Chapter 1: Same Atomic Number

The atomic number is a fundamental property of an element that identifies the element on the periodic table.
It indicates the number of protons in the nucleus.
In a neutral atom, the atomic number also equals the number of electrons surrounding the nucleus.
Therefore, for a neutral atom, the numbers of protons, electrons, and the atomic number are all equal:
- Z = #\text{protons} = #\text{electrons (in a neutral atom)}
The atomic number is the key descriptor we use to identify elements, and it remains constant for all atoms of a given element.

Isotopes are atoms of the same element that differ in the number of neutrons, giving different masses.
They share the same atomic number (and thus the same number of protons and, in a neutral atom, the same number of electrons) but have different mass numbers.
This means:
- Same Z, different A and N.
Definitions:
- Mass number: A = Z + N where N is the number of neutrons.
- Neutrons: N = A - Z
The idea emphasized in the transcript: carbon can be described as carbon across isotopes, still with the same chemical identity, yet with a different mass.

Carbon has atomic number Z = 6 (six protons).
Isotopes mentioned: carbon-12 and carbon-13.
- For carbon-12: A = 12, ext{ so } N = A - Z = 12 - 6 = 6. Thus, carbon-12 has 6 neutrons.
- For carbon-13: A = 13, ext{ so } N = A - Z = 13 - 6 = 7. Thus, carbon-13 has 7 neutrons.
Both isotopes are still carbon because they have the same atomic number (Z = 6) and the same number of protons, but they differ in mass due to the different number of neutrons.
Isotopes notation examples:
- ^{12}\mathrm{C} and ^{13}\mathrm{C}
Summary: the difference in mass between these isotopes arises from the difference in neutrons, while the proton count (and thus chemical behavior) remains the same.

Mass number relates to the total count of nucleons (protons + neutrons):
- A = Z + N
If you know any two of the quantities (A, Z, N), you can determine the third:
- N = A - Z
- Z = A - N
These relationships underpin why isotopes have identical chemical properties (driven largely by Z and electron structure) but different physical properties (mass, stability).

The transcript references an historical framing (eighteenth century) in relation to protons and the atomic number.
Conceptual connections:
- Atomic number (Z) anchors the identity of the element and the electron count in neutral atoms.
- Isotopes illustrate that mass can vary independently of chemical identity because mass is influenced by neutrons (N) as captured by A = Z + N.
- The idea that isotopes share chemical properties despite differences in mass stems from identical electron configurations (set by Z).

Isotopes have critical applications in science and technology:
- Radiometric dating and tracing using specific isotopes with known half-lives (e.g., ^{14}C dating) – not explicitly in the transcript but a practical extension of the isotope concept.
- Medical imaging and treatment often use isotopes with particular neutron numbers to achieve desired stability or radioactivity.
Understanding Z, A, and N is foundational for spectroscopy, nuclear physics, and chemistry, enabling predictions about chemical identity, bonding behavior, and nuclear properties.