Question

SOLUTION Analyze We must write balanced nuclear equations in which the masses and charges of reactants and products are equal. Plan We can begin by writing the complete chemical symbols for the nuclei and decay particles that are given in the problem. Solve (a) The information given in the question can be summarized as $$ { }_{80}^{201} \mathrm{Hg}+{ }_{-1}^{0} \mathrm{e} \longrightarrow{ }_{Z}^{A} \mathrm{X} $$ The mass numbers must have the same sum on both sides of the equation: $$ 201+0=A $$ Thus, the product nucleus must have a mass number of 201. Similarly, balancing the atomic numbers gives $$ 80-1=Z $$ Thus, the atomic number of the product nucleus must be 79 , which identifies it as gold (Au): $$ { }_{80}^{201} \mathrm{Hg}+{ }_{-1}^{0} \mathrm{e} \longrightarrow{ }_{79}^{201} \mathrm{Au} $$ (b) In this case we must determine what type of particle is emitted in the course of the radioactive decay: $$ { }_{90}^{231} \mathrm{Th} \longrightarrow{ }_{91}^{231} \mathrm{~Pa}+{ }_{\mathrm{Z}}^{\mathrm{X}} \mathrm{X} $$ From \(231=231+A\) and \(90=91+Z\), we deduce \(A=0\) and \(Z=-1\). According to Table \(21.2\), the particle with these characteristics is the beta particle (electron). We therefore write $$ { }_{90}^{231} \mathrm{Th} \longrightarrow{ }_{91}^{231} \mathrm{~Pa}+{ }_{-1}^{0} \mathrm{e} \text { or } \quad{ }_{90}^{231} \mathrm{Th} \longrightarrow{ }_{91}^{231} \mathrm{~Pa}+\beta^{-} $$

Short answer

(a) In the electron capture decay of \(_{80}^{201}\textrm{Hg}\), the balanced nuclear equation is: \[ _{80}^{201}\textrm{Hg} + _{-1}^{0}\textrm{e} \longrightarrow _{79}^{201}\textrm{Au} \] (b) In the radioactive decay of \(_{90}^{231}\textrm{Th}\) to \(_{91}^{231}\textrm{Pa}\), the balanced nuclear equation is: \[ _{90}^{231}\textrm{Th} \longrightarrow _{91}^{231}\textrm{Pa} + _{-1}^{0}\textrm{e} \] or \[ _{90}^{231}\textrm{Th} \longrightarrow _{91}^{231}\textrm{Pa} + \beta^{-} \]

Step-by-step solution

(a) Writing the initial given equation of decay

Given information: \(_{80}^{201}\textrm{Hg}\) decays by electron capture (\(_{-1}^{0}\textrm{e}\)). We can represent this as: \[ _{80}^{201}\textrm{Hg}+_{-1}^{0}\textrm{e} \longrightarrow _{Z}^{A}\textrm{X} \]

(a) Balancing equations and finding the product nucleus

The mass numbers and atomic numbers should be equal on both sides of the equation. For mass number (A): \[201+0=A\] Which means A=201. For atomic number (Z): \[80-1=Z\] Which means Z=79. The atomic number 79 corresponds to gold (Au). The balanced equation is: \[ _{80}^{201}\textrm{Hg} + _{-1}^{0}\textrm{e} \longrightarrow _{79}^{201}\textrm{Au} \]

(b) Writing the initial given equation of decay

Given information: \(_{90}^{231}\textrm{Th}\) decays to \(_{91}^{231}\textrm{Pa}\). We need to find the emitted particle (X). The equation can be written as: \[ _{90}^{231}\textrm{Th} \longrightarrow _{91}^{231}\textrm{Pa} + _{Z}^{A}\textrm{X} \]

(b) Balancing equations and finding the emitted particle

Balancing mass and atomic numbers: For mass number (A): \[231=231+A\] Which means A=0. For atomic number (Z): \[90=91+Z\] Which means Z=-1. A particle with a mass number of 0 and an atomic number of -1 is a beta particle (electron, e). Therefore, the balanced equation is: \[ _{90}^{231}\textrm{Th} \longrightarrow _{91}^{231}\textrm{Pa} + _{-1}^{0}\textrm{e} \] or \[ _{90}^{231}\textrm{Th} \longrightarrow _{91}^{231}\textrm{Pa} + \beta^{-} \]

Key concepts

Nuclear Chemistry

Nuclear chemistry is an area of chemistry that involves the study of changes that occur within atomic nuclei. This field explores various types of nuclear reactions, including radioactive decay, nuclear fission, and nuclear fusion. Understanding nuclear chemistry is pivotal in many applications such as nuclear power generation, medical diagnostics and treatments, as well as in radiocarbon dating.

Nuclear reactions can lead to changes in an atom's nucleus, resulting in the production of a different element or a different isotope of the same element. These transformations are governed by the laws of conservation of mass and energy, implying that the total mass and total charge before and after a nuclear reaction must be equal. Writing balanced nuclear equations is essential in nuclear chemistry to depict these reactions accurately.

Radioactive Decay

Radioactive decay is a spontaneous process by which unstable atomic nuclei release energy by emitting radiation in the form of particles or electromagnetic waves. There are several types of radioactive decay, including alpha decay, beta decay, gamma decay, and electron capture. Each type of decay results in the transmutation of elements and can be represented by balanced nuclear equations.

In radioactive decay, atoms of one element can change into atoms of another element altogether, or into a different isotope of the same element. The rate of decay is characterized by half-lives, which indicate the time required for half of the radioactive atoms in a sample to decay. Radioactive decay is a natural and random process, and cannot be influenced by changes in temperature, pressure, or chemical state.

Electron Capture

Electron capture is a type of beta decay, which is a process where an unstable atom becomes more stable. During electron capture, a proton within the atomic nucleus captures an orbiting electron and combines with it to form a neutron. As a result, the atomic number decreases by one while the mass number remains the same, because the proton and neutron have approximately equal mass.

The captured electron typically comes from the innermost electron shell (called the K shell), and the process can be represented in a nuclear equation by showing an electron being added to the nucleus of the parent atom. Electron capture is an important process in the balance of nuclear equations since it helps to illustrate the transformation of an element into another with a lower atomic number.

Beta Particle Emission

Beta particle emission is another form of radioactive decay. In beta-minus (β−) decay, a neutron in the nucleus is transformed into a proton, an electron (called a beta particle), and an antineutrino. The beta particle is ejected from the nucleus, resulting in an increase in the atomic number by one, while the mass number remains unchanged.

Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The nuclear equation representing beta particle emission shows the nucleus of the parent atom on one side and the nucleus of the new atom and the emitted beta particle on the other. Beta particle emission is central to many applications, including medical therapies and radiometric dating. Balancing nuclear equations for beta particle emission helps in visualizing the identity of the newly formed atom after this type of decay.