Atomic Structure & Isotopes — Quick Reference
Subatomic structure
Matter: pure substances (elements, compounds) and mixtures; atoms are the basic unit of elements.
Atom composition: nucleus (protons +, neutrons 0) and surrounding electrons (negative).
Nucleus: contains protons and neutrons, which are significantly heavier than electrons and account for most of the atom's mass.
Proton mass: approximately 1.007 \text{ amu}, charge: +1.
Neutron mass: approximately 1.009 \text{ amu}, charge: 0.
Electrons: much lighter particles orbiting the nucleus in specific energy levels or shells.
Electron mass: approximately 0.00055 \text{ amu}, charge: -1. Due to their negligible mass, electrons contribute very little to the atom's overall mass.
Neutral atom: contains an equal number of protons and electrons, resulting in an overall charge of 0.
Electrostatic attraction between opposite charges (the positive nucleus and negative electrons) keeps the electrons bound to the atom.
Within the nucleus, the strong nuclear force, one of the four fundamental forces, acts to overcome the electrostatic repulsion between positively charged protons, holding the nucleus intact over very short distances.
Central force analogy: a centripetal force can keep a particle in orbit around a center, similar to how electrons orbit the nucleus.
Protons, electrons, neutrons are subatomic particles, fundamental building blocks of atoms.
Atomic number Z = number of protons; in a neutral atom, Z = number of electrons, determining the element's identity.
Mass concepts: atomic mass unit (amu) is used instead of grams for individual atoms to work with more manageable numbers; 1 amu is precisely defined as 1/12 the mass of a carbon-12 atom.
Example: sodium (Na) has Z = 11 \Rightarrow 11 protons and (in a neutral atom) 11 electrons, identifying it as sodium.
Atomic number, mass number, and neutrons
Mass number A = total number of nucleons = protons + neutrons. This number represents the approximate mass of the atom in amu.
Neutron count N = A - Z. This calculation allows for determining the number of neutrons in a specific isotope when the mass number and atomic number are known.
Isotopes: atoms with the same atomic number (Z) but different mass numbers (A). This difference arises from a varying number of neutrons (e.g., Carbon-12 has 6 neutrons, Carbon-13 has 7 neutrons, and Carbon-14 has 8 neutrons, all having 6 protons).
Isotopes differ in their number of neutrons and thus their atomic mass, but because they have the same number of protons (and consequently electrons in a neutral atom), they exhibit nearly identical chemical behavior. Chemical properties are primarily determined by electron configuration.
Isotope notation typically uses the mass number on top and the atomic number on bottom, preceding the element symbol, e.g., ^{A}{Z}\text{X}. For instance, Carbon-12 is written as ^{12}{6}\text{C}.
Isotopes and atomic mass
Natural elements consist of a mixture of isotopes with different natural abundances; the atomic mass listed on the periodic table is a weighted average of these isotopic masses.
The atomic mass is a decimal number because it reflects the weighted averages of the naturally occurring fractional abundances of an element's isotopes. It's not the mass of a single atom, but an average.
Abundances are fractions (or percentages divided by 100) that sum to 1: \sum{i} f{i} = 1\,.
Weighted average mass: M = \sum{i} f{i} m{i}, \quad \sum{i} f{i} = 1, where f{i} is the fractional abundance of isotope i, and m_{i} is the exact mass of isotope i.
Mass spectrometry (isotope masses and identification)
Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions, which allows for the determination of isotopic masses and their relative abundances.
General process:
Vaporize sample: Solid or liquid samples are converted into a gaseous state.
Ionize: The gaseous atoms or molecules are bombarded with electrons, causing them to lose or gain electrons and become charged ions.
Fragment (optional): Larger molecules may break into smaller charged fragments.
Accelerate ions: The ions are accelerated through an electric field to give them a uniform kinetic energy.
Separate by mass-to-charge ratio: The accelerated ions pass through a magnetic field, which deflects them. Lighter ions with higher charge-to-mass ratios are deflected more strongly than heavier ions or those with lower charge-to-mass ratios. This separates ions based on their mass.
Detect: A detector records the abundance of each ion at different mass-to-charge ratios, producing a mass spectrum.
Mass spectrometry is widely used to determine an element's isotopic composition, identify unknown substances, and characterize the structure of molecules in fields like chemistry, biology, and forensic science.
Example problem: copper isotopes
Given Isotopes: Copper-63 (Cu-63) and Copper-65 (Cu-65) with known exact atomic masses m{63} = 62.9296 \text{ amu} and m{65} = 64.9278 \text{ amu}. The atomic mass of copper found on the periodic table is the weighted average atomic mass, M = 63.546 \text{ amu}.
Let x be the fractional abundance of Cu-63, and y be the fractional abundance of Cu-65. Since these are the only two isotopes contributing to the natural abundance, their fractions must sum to 1: y = 1 - x.
Using the weighted average mass formula, the equation becomes: 63.546 = (62.9296) x + (64.9278) (1 - x).
To solve for x:
63.546 = 62.9296 x + 64.9278 - 64.9278 x
63.546 = 64.9278 - (64.9278 - 62.9296) x
63.546 = 64.9278 - 1.9982 x
1.9982 x = 64.9278 - 63.546
1.9982 x = 1.3818
x = 1.3818 / 1.9982 \approx 0.6915Therefore, y = 1 - x = 1 - 0.6915 \approx 0.3085.
The natural abundances are: Cu-63 \% \approx 69.15\%\, and Cu-65 \% \approx 30.85\%. These percentages represent the relative proportion of each isotope found in a naturally occurring sample of copper.
Significant figures in isotope calculations
When adding/subtracting numbers, the result should be rounded to the same number of decimal places as the measurement with the fewest decimal places. This is because the precision of the sum or difference is limited by the least precise input value (the one with the largest uncertainty in its decimal place).
When multiplying/dividing numbers, the result should be rounded to the same number of significant figures as the measurement with the fewest significant figures. The accuracy of a product or quotient is limited by the component with the fewest significant digits.
Example considerations: It is crucial that the final result does not imply more precision than the input data allows. For abundance calculations, ensure that the sum of all fractional abundances precisely equals 1 (or 100% when expressed as percentages) to maintain accuracy and consistency.
Quick recap: key relationships
Atomic number: Z = \text{# protons} = \text{# electrons (in neutral atom)}. This value defines the element.
Mass number: A = Z + N, where N = A - Z is the number of neutrons. This value indicates the specific isotope.
Isotopes: Atoms of the same element (same Z) but with different numbers of neutrons (different A); their masses are weighted by their natural abundances to give the average atomic mass found on the periodic table.
Isotope mass calculation: The average atomic mass M is calculated as the sum of the products of each isotope's fractional abundance (f{i}) and its exact mass (m{i}), with the condition that the sum of all fractional abundances must be 1: M = \sum{i} f{i} m{i}, \quad \sum{i} f_{i} = 1.
Practical metric: Mass spectrometry is the primary experimental technique used to accurately determine both the exact masses of individual