11.2: Basic Equations (2023)

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    Learning objectives
    • Identify common particles and energies involved in nuclear reactions.
    • Write and balance nuclear equations.

    Changes in nuclei that result in changes in their atomic numbers, mass numbers, or energy states arenuclear reactions. To describe a nuclear reaction, we use an equation that identifies the nuclides involved in the reaction, their mass numbers and atomic numbers, and the other particles involved in the reaction.

    Types of particles in nuclear reactions

    Many entities can be involved in nuclear reactions. The most common are protons, neutrons, alpha particles, beta particles, positrons, and gamma rays, as shown in Table \(\PageIndex{1}\).

    Table \(\PageIndex{1}\) A summary of the names, symbols, representations, and descriptions of the most common particles in nuclear reactions.

    11.2: Basic Equations (1)

    equilibrium of nuclear reactions

    A balanced chemical reaction equation reflects the fact that, during a chemical reaction, bonds are broken and formed, and atoms rearrange, but the total number of atoms of each element is conserved and does not change. A balanced nuclear reaction equation indicates that there is rearrangement during a nuclear reaction, but of subatomic particles instead of atoms. Nuclear reactions also follow conservation laws and balance in two ways:

    1. The sum of the mass numbers of the reactants is equal to the sum of the mass numbers of the products.
    2. The sum of the charges of the reactants is equal to the sum of the charges of the products.

    If the atomic number and mass number of all but one of the particles in a nuclear reaction are known, we can identify the particle by balancing the reaction. For example, we can determine that \(\ce{^{17}_8O}\) is a product of the nuclear reaction of \(\ce{^{14}_7N}\) and \(\ce{^4_2He} \ ) if we knew that a proton, \(\ce{^1_1H}\), is one of the two products. The example \(\PageIndex{1}\) shows how we can identify a nuclide by balancing the nuclear reaction.

    Nuclear decay processes

    Radioactive decay involves the emission of a particle and/or energy when one atom transforms into another. In most cases, the atom changes its identity to become a new element. There are four different types of emissions that occur.

    alpha emission

    Alpha \(\left( \alpha \right)\) decadenciaIt involves the release of helium ions from the nucleus of an atom. This ion consists of two protons and two neutrons and has charge \(2+\). The release of a \(\alpha\) particle produces a new atom that has an atomic number two less than the original atom and an atomic weight four less. A typical alpha decay reaction is the conversion of uranium-238 to thorium:

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    \[\ce{^{238}_{92}U} \rightarrow \ce{^{234}_{90}Th} + \ce{^4_2 \alpha}^+ \nonumber \]

    We see a decrease of two in atomic number (uranium to thorium) and a decrease of four in atomic weight (238 to 234). Emission is usually not written with the indicated atomic number and weight, since it is a common particle whose properties must be memorized. Alpha emission is often accompanied by gamma radiation \(\left( \gamma \right)\), a form of energy release. Many of the largest elements on the periodic table are alpha emitters.

    11.2: Basic Equations (2)

    Chemists often use the namesparent isotopemidaughter isotopeto represent the original atom and the different product of the alpha particle. In the example above, \[_{92}^{238}\textrm{U} \nonumber \] is the principal isotope and \[_{90}^{234}\textrm{Th} \nonumber \] is the isotope son. When one element is transformed into another in this way, it suffersradioactive decay.

    Example \(\PageIndex{1}\)

    Write the nuclear equation representing the radioactive decay of radon-222 by the emission of alpha particles, and identify the daughter isotope.


    Radon has an atomic number of 86, so the parent isotope is represented as \[_{86}^{222}\textrm{Rn} \nonumber \]

    We represent alpha particle as

    \[_{2}^{4}\textrm{He} \nonumber \]

    Use subtraction (222 − 4 = 218 and 86 − 2 = 84) to identify the secondary isotope as polonium:

    \[_{86}^{222}\textrm{Rn}\rightarrow \; _ {2}^{4}\textrm{Ele}+\:_{84}^{218}\textrm{Th} \nonumber \]

    Exercise \(\PageIndex{1}\)

    Write the nuclear equation representing the radioactive decay of polonium-208 by the emission of alpha particles, and identify the daughter isotope.

