Atomic Numbers and Mass Numbers – Study Notes

Atomic Numbers and Mass Numbers

This section clarifies the key numbers used to describe atomic structure: atomic number (Z), mass number (A), and neutron number (N). It also distinguishes mass number from atomic weight and introduces isotopes and notation.

Atomic number (Z): the number of protons in the nucleus. It determines the identity of the element and, in a neutral atom, equals the number of electrons.
Mass number (A): the total number of nucleons (protons + neutrons) in the nucleus. A is an integer for a given isotope.
Neutron number (N): the number of neutrons in the nucleus. N = A - Z
Isotopes: atoms of the same element (same Z) that have different A due to different numbers of neutrons (N). Chemically similar, but physical properties can differ due to mass differences.
Relationship to the nucleus and electrons:
- In a neutral atom, number of electrons equals Z.
- Ionization changes the electron count but not Z.

Isotope notation: ^{A}_{Z}X where X is the chemical symbol, A is the mass number, and Z is the atomic number.
Examples:
- Carbon-12: ^{12}_{6}\mathrm{C}
- Uranium-238: ^{238}_{92}\mathrm{U}
Alternate notation sometimes used is X with a superscript and subscript, but the conventional form is ^{A}_{Z}X.

Mass-number relation:
- A = Z + N
- N = A - Z
Atomic weight vs mass number:
- Mass number A is the integer count of nucleons in a specific isotope.
- Atomic weight (also called relative atomic mass) is the weighted average of the masses of all isotopes of an element as they occur naturally.
- Atomic weight is expressed in atomic mass units (amu) and corresponds to the molar mass in g/mol.
Atomic mass unit:
- 1\ \text{amu} = \frac{1}{12}\,m({}^{12}\mathrm{C})
- The molar mass M (in g/mol) equals the atomic weight (in amu): M\,\text{(g/mol)} = \text{Atomic weight (amu)}
Weighted average for atomic weight:
- If isotopes i have mass numbers $Ai$ and fractional abundances $fi$ (with \sumi fi = 1):
- \text{Atomic weight} = \sumi fi A_i
Example abundances:
- Hydrogen: isotopes H-1, H-2, H-3 with respective abundances that yield an average ≈ 1.008\ \text{amu}
Practical note:
- The atomic weight is not necessarily an integer; A is. A is an integer for a given isotope, while the atomic weight is a weighted average over isotopes.

Example calculation of atomic weight from isotopic abundances:
- Suppose isotope i has mass number $Ai$ and fractional abundance $fi$ with \sumi fi = 1. Then:
- \text{Atomic weight} = \sumi fi A_i
Example with two isotopes:
- If an element has $f{A=12}=0.9893$ and $f{A=13}=0.0107$ (approximate natural abundances for carbon), then:
- \text{Atomic weight} = 0.9893\times 12 + 0.0107\times 13 = 12.011\ \text{amu}

Isotopes are used in:
- Radiometric dating (e.g., Carbon-14 dating) to estimate ages of ancient materials.
- Medical imaging and therapy with specific isotopes.
- Tracing chemical and biological processes in research.
Chemical calculations:
- To convert mass to moles: n = \dfrac{m}{M} where $M$ is the molar mass (in g/mol) equal to the atomic weight in amu.
- One mole contains Avogadro's number $N_A \approx 6.022\times 10^{23}$ entities.

Do not confuse A (mass number) with atomic weight. A is an integer for a given isotope; atomic weight is a weighted average.
Z counts protons; N counts neutrons; For a given isotope, A = Z + N.
In a neutral atom, electron count equals Z; in ions, electron count differs, but Z remains the same.
Binding-energy effects (mass defect) do not change the count of nucleons A; they affect the actual mass slightly, not the integer A.

Problem 1: If Z = 8 and A = 16, find N.
- Solution: N = A - Z = 16 - 8 = 8.
Problem 2: Write the isotope notation for Oxygen-18.
- Solution: ^{18}_{8}\mathrm{O}
Problem 3: If a two-isotope system has abundances 0.9 for A = 12 and 0.1 for A = 13, what is the atomic weight?
- Solution: \text{Atomic weight} = 0.9\times 12 + 0.1\times 13 = 12.1\text{ amu}