Comprehensive Study Notes on Radioactive Decay Equations

Fundamentals of Nuclear Decay Equations

• The primary driving force behind radioactive decay is the drive of an unstable isotope to achieve a stable atomic structure.

• A nuclear decay equation represents the first specific type of chemical reaction analyzed in nuclear chemistry.

• Standard components of a nuclear decay equation:

  • Reactants are located on the left side of the reaction arrow.

  • The reaction arrow represents the directional transformation from reactants to products.

  • Products are located on the right side of the reaction arrow.

• A complete decay equation for a single emission step yields two distinct products: the emitted decay particle and the transformed daughter isotope.

Types of Radioactive Decay

Alpha Decay

• Alpha decay involves the ejection of an alpha particle from an unstable parent nucleus.

• An alpha particle is identical to a helium nucleus and must be written as ${}_{2}^{4}\text{He}$.

• When an isotope undergoes alpha decay, it experiences a loss of mass and protons:

  • Loss of atomic mass: $4$ mass units.

  • Loss of atomic number: $2$ protons.

• General arithmetic rule for alpha decay products:

  • New Atomic Mass = $\text{Parent Mass} - 4$

  • New Atomic Number = $\text{Parent Atomic Number} - 2$

• Detailed Example:

  • Parent Isotope: Bohrium-266 (${}_{107}^{266}\text{Bh}$).

  • Calculation of mass: $266 - 4 = 262$.

  • Calculation of atomic number: $107 - 2 = 105$.

  • Periodic table lookup for atomic number $105$ yields Dubnium ($\text{Db}$).

  • Complete nuclear equation:

${}_{107}^{266}\text{Bh} \rightarrow {}_{2}^{4}\text{He} + {}_{105}^{262}\text{Db}$

• Dubnium reaches a stable state at an atomic mass of $262$.

Beta Decay

• Beta decay involves the conversion of a neutron into a proton, accompanied by the emission of an electron.

• The emitted particle is a beta particle, written symbolically as ${}_{-1}^{0}\text{e}$.

• A beta particle has an atomic mass of $0$ and an atomic charge/number of $-1$.

• During beta decay, the overall mass of the nucleus remains constant while the atomic number increases by $1$ unit.

• General arithmetic rule for beta decay products:

  • New Atomic Mass = $\text{Parent Mass} - 0 = \text{Parent Mass}$

  • New Atomic Number = $\text{Parent Atomic Number} - (-1) = \text{Parent Atomic Number} + 1$

• Detailed Example:

  • Parent Isotope: Iridium-195 (${}_{77}^{195}\text{Ir}$).

  • Calculation of mass: $195 - 0 = 195$.

  • Calculation of atomic number: $77 - (-1) = 78$.

  • Periodic table lookup for atomic number $78$ yields Platinum ($\text{Pt}$).

  • Complete nuclear equation:

${}_{77}^{195}\text{Ir} \rightarrow {}_{-1}^{0}\text{e} + {}_{78}^{195}\text{Pt}$

• Platinum is typically stable at an atomic mass around $195$.

Positron Emission (Beta Plus Decay)

• Positron emission, also referred to as beta plus decay, involves the conversion of a proton into a neutron with the release of a positron.

• A positron is an electron antiparticle with identical zero mass but a positive charge, written symbolically as ${}_{+1}^{0}\text{e}$.

• During positron emission, the nucleus loses no mass, but its atomic number decreases by $1$ unit.

• General arithmetic rule for positron emission products:

  • New Atomic Mass = $\text{Parent Mass} - 0 = \text{Parent Mass}$

  • New Atomic Number = $\text{Parent Atomic Number} - 1$

• Detailed Example:

  • Parent Isotope: Tantalum-170 (${}_{73}^{170}\text{Ta}$).

  • Calculation of mass: $170 - 0 = 170$.

  • Calculation of atomic number: $73 - 1 = 72$.

  • Periodic table lookup for atomic number $72$ yields Hafnium ($\text{Hf}$).

