Beta decay is a type of radioactive decay where a beta particle (an electron or a positron) is emitted. If a positron is emitted, it's called beta-plus decay (尾鈦 decay). If an electron is emitted, it's called beta-minus decay (尾鈦 decay). In 尾鈦 decay, a neutron converts into a proton while an electron and an antineutrino are emitted. In 尾鈦 decay, a proton converts into a neutron while a positron and a neutrino are emitted.
02

Analyze Neutron-to-Proton Ratio

The neutron-to-proton ratio determines the type of beta decay. Isotopes that have an excess of protons usually undergo 尾鈦 decay to reduce the proton count, whereas isotopes with more neutrons undergo 尾鈦 decay to reduce the neutron count.
03

Determine the Decay Type for ^{14}O

^{14}O has 14 nucleons and 8 protons, meaning it has 6 neutrons. The N/P ratio (neutrons/protons) is 6/8 or 0.75, which is relatively low, indicating a proton-rich nucleus that will likely undergo 尾鈦 decay to achieve a more stable state.
04

Determine the Decay Type for _{8}^{19}O

_{8}^{19}O has 19 nucleons and 8 protons, meaning it has 11 neutrons. The N/P ratio is 11/8 or 1.375, which is relatively high, indicating a neutron-rich nucleus that will likely undergo 尾鈦 decay to achieve a more stable state.

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Key Concepts

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

Beta-plus Decay
Beta-plus decay, also known as 尾鈦 decay, is a fascinating process that occurs in the heart of an atom. Here, inside the nucleus, a proton is converted into a neutron. This change is accompanied by the emission of a positron (a particle with the same mass as an electron but with a positive charge) and a neutrino, which is a neutral and nearly massless particle.

Beta-plus decay happens when an atom has too many protons in relation to neutrons, making its nucleus unstable. To reach a more stable state, beta-plus decay helps by decreasing the excess protons.

In the case of \({ }^{14}O\), the nucleus is proton-rich, leading to 尾鈦 decay. This process helps \({ }^{14}O\) transition towards a balanced neutron-to-proton ratio, creating a more stable atomic configuration.
Beta-minus Decay
In contrast to beta-plus decay, beta-minus decay, or 尾鈦 decay, occurs when a neutron in an unstable nucleus is transformed into a proton. Alongside this transformation, an electron (referred to as a beta particle in this context) and an antineutrino are emitted.

This type of decay is typical in isotopes with a higher number of neutrons compared to protons. It's nature's way of balancing an atom's internal structure, reducing the number of neutrons. By converting one neutron to a proton, the balance tips closer to stability.

For example, in the isotope \({ }_{8}^{19}O\), there is an excess of neutrons, resulting in a 尾鈦 decay. By undergoing this process, \({ }_{8}^{19}O\) aligns itself with a more stable neutron-to-proton ratio.
Neutron-to-Proton Ratio
The neutron-to-proton (N/P) ratio is a critical factor in determining the type of beta decay an atom will undergo. Essentially, this ratio influences the stability of an atom's nucleus.

Atoms are stable when their N/P ratio is balanced. An excess of neutrons suggests a shift towards beta-minus decay, where neutrons transform into protons to achieve equilibrium. Conversely, if there are too many protons, beta-plus decay can occur to turn protons into neutrons.

For instance, \({ }^{14}O\) has a N/P ratio of 0.75, indicating a proton-rich environment, which suggests it will undergo 尾鈦 decay. Meanwhile, \({ }_{8}^{19}O\) has a N/P ratio of 1.375, showing it is neutron-rich and will thus experience 尾鈦 decay. This delicate balance maintained by the N/P ratio is fundamental for nuclear stability across the periodic table.

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Most popular questions from this chapter

Show that the nuclear density of \({ }_{1}^{1} \mathrm{H}\) is over \(10^{14}\) times greater than its atomic density. (Assume the atom to have the radius of the first Bohr orbit.)

According to the Fermi gas model of the nucleus, its protons and neutrons exist in a box of nuclear dimensions and fill the lowest available quantum states to the extent permitted by the exclusion principle. Since both protons and neutrons have spins of \(\frac{1}{2}\) they are fermions and obey Fermi- Dirac statistics. (a) Find an equation for the Fermi energy in'a nucleus under the assumption that \(A=\) 2Z. Note that the protons and neutrons must be considered separately. (b) What is the Fermi energy in such a nucleus for \(R_{0}=\) \(1.2 \mathrm{fm}\) ? \((c)\) In heavier nuclei, \(A>2 Z\). What effect will this have on the Fermi energies for each type of particle?

$$ \begin{aligned} &\text { state the number of neutrons and protons in each of the }\\\ &\text { following: }{ }_{3}^{6} \mathrm{Li} ;{ }_{10}^{22} \mathrm{Ne} ;{ }_{40}^{94} \mathrm{Zr},{ }_{12}^{180} \mathrm{Hf} \end{aligned} $$

Van der Waals forces are limited to very short ranges and do not have an inverse-square dependence on distance, yet nobody suggests that the exchange of a special mesonlike particle is responsible for such forces. Why not?

A simplified model of the deuteron consists of a neutron and a proton in a square potential well \(2 \mathrm{fm}\) in radius and \(35 \mathrm{MeV}\) deep. Is this model consistent with the uncertainty principle?

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