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    \[_{80}^{208}\textrm{Po}\rightarrow \; _ {2}^{4}\textrm{Ele}+\:_{82}^{204}\textrm{Pb} \no number\]

    \[_{82}^{204}\textrm{Pb} \nonumber \]

    beta problems

    Beta \(\left( \beta \right)\) decadenciait is a more complicated process. Unlike \(\alpha\) emission, which simply ejects a particle, \(\beta\) emission involves the transformation of a neutron in the nucleus into a proton and an electron. Then the electron is ejected from the nucleus. In the process, the atomic number increases by one while the atomic weight remains the same. As is the case with \(\alpha\) emissions, \(\beta\) emissions are often accompanied by \(\gamma\) radiation.

    11.2: Basic Equations (3)

    A typical beta decay process involves carbon-14, which is often used in radioactive dating techniques. The reaction forms nitrogen-14 and one electron:

    \[\ce{^{14}_6C} \rightarrow \ce{^{14}_7N} + \ce{^0_{-1}e} \nonúmero \]

    Again, beta emission is usually denoted simply by the Greek letter \(\beta\); memorization of the process is necessary to follow nuclear calculations in which the Greek letter \(\beta\) appears without any other notation.

    Example \(\PageIndex{2}\)

    Write the nuclear equation representing the radioactive decay of boron-12 by the emission of beta particles, and identify the daughter isotope. A gamma ray is emitted simultaneously with the beta particle.


    The parent isotope is \[B512," id="MathJax-Element-16-Frame" role="presentation" style="position:relative;" tabindex="0">_ {2}^{4}\textrm{He} \nonumber \]

    B512," role="presentation" style="position:relative;" tabindex="0">while one of the products isB512," role="presentation" style="position:relative;" tabindex="0">\[_{-1}^{0}\textrm{e} \nonumber \]

    For the mass and atomic numbers to have the same value on both sides, the mass number of the daughter isotope must be 12 and its atomic number must be 6. The element with atomic number 6 is carbon. Therefore, the complete nuclear equation is the following:

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    \[_{5}^{12}\textrm{B}\rightarrow \;_{6}^{12}\textrm{C}+_{-1}^{0}\textrm{e}+\gamma \without number \]

    The daughter isotope is carbon-12.

    Exercise \(\PageIndex{2}\)

    Write the nuclear equation representing the radioactive decay of rubidium-87 by the emission of beta particles, and identify the daughter isotope.


    \[_{37}^{87}\textrm{Rb}\rightarrow \;_{38}^{87}\textrm{Sr}+_{-1}^{0}\textrm{e} \nonumber \ ]

    \[_{38}^{87}\textrm{Sr} \nonúmero\]

    gamma emission

    gamma radiation \(\left( \gamma \right)\)it is simply energy. It can be released by itself or, more commonly, in association with other radiation events. There is no change in atomic number or atomic weight in a single emission of \(\gamma\). Often, an isotope can produce \(\gamma\) radiation as a result of a transition to a metastable isotope. This type of isotope can only "settle", with a displacement of the particles in the nucleus. The composition of the atom does not change, but the nucleus can be considered more "comfortable" after the displacement. This change increases the stability of the isotope from the energetically unstable (or "metastable") isotope to a more stable form of the nucleus. gamma (\(\gamma\)) emission can occur virtually instantaneously, as occurs in the alpha decay of uranium-238 to thorium-234, where the asterisk denotes an excited state:

    \[^{238}_{92}\textrm{U}\rightarrow \, \underset{\textrm{excited} \\ \textrm{nuclear} \\ \textrm{state}}{^{234}_{90 }\textrm{Th*}} + ^{4}_{2}\alpha\xrightarrow {\textrm{relaxation}\,}\,^{234}_{90}\textrm{Th}+^{0} _{0}\gamma\label{Eq13} \]