  • Complete nuclear equation:

${}_{73}^{170}\text{Ta} \rightarrow {}_{+1}^{0}\text{e} + {}_{72}^{170}\text{Hf}$

Decay Prevalence and Stability Dynamics

• Alpha decay and beta decay are by far the most common forms of radioactive decay.

• Beta decay is the single most prevalent decay mechanism among radioactive isotopes found at nuclear locations such as Chernobyl.

• Outcome variations in nuclear reactions:

  • A single decay process does not always result in a stable nucleus.

  • If the product nucleus remains unstable, it can undergo additional consecutive decay steps.

• Selective Pathway Analysis for Carbon-14 (${}_{6}^{14}\text{C}$):

  • Hypothetical Beta Decay Pathway:

${}_{6}^{14}\text{C} \rightarrow {}_{-1}^{0}\text{e} + {}_{7}^{14}\text{N}$

• Hypothetical Positron Emission Pathway:

${}_{6}^{14}\text{C} \rightarrow {}_{+1}^{0}\text{e} + {}_{5}^{14}\text{B}$

• Selectivity Determination: Carbon-14 exclusively undergoes beta decay to form Nitrogen-14 (${}_{7}^{14}\text{N}$).

• Reason: Nitrogen-14 is a stable atom corresponding to the standard mass of nitrogen on the periodic table. Conversely, Boron-14 (${}_{5}^{14}\text{B}$) produced via positron emission is extremely unstable and volatile.

Step-by-Step Practice Problems

• Problem 1: Alpha Decay of Seaborgium-263 (${}_{106}^{263}\text{Sg}$)

  • Reactant: Seaborgium-263 (${}_{106}^{263}\text{Sg}$).

  • Emitted particle: ${}_{2}^{4}\text{He}$.

  • Mass arithmetic: $263 - 4 = 259$.

  • Atomic number arithmetic: $106 - 2 = 104$.

  • Daughter isotope lookup: Atomic number $104$ is Rutherfordium ($\text{Rf}$).

  • Complete equation:

${}_{106}^{263}\text{Sg} \rightarrow {}_{2}^{4}\text{He} + {}_{104}^{259}\text{Rf}$

• Problem 2: Positron Emission of Magnesium-23 (${}_{12}^{23}\text{Mg}$)

  • Reactant: Magnesium-23 (${}_{12}^{23}\text{Mg}$).

  • Emitted particle: ${}_{+1}^{0}\text{e}$.

  • Mass arithmetic: $23 - 0 = 23$.

  • Atomic number arithmetic: $12 - 1 = 11$.

  • Daughter isotope lookup: Atomic number $11$ is Sodium ($\text{Na}$).

  • Complete equation:

${}_{12}^{23}\text{Mg} \rightarrow {}_{+1}^{0}\text{e} + {}_{11}^{23}\text{Na}$

• Problem 3: Beta Emission of Carbon-14 (${}_{6}^{14}\text{C}$)

  • Reactant: Carbon-14 (${}_{6}^{14}\text{C}$).

  • Emitted particle: ${}_{-1}^{0}\text{e}$.

  • Mass arithmetic: $14 - 0 = 14$.

  • Atomic number arithmetic: $6 - (-1) = 7$.

  • Daughter isotope lookup: Atomic number $7$ is Nitrogen ($\text{N}$).

  • Complete equation:

${}_{6}^{14}\text{C} \rightarrow {}_{-1}^{0}\text{e} + {}_{7}^{14}\text{N}$

Questions and Discussion

• Question: Is beta emission identical to beta decay?

  • Response: Yes, the terms "beta emission" and "beta decay" refer to the exact same nuclear process and are completely interchangeable.

• Educational Partner (EP) Tutoring Information:

  • Credit requirements are satisfied by meeting with an assigned EP.

  • Contact options include Sean at the Brighton location or the primary course instructor.

  • Tutoring session availability occurs on Tuesdays, Wednesdays, and Fridays.