Study Guide: Nuclear Stability and Decay

Introduction to Nuclear Stability and Decay

• This material covers Module 2, Lesson 2, focusing on the study of the nucleus of the atom and nuclear chemistry.

• Nuclear chemistry is often taught at the end of a chemistry course, but fits naturally with the study of atomic structure due to the shared notation.

Key Terminology and Abbreviations

• Nucleon: A term used to describe either a proton or a neutron.

• Nuclide: Another term for the nucleus of an atom.

• Standard Abbreviations for Note-Taking:

  • $E$: Energy.

  • $RXN$: Reaction.

Forces Affecting the Nucleus

• Electrostatic Force: A force that causes like charges, such as two negative charges or two positive charges, to repel one another.

• Strong Nuclear Force: The force that holds the nucleus together; neutrons help counteract the electrostatic repulsion between protons.

• Nuclear Stability: The ability for a nucleus to stay intact, determined by the balance between the electrostatic force and the strong nuclear force.

Factors Contributing to Nuclear Instability

• Neutron-to-Proton Ratio:

  • For small atoms, a roughly $1:1$ ratio of neutrons to protons is stable.

  • As atoms increase in size, more neutrons are required to maintain stability.

  • Very large atoms require approximately a $1.5:1$ neutron-to-proton ratio.

• Large Nucleus: Any atom with a nucleus containing $84$ or more protons is inherently unstable.

• Neutron Excess: Having too many neutrons can lead to an unstable nucleus.

• Belt of Stability: A graphical region where stable isotopes are found based on their proton and neutron counts. Atoms falling outside this belt (either having too few or too many neutrons) are unstable.

Isotopes and Radioactivity

• Definition: Isotopes of an element possess the same number of protons but a different number of neutrons.

• Stability of Isotopes: Most elements on the periodic table have both stable and unstable isotopes.

  • Example: Carbon has $15$ known isotopes, but only two are stable: Carbon-12 ($^{12}C$) and Carbon-13 ($^{13}C$).

  • Carbon-12 is the most abundant isotope of carbon on Earth.

• Radioisotopes: Unstable isotopes that undergo radioactive decay.

• Radioactive Decay: The spontaneous disintegration of a nucleus into a stable nucleus by emitting particles, radiation, or both.

Types of Nuclear Decay

Alpha ($\alpha$) Decay

• Characterized by the emission of an alpha particle.

• An alpha particle is a cluster of two protons and two neutrons, equivalent to a Helium-4 nucleus ($^4_2He$).

• Common in heavy nuclei which are unstable due to their large size (too many total nucleons).

• Shedding protons and neutrons via alpha decay helps stabilize the isotope.

Beta (Beta Minus, $\beta^-$) Decay

• Characterized by the emission of an electron (beta particle).

• Internally, a neutron in the parent nucleus is converted into a proton and an electron.

Gamma ($\gamma$) Decay

• Emits a photon, which is the smallest quantity of light.

• Gamma rays are a form of electromagnetic radiation, not matter.

Additional Decay Types

• Positron Decay (Beta Plus, $\beta^+$): The emission of a positron, which is the size of an electron but carries a positive charge.

• Electron Capture: An inner electron is captured by the nucleus, combining with a proton to convert it into a neutron.

Characteristics of Emitted Particles

• Notation:

  • Alpha particle: $^4_2He$

  • Beta particle (electron): $^0_{-1}e$ (Mass number $0$, charge $-1$)

  • Beta particle (positron): $^0_{+1}e$ (Mass number $0$, charge $+1$)

  • Gamma rays: Usually not included in nuclear formulas as they do not affect mass or atomic numbers.

Comparison of Chemical and Nuclear Reactions

• Einstein’s Formula: $E = mc^2$ is used to calculate energy produced when mass is consumed.

  • $E$ = Energy

  • $m$ = Mass

  • $c$ = Speed of light ($3.00 \times 10^8 \, m/s$)

Modeling Nuclear Reactions

• Parent Nucleus: The initial unstable radioactive nucleus.

• Transmutation: The process of changing from one element to another.

• Daughter Nucleus: The new, more stable nucleus produced by the reaction.

• General Reaction Equation: $\text{Parent Nucleus} \rightarrow \text{Daughter Nucleus} + \text{Particle} + \text{Energy}$.

• Reactants: Substances on the left-hand side of the arrow.

• Products: Substances on the right-hand side of the arrow.

• Decay Series: A sequence of radioactive nuclides produced by multiple rounds of decay until a final stable nuclide is reached.

  • Example: Uranium-238 ($^{238}U$) undergoes a series of steps to eventually become stable Lead-206 ($^{206}Pb$).

Balancing Nuclear Equations

• Symbol Notation: ${^A_Z\text{X}}$

  • $A$ = Mass Number (Protons + Neutrons)

  • $Z$ = Atomic Number (Protons)

• Balancing Rule: The sum of the mass numbers ($A$) and the sum of the atomic numbers ($Z$) on the reactant side must equal the sum of those on the product side.

Practice: Alpha Decay Balancing

• Example 1: $^{226}_{88}Ra \rightarrow ^{222}_{86}Rn + ^4_2He$

  • Mass Check: $226 = 222 + 4$

  • Atomic Check: $88 = 86 + 2$

• Example 2: Finding the missing daughter for Polonium-218 ($Po$):

  • $^{218}_{84}Po \rightarrow {^A_Z\text{?}} + ^4_2He$

  • $A = 218 - 4 = 214$

  • $Z = 84 - 2 = 82$

  • Element $82$ is Lead ($Pb$). Result: $^{214}_{82}Pb$.

• Example 3: Uranium-238 ($U$) alpha decay:

  • $^{238}_{92}U \rightarrow {^A_Z\text{?}} + ^4_2He$

  • $A = 238 - 4 = 234$

  • $Z = 92 - 2 = 90$

  • Element $90$ is Thorium ($Th$). Result: $^{234}_{90}Th$.

Practice: Beta Minus Decay Balancing

• During $\beta^-$ decay, the atomic number of the parent nuclide increases by $1$.

• Example 1: Carbon-14 ($C$):

  • $^{14}_6C \rightarrow {^A_Z\text{?}} + ^0_{-1}e$

  • $A = 14 - 0 = 14$

  • $Z = 6 - (-1) = 7$

  • Element $7$ is Nitrogen ($N$). Result: $^{14}_7N$.

• Example 2: Cobalt-60 ($Co$):

  • $^{60}_{27}Co \rightarrow {^A_Z\text{?}} + ^0_{-1}e$

  • $A = 60 - 0 = 60$

  • $Z = 27 - (-1) = 28$

  • Element $28$ is Nickel ($Ni$). Result: $^{60}_{28}Ni$.

Questions & Discussion

Q: In the mixed practice problem, radium decays into radon ($^{226}_{88}Ra \rightarrow ^{222}_{86}Rn + \text{particle}$). What kind of particle is emitted and what kind of decay is this?

• A:

  • The change in mass number is $226 - 222 = 4$.

  • The change in atomic number is $88 - 86 = 2$.

  • The particle with a mass of $4$ and an atomic number of $2$ is a Helium nucleus ($^4_2He$).

  • This is an alpha particle, and the process is alpha decay.