Isotopes and Mass Number

Misconception in the transcript: the question started with calculating the weight of the isomers, but the topic is isotopes, not isomers. Important distinction:
- Isotopes: atoms of the same element (same number of protons, Z) that have different numbers of neutrons (N). They share chemical properties but differ in mass and nuclear properties.
- Isomers (in chemistry): different structural arrangements of the same molecular formula. In nuclear discussions, isomers can refer to different energy states of a nucleus, but this is separate from isotopes.
Core idea: isotopes are variants of an element with the same Z but different A = Z + N.

Definitions:
- Z = atomic number (number of protons)
- N = number of neutrons
- A = mass number = Z + N
Neutrons count relation:
$N = A - Z$
Isotopic notation (nuclear notation): $^{A}_{Z}\mathrm{X}$ where X is the element symbol. Examples:
- Carbon-12: $^{12}_{6}\mathrm{C}$
- Carbon-13: $^{13}_{6}\mathrm{C}$
- Carbon-14: $^{14}_{6}\mathrm{C}$

For carbon, Z = 6 (six protons).
C-12: A = 12, N = A - Z = 12 - 6 = 6 neutrons.
C-13: A = 13, N = 13 - 6 = 7 neutrons.
C-14: A = 14, N = 14 - 6 = 8 neutrons.
Benefit of using Carbon-12 as a standard: it defines the atomic mass unit (amu).

Definition: The unified atomic mass unit (amu) is defined relative to Carbon-12:
$1\ \mathrm{amu} = \frac{1}{12} m(^{12}_{6}\mathrm{C})$
This means the mass of a Carbon-12 atom is exactly 12 amu.
In practice, the atomic mass of an element is a weighted average of the isotopic masses, not simply the integer A values.
The actual mass of an isotope is very close to its mass number, but not exactly equal due to binding energy and mass defects.

The atomic mass of an element is the weighted average of its isotopic masses according to natural abundances: $\overline{m} = \sum<em>i f</em>i m_i$ where:
- $f_i$ are the fractional abundances (summing to 1), and
- $m_i$ are the isotopic masses (in amu).
If you have percentages, convert to fractions by dividing by 100.
Example for a simplified carbon system (ignoring very small C-14):
- C-12 abundance ≈ 98.93% and mass ≈ 12 amu
- C-13 abundance ≈ 1.07% and mass ≈ 13 amu
  Then:
  $\overline{m} \approx 0.9893 \times 12 + 0.0107 \times 13 \approx 12.0107 \ \text{amu}$
  This aligns with the standard atomic mass of carbon ~ $12.011\,\text{amu}$ .
Note: C-14 is present in trace amounts and has a negligible effect on the standard atomic mass of carbon.

For a neutral atom, the number of electrons equals Z (same as protons):
$\text{electrons} = Z$
Example with carbon isotopes (Z = 6):
- C-12: electrons = 6, neutrons = 6
- C-13: electrons = 6, neutrons = 7
- C-14: electrons = 6, neutrons = 8

Isotopes are used in real-world applications:
- Radiocarbon dating uses C-14 to date ancient organic materials.
- Isotopic labeling in chemistry and biochemistry helps track reaction pathways.
- Mass spectrometry relies on precise isotopic masses to identify elements and compounds.
Important conceptual distinction: weight vs mass:
- In chemistry, people often refer to atomic weight or atomic mass, both tied to the amu scale defined by C-12.
Summary of key ideas:
- Isotopes have the same Z but different A and N.
- Mass number A = Z + N; N = A - Z.
- Atomic mass is a weighted average of isotope masses:
  $\overline{m} = \sum<em>i f</em>i m_i$
- 1 amu is defined via Carbon-12 as above, anchoring the mass scale.

Determine N for each carbon isotope:
- C-12: (N = A - Z = 12 - 6 = 6)
- C-13: (N = 13 - 6 = 7)
- C-14: (N = 14 - 6 = 8)
If a sample contains 60% C-12 and 40% C-13, approximate the atomic mass:
$\overline{m} = 0.60 \times 12 + 0.40 \times 13 = 12.4 \text{ amu}$
Notation recap: Carbon-12 is written as $^{12}_{6}\mathrm{C}$ .
Quick check on units: all masses are in amu unless stated otherwise.