Isotopes, Radioisotopes, and Atomic Mass (Lecture Notes)

Isotopes: Definition and Carbon Examples

Atoms of the same element can have different numbers of neutrons but the same number of protons. Protons define the element; electrons balance charge in a neutral atom.
Isotopes are atoms of the same element with different neutron counts.
Example with carbon:
- All three shown variants have 6 protons (and 6 electrons if neutral).
- They differ in neutrons: carbon-12 (6 neutrons), carbon-13 (7 neutrons), carbon-14 (8 neutrons).
- Mass numbers reflect protons + neutrons: Mg. For carbon-12, mass = 12; for carbon-13, mass = 13; for carbon-14, mass = 14.
The decimal masses shown on the periodic table come from averaging the masses of all naturally occurring isotopes of an element (isotopic abundance).
Not every isotope is radioactive; many are stable.
Isotopic abundance (or isotopic abundance, abundance) is the frequency or percentage of a given isotope found in nature.
The concept that electrons can transfer between objects (electronic rearrangement) shows that electron numbers can change without altering the identity of the element (protons fixed).
Magnesium is used as a practical example of multiple isotopes in nature (Mg-24, Mg-25, Mg-26) with particular natural abundances.
Practical lab note: magnesium is a common reactive metal used in Grade 11/12 labs due to its reactivity with acids and oxygen.
The “mass on the periodic table” (e.g., 24.01 for magnesium) is a weighted average of all isotopic masses, reflecting how common each isotope is in nature.

Isotopic Abundance and Atomic Mass

Isotopic abundance is typically expressed as a percentage; to use it in calculations it must be converted to a fraction (decimal).
Example: magnesium isotopes and abundances:
- Mg-24: 78.7%
- Mg-25: 10.1%
- Mg-26: 11.2%
Atomic mass on the periodic table is the weighted average of the isotopic masses using their fractional abundances:
- For magnesium, the table lists a mass around 24.01 amu, which is the weighted average of Mg-24 (mass ~24), Mg-25 (mass ~25), Mg-26 (mass ~26).
Concept of age-weighted average:
- If isotopes had equal abundances, the simple average would be the mass; however, real abundances are not equal, so you must weight by abundance.
Mass calculation concept: the average mass A (often called atomic or molar mass) is computed as:
- A = igg( f{1} imes m{1} igg) + igg( f{2} imes m{2} igg) + igg( f{3} imes m{3} igg) + \, \ ext{where } f{i} ext{ are fractional abundances (as decimals) and } m{i} ext{ are isotope masses}.
The general notation used in chemistry: A is atomic mass (mass of an atom or relative to a reference), Z is atomic number, and the sum notation expresses the weighted contribution of each isotope:
- $A = abla \,?$
- In practice: A = oxed{ igg( f{1} m{1} + f{2} m{2} + f{3} m{3} + \, igg) }
Example calculation for magnesium (to illustrate the weighted average):
- Using the given abundances (convert percentages to decimals):
- Mg-24: 0.787
- Mg-25: 0.101
- Mg-26: 0.112
- Masses: 24, 25, 26 respectively.
- Compute: $A_{Mg} = (0.787 \times 24) + (0.101 \times 25) + (0.112 \times 26) \approx 18.888 + 2.525 + 2.912 = 24.325\,\text{amu} \approx 24.3\,\text{amu}.$
The resulting atomic mass (e.g., ~24.3 amu for magnesium) is a relative mass unit; it reflects how heavy an atom is relative to 1/12 of a carbon-12 atom.
Atomic mass units vs grams per mole:
- amu is a relative unit used for single atoms.
- Grams per mole (g/mol) is the molar mass used for amounts of substance in chemistry.
The concept of the mole is introduced to connect atomic-scale masses to macroscopic quantities.

Mass Spectrometry: How We Measure Isotopes

A mass spectrometer is a device used to identify isotopes and their relative abundances by measuring masses.
Basic workflow:
- The sample is vaporized to a gas.
- The sample is ionized to form charged particles (ions).
- Ions are accelerated by electric fields.
- Ions enter a magnetic field, which deflects them based on their mass-to-charge ratio (m/z).
- Lighter and differently charged ions are deflected more than heavier ones; the magnet causes separation by mass.
- By detecting the ions, scientists determine the masses and the relative abundances of isotopes.
Real-world application: NASA uses mass spectrometers on Mars rovers to analyze soil and rock composition remotely.
This technique provides precise information about isotopic composition and helps confirm the presence of specific isotopes in planetary samples.

Radiation and Radioisotopes

Not all isotopes are radioactive; radioisotopes are those that decay spontaneously, emitting radiation as they transform into different elements.
Radioactivity involves the emission of nuclear radiation (energy) or particles from an unstable nucleus.
Isotopes that decay over time are called radioactive; the rate of decay depends on the isotope and is characterized by its half-life.
Nuclear decay is the process by which unstable nuclei release energy, often changing the identity of the element via transmutation.
The study of radioactive decay falls under nuclear chemistry and physics and has wide-ranging applications in medicine, dating, energy, and industry.

Types of Nuclear Radiation and Their Properties

Alpha radiation (alpha particles): two protons and two neutrons (a helium-4 nucleus).
- Very low penetrating power; stopped by a sheet of paper or skin.
- Not highly dangerous externally, but can be hazardous if ingested.
Beta radiation (beta particles): high-energy electrons or positrons.
- More penetrating than alpha; can pass through paper but is blocked by aluminum.
- Can cause internal damage if inhaled or ingested.
Gamma radiation (gamma rays): high-energy electromagnetic radiation.
- Highly penetrating; can pass through paper and aluminum and requires dense shielding (e.g., lead) to reduce exposure.
- Extremely hazardous due to deep penetrating power.
Shielding and safety: gamma radiation requires substantial shielding; nuclear waste is often stored on-site and shielded (e.g., underwater with lead walls) to limit radiation exposure.
Applications and caveats: while some radiation is dangerous, radioisotopes have useful applications in medicine and industry when handled properly and with appropriate shielding.

Practical and Real-World Implications of Radioisotopes

Medical uses: radioisotopes are used in imaging techniques (e.g., PET scans, sometimes CT with radiopharmaceuticals) to visualize internal body processes.
Radioisotopes in everyday life: some isotopes occur naturally in the body and environment and participate in various biological and geological processes.
Ethical and safety considerations: handling, storage, and disposal of radioactive materials require strict safety protocols due to health and environmental risks.

Carbon-14 Dating: Half-Life, Decay, and Dating Uses

Carbon-14 is a natural radioisotope produced in the atmosphere and incorporated into living organisms via the carbon cycle.
Decay process (beta decay): one neutron in carbon-14 converts to a proton, emitting a beta particle (electron) and an antineutrino, turning carbon-14 into nitrogen-14:
- $^{14}{6}\mathrm{C} \rightarrow ^{14}{7}\mathrm{N} + e^{-} + \bar{\nu}_{e}.$
Carbon-14 dating relies on the known half-life of carbon-14 to estimate the time since an organism died.
Half-life concept:
- The half-life T_{1/2} is the time required for half of a given amount of carbon-14 to decay.
- For carbon-14, $T_{1/2} = 5730\ \text{years}.$ (five thousand seven hundred thirty years)
Decay progression example:
- At t = 0, 100% of the carbon-14 is present.
- After one half-life (5730 years), 50% remains.
- After two half-lives (11460 years), 25% remains.
- After three half-lives (17190 years), 12.5% remains.
Dating principle:
- By measuring the current fraction of carbon-14 in a fossil or sample, and knowing the initial carbon-14 amount (in the environment), one can calculate the time since death using the decay relationship.
- General relationship between remaining N and original N0:
- $N = N0 \left(\frac{1}{2}\right)^{\frac{t}{T{1/2}}}.$
- Equivalently, solving for time: $t = T{1/2} \cdot \frac{\log\left(\frac{N}{N0}\right)}{\log\left(\frac{1}{2}\right)}.$
Real-world narrative example: fossils (e.g., saber-toothed tiger) containing carbon-14 reveal their age when carbon-14 has decayed to measurable levels, allowing us to date specimens from millions of years ago.
Limitations and calibration: carbon dating is most effective for dates up to about 50,000–60,000 years; variations in atmospheric C-14 production, contamination, and preservation conditions affect accuracy and require calibration.

Key Equations and Concepts to Remember

Isotopes differ in neutron number but have the same number of protons:
- Example: Carbon isotopes C-12, C-13, C-14 have 6 protons but 6, 7, and 8 neutrons respectively.
Isotopic abundance and atomic mass:
- Atomic mass (weighted average):
- $A = \sumi fi m_i,$
- where $fi$ are fractional abundances (as decimals) and $mi$ are isotopic masses.
Example weighted average for magnesium (illustrative):
- $A_{Mg} = (0.787\times 24) + (0.101\times 25) + (0.112\times 26) \approx 24.325\ \text{amu} \approx 24.3\ \text{amu}.$
Mass spectrometry workflow (conceptual): vaporize → ionize → accelerate → separate by mass-to-charge ratio in a magnetic field → detect and quantify isotopes.
Atomic mass unit versus grams per mole:
- 1 amu is defined relative to 1/12 the mass of a carbon-12 atom; practical lab use often involves g/mol to quantify amounts of substance.
Nuclear decay and radiation types:
- Alpha: helium-4 nucleus; blocked by paper/skin; low penetration.
- Beta: high-energy electron; blocked by aluminum; moderate penetration.
- Gamma: high-energy photons; requires dense shielding (lead); high penetration.
Carbon-14 dating relationship:
- $N = N0 \left(\frac{1}{2}\right)^{\frac{t}{T{1/2}}}$ with $T_{1/2} = 5730\,\text{years}.$
Carbon-14 decay equation (beta decay):
- $^{14}{6}\mathrm{C} \rightarrow ^{14}{7}\mathrm{N} + e^{-} + \bar{\nu}_e.$
Societal and ethical implications of radiative technologies include safety, long-term storage of nuclear waste, environmental impact, and medical benefits when properly managed.

Quick Practice Prompts (to test understanding)

If a sample of magnesium has isotopic abundances 0.787 (Mg-24), 0.101 (Mg-25), 0.112 (Mg-26), what is the expected atomic mass in amu? Show the calculation steps.
Explain why the atomic mass on the periodic table is not simply the average of the isotope masses.
Describe how a mass spectrometer separates isotopes and what information is obtained from the spectrum.
Compare alpha, beta, and gamma radiation in terms of penetrating ability and shielding requirements.
Outline the carbon-14 dating method and interpret what the 50%, 25%, and 12.5% remaining fractions mean for age estimates.