Question

Write balanced nuclear equations for the following transformations: (a) bismuth- 213 undergoes alpha decay; (b) nitrogen-13 undergoes electron capture; (c) technicium-98 undergoes electron capture; (d) gold-188 decays by positron emission.

Short answer

The balanced nuclear equations for the given transformations are: (a) Bismuth-213 undergoes alpha decay: \[_{83}^{213}\text{Bi} \rightarrow _2^4\text{He} + _{81}^{209}\text{Tl}\] (b) Nitrogen-13 undergoes electron capture: \[_7^{13}\text{N} + e^- \rightarrow _{6}^{13}\text{C}\] (c) Technetium-98 undergoes electron capture: \[_{43}^{98}\text{Tc} + e^- \rightarrow _{42}^{98}\text{Mo}\] (d) Gold-188 decays by positron emission: \[_{79}^{188}\text{Au} \rightarrow e^+ + _{78}^{188}\text{Pt}\]

Step-by-step solution

Bismuth-213 undergoes alpha decay

An alpha particle consists of 2 protons and 2 neutrons, so its atomic and mass numbers are 2 and 4, respectively. In alpha decay, a nucleus emits an alpha particle. For bismuth-213 (\(_{83}^{213}\)Bi), emit an alpha particle (\(_2^4\)He) and identify the remaining nucleus: \[_{83}^{213}\text{Bi} \rightarrow _2^4\text{He} + _{81}^{209}\text{Tl}\] The balanced nuclear equation for the alpha decay of bismuth-213 is: \[_{83}^{213}\text{Bi} \rightarrow _2^4\text{He} + _{81}^{209}\text{Tl}\]

Nitrogen-13 undergoes electron capture

In electron capture, a nucleus absorbs an electron and converts a proton to a neutron. For nitrogen-13 (\(_7^{13}\)N), capture an electron (e) and identify the resulting nucleus: \[_7^{13}\text{N} + e^- \rightarrow _{6}^{13}\text{C}\] The balanced nuclear equation for the electron capture of nitrogen-13 is: \[_7^{13}\text{N} + e^- \rightarrow _{6}^{13}\text{C}\]

Technetium-98 undergoes electron capture

For technetium-98 (\(_{43}^{98}\)Tc), capture an electron (e) and identify the resulting nucleus: \[_{43}^{98}\text{Tc} + e^- \rightarrow _{42}^{98}\text{Mo}\] The balanced nuclear equation for the electron capture of technetium-98 is: \[_{43}^{98}\text{Tc} + e^- \rightarrow _{42}^{98}\text{Mo}\]

Gold-188 decays by positron emission

In positron emission, a nucleus emits a positron (a positively charged electron, e^+) and converts a proton to a neutron. For gold-188 (\(_{79}^{188}\)Au), emit a positron (e^+) and identify the resulting nucleus: \[_{79}^{188}\text{Au} \rightarrow e^+ + _{78}^{188}\text{Pt}\] The balanced nuclear equation for the positron emission of gold-188 is: \[_{79}^{188}\text{Au} \rightarrow e^+ + _{78}^{188}\text{Pt}\]

Key concepts

Alpha Decay

Alpha decay is a type of radioactivity in which an unstable atomic nucleus emits an alpha particle, comprising two protons and two neutrons—the same as a helium-4 nucleus. This process reduces the mass number of the original nucleus by four and the atomic number by two, resulting in a new element. For instance, in the alpha decay of bismuth-213, a helium-4 particle is emitted, leaving behind a stable nucleus of thallium-209.

During this transformation, the equation must conserve both mass and atomic numbers. Hence, the sum of mass numbers and atomic numbers on the reactant side should equal the sum on the product side. This ensures that the nuclear equation for the decay process is balanced.

Practical Implications of Alpha Decay:

• Alpha decay is significant in the fields of nuclear physics and geology, especially in the dating of rocks and minerals.
• It can also pose a health risk if alpha-emitting substances are inhaled or ingested, as they can cause biological damage.

Electron Capture

Electron capture is a process by which an atomic nucleus with excess proton absorbs an inner atomic electron, usually from the K-shell, which leads to the conversion of a proton into a neutron. As a result, the atomic number decreases by one while the mass number remains unchanged. This process is represented in nuclear equations by adding an electron (e⁻) on the reactant side and showing the transformation into a different element on the product side.

For example, when nitrogen-13 undergoes electron capture, it absorbs an electron to become carbon-13. This transformation follows the law of conservation of charge and mass number: the overall charge and the mass number before and after the reaction are equal.

Electron Capture in Nature and Technology:

• Electron capture plays a role in the energy production in stars.
• It is used in some types of neutrino detectors and medical diagnostic tests like PET scans.

Positron Emission

Positron emission, also known as beta-plus decay, is a process where an unstable nucleus with an excess of protons emits a positron (e⁺), which is the antiparticle of the electron. During this process, one of the protons in the nucleus is converted to a neutron, causing the atomic number to decrease by one, while the mass number remains unchanged.

A classic example of this can be seen in the decay of gold-188 to platinum-188, where a positron is emitted. In the balanced nuclear equation, a positron appears on the product side to ensure conservation of charge. After emission, the positron will usually annihilate with an electron, releasing gamma radiation.

Applications of Positron Emission:

• It is a critical phenomenon in the use of positron emission tomography (PET) scans in medicine.
• Understanding positron emission is essential for research in particle physics and astrophysics.

Nuclear Chemistry

Nuclear chemistry is the subdiscipline of chemistry that deals with the chemical and physical properties of elements as influenced by changes in the structure of the atomic nucleus. It is an essential field for understanding various types of nuclear reactions, such as alpha decay, electron capture, and positron emission. Nuclear chemistry has both theoretical and practical applications, which include the study of nuclear reactions and their applications.

Applications and implications of nuclear chemistry are widespread, from the generation of electricity in nuclear power plants to the development of medical diagnostic techniques like PET scans and the treatment of cancer using radiotherapy.

Key aspects of Nuclear Chemistry:

• Understanding radioactive decay processes and the methods for balancing nuclear equations.
• Learning about the safety measures and environmental impacts associated with the use of nuclear technology.
• Exploring the role of nuclear chemistry in ancient artifacts dating and in space exploration through the use of radioisotope thermoelectric generators.