    If we ignore the decay event that created the excited nucleus, then

    \[^{234}_{88}\textrm{Th*} \rightarrow\, ^{234}_{88}\textrm{Th}+^{0}_{0}\gamma\label{Eq14} \ ]

    or more generally,

    \[^{A}_{Z}\textrm{X*} \rightarrow\, ^{A}_{Z}\textrm{X}+^{0}_{0}\gamma\label{Eq15} \ ]

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    Gamma emission can also occur after a significant delay. For example, technetium-99metrohas a half-life of approximately 6 hours before emitting a γ-ray to form technetium-99 (themetrois for metastable). Since γ-rays are energy, their emission does not affect the mass number or atomic number of the daughter nuclide. Gamma ray emission is thus the only type of radiation that does not necessarily involve the conversion of one element to another, although it is almost always observed in conjunction with some other nuclear decay reaction.

    positron emission

    Apositronit is a positive electron (a form of antimatter). This rare type of emission occurs when a proton becomes a neutron and a positron in the nucleus, with ejection of the positron. The atomic number will decrease by one as long as the atomic weight does not change. A positron is often called \(\beta^+\).

    Carbon-11 emits a positron to become boron-11:

    \[\ce{^{11}_6C} \rightarrow \ce{^{11}_5B} + \ce{^0_{+1} \beta} \sin number \]

    electron capture

    An alternative way for a nuclide to increase its ratio of neutrons to protons is through a phenomenon called electron capture. In electron capture, the nucleus of the atom captures an electron from an inner orbital and combines it with a proton to form a neutron. For example, silver-106 undergoes electron capture to become palladium-106.

    \[\ce{^{106}_{47}Ag} + \ce{^0_{-1}e} \rightarrow \ce{^{106}_{46}Pd} \nonumber \]

    Note that the overall result of electron capture is identical to positron emission. The atomic number decreases by one while the mass number remains the same.

    Table \(\PageIndex{2}\) Different types of decomposition and changes in atomic and mass numbers.

    11.2: Basic Equations (4)

    The following are the equations of various nuclear reactions that have played an important role in the history of nuclear chemistry:

    • The first naturally-occurring unstable element to be isolated, polonium, was discovered by the Polish scientist Marie Curie and her husband Pierre in 1898. It decays and emits α particles: \[\ce{^{212}_{84}Po⟶ ^ { 208}_{82}Pb + ^4_2He} \unnumbered\]
    • The first nuclide to be prepared by artificial means was an isotope of oxygen,17O. It was made by Ernest Rutherford in 1919 by bombarding nitrogen atoms with α particles: \[\ce{^{14}_7N + ^4_2α⟶ ^{17}_8O + ^1_1H} \nonumber \]
    • James Chadwick discovered the neutron in 1932, as a previously unknown neutral particle produced along with12C by the nuclear reaction between9be and4Them: \[\ce{^9_4Be + ^4_2He⟶ ^{12}_6C + ^1_0n} \nonumber \]
    • The first element to be prepared that does not exist naturally on Earth, technetium, was created by bombarding molybdenum with deuterons (heavy hydrogen, \(\ce{^2_1H}\)), by Emilio Segre and Carlo Perrier in 1937: \ [ \ce{^2_1H + ^{97}_{42}Mo⟶2^1_0n + ^{97}_{43}Tc} \nonumber\]
    • The first controlled nuclear chain reaction was carried out in a reactor at the University of Chicago in 1942. One of the many reactions involved was: \[ \ce{^{235}_{92}U + ^1_0n⟶ ^{87} _ {35}Br + ^{146}_{57}La + 3^1_0n} \unnumbered\]


    • Nuclei can undergo reactions that change their number of protons, number of neutrons, or energy state.
    • Many different particles can be involved in nuclear reactions. The most common are protons, neutrons, positrons (which are positively charged electrons), electrons, alpha (α) particles (which are high-energy helium nuclei), beta (β) particles (which are high-energy electrons), and gamma. (γ). rays (which make up high-energy electromagnetic radiation).
    • Just like with chemical reactions, nuclear reactions are always balanced. When a nuclear reaction occurs, the total mass (number) and the total charge remain unchanged.
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