Atomic Structure, Isotopes, and Average Atomic Mass

Atomic Number ( $Z$ ):
- Definition: The total number of protons located within the nucleus of every atom of a specific element.
- Element Identity: Serves as the defining characteristic of an element. If the number of protons is known, the chemical identity of the atom is determined.
- Representation on Periodic Table: Displayed as a whole number above or alongside each element's symbol on the periodic table.
- Electrical Neutrality:
- In a neutral atom, the overall electrical charge is zero.
- Equation for neutral atoms: $\text{Number of Protons} = \text{Number of Electrons}$ .
- Operating Assumption: Unless stated otherwise (e.g., when dealing with ions), atoms are assumed to be neutral.
Mass Number ( $A$ ):
- Definition: The total sum of protons and neutrons contained within the nucleus of an atom.
- Equation: $\text{Mass Number } (A) = \text{Number of Protons} + \text{Number of Neutrons}$ .
- Absence from Periodic Table: Mass number does not appear on the periodic table because it pertains exclusively to individual atoms or specific isotopes, rather than an element-wide average.
- Calculation of Neutrons: $\text{Number of Neutrons} = \text{Mass Number } (A) - \text{Atomic Number } (Z)$ .

Element Symbol Notation (Nuclear Notation):
- Structure: The element's chemical symbol is written with the mass number ( $A$ ) as a superscript and the atomic number ( $Z$ ) as a subscript, both positioned directly in front of the symbol: ${}^{A}_{Z}\text{X}$
- Omission of Atomic Number: The atomic number ( $Z$ ) is sometimes omitted from this notation ( ${}^{A}\text{X}$ ) because the element symbol itself uniquely identifies the atomic number on the periodic table.
- Examples:
- Platinum-195: ${}^{195}_{78}\text{Pt}$
- Cobalt-60: ${}^{60}_{27}\text{Co}$
- Magnesium-24: ${}^{24}_{12}\text{Mg}$
Hyphen Notation:
- Structure: Written as the full name of the element followed by a hyphen and the mass number ( $A$ ): $\text{element name}-\text{mass number}$
- Examples:
- $\text{platinum}-195$
- $\text{cobalt}-60$
- $\text{magnesium}-24$
- $\text{hydrogen}-1$
- $\text{hydrogen}-2$
- $\text{hydrogen}-3$

Definition of Isotopes:
- Atoms belonging to the same element that have identical atomic numbers ( $Z$ , same number of protons) but different numbers of neutrons, resulting in distinct mass numbers ( $A$ ).
- Absence of a Single "True" Atom: Every chemical element consists of naturally occurring isotopes; no single isotope represents the sole "true" version of an element.
Chemical and Physical Properties of Isotopes:
- Chemical Reactivity: Different isotopes of the same element behave identically in chemical reactions because chemical behavior is determined by electron configuration and proton count.
- Exceptions: Only a few rare, specific physical/rate-limited situations show slight differences in behavior due to mass disparities.
Isotopes of Hydrogen:
- Hydrogen-1 ( ${}^1_1\text{H}$ , Protium):
- Composition: $1\text{ proton}$ , $0\text{ neutrons}$ .
- Abundance: Comprises approximately $98.98\text{\%}$ (or $\sim 99\%$ ) of natural hydrogen.
- Hydrogen-2 ( ${}^2_1\text{H}$ , Deuterium):
- Composition: $1\text{ proton}$ , $1\text{ neutron}$ .
- Abundance: Comprises less than $2\%$ of natural hydrogen.
- Hydrogen-3 ( ${}^3_1\text{H}$ , Tritium):
- Composition: $1\text{ proton}$ , $2\text{ neutrons}$ .
- Abundance: Occurs in trace amounts in nature.
- Natural Distribution Note: Hydrogen isotopes are not evenly distributed in nature; Hydrogen-1 is overwhelmingly dominant.

Symbol	Atomic # ( $Z$ )	Mass # ( $A$ )	# Protons	# Neutrons	# Electrons
$\text{Cl}$	$17$	$37$	$17$	$20$	$17$
$\text{Pt}$	$78$	$195$	$78$	$117$	$78$
$\text{P}$	$15$	$31$	$15$	$16$	$15$
$\text{P}$	$15$	$32$	$15$	$17$	$15$
$\text{Co}$	$27$	$60$	$27$	$33$	$27$

Step-by-Step Breakdown of Table Calculations:
- Chlorine-37 ( ${}^{37}_{17}\text{Cl}$ ):
- Atomic Number ( $Z$ ) = $17$
- Mass Number ( $A$ ) = $37$
- Protons = $17$
- Electrons = $17$ (assuming neutral atom)
- Neutrons = $37 - 17 = 20$
- Platinum-195 ( ${}^{195}_{78}\text{Pt}$ ):
- Atomic Number ( $Z$ ) = $78$
- Mass Number ( $A$ ) = $195$
- Protons = $78$
- Electrons = $78$
- Neutrons = $195 - 78 = 117$
- Phosphorus-31 ( ${}^{31}_{15}\text{P}$ ):
- Atomic Number ( $Z$ ) = $15$
- Mass Number ( $A$ ) = $31$
- Protons = $15$
- Electrons = $15$
- Neutrons = $31 - 15 = 16$
- Phosphorus-32 ( ${}^{32}_{15}\text{P}$ ):
- Atomic Number ( $Z$ ) = $15$
- Mass Number ( $A$ ) = $32$
- Protons = $15$
- Electrons = $15$
- Neutrons = $32 - 15 = 17$
- Cobalt-60 ( ${}^{60}_{27}\text{Co}$ ):
- Atomic Number ( $Z$ ) = $27$
- Mass Number ( $A$ ) = $60$
- Protons = $27$
- Electrons = $27$
- Neutrons = $60 - 27 = 33$

Atomic Mass Unit ( $\text{amu}$ ):
- Definition: Defined as exactly $\frac{1}{12}$ of the mass of a single carbon-12 ( ${}^{12}\text{C}$ ) atom.
- Scale Relationships:
- $1\,\text{amu} \approx \text{mass of a proton} \approx \text{mass of a neutron}$ .
- Mass in grams: $1\,\text{amu} \approx 1.66 \times 10^{-24}\,\text{g}$ .
- Scope of Usage: Used for calculating and expressing the mass of individual atoms or subatomic particles.
Average Atomic Mass:
- Definition: The weighted average of the atomic masses of all naturally occurring isotopes of an element.
- Periodic Table Representation: Displayed as a decimal number for each element on the periodic table.
- Nature of Weighted Average: Unlike a simple arithmetic mean, a weighted average accounts for both the mass and relative natural percentage abundance of each isotope.
- Example - Carbon:
- Carbon exists primarily as ${}^{12}\text{C}$ (abundance $\approx 99\%$ ) and ${}^{13}\text{C}$ (abundance $\approx 1\%$ ).
- Because ${}^{12}\text{C}$ is overwhelmingly predominant, the average atomic mass of carbon on the periodic table is close to $12.011\,\text{amu}$ .

Formula for Weighted Average Atomic Mass: $\text{Average Atomic Mass} = \frac{(\text{percent abundance}_1 \times \text{mass}_1) + (\text{percent abundance}_2 \times \text{mass}_2) + \dots}{100}$
Worked Example Problem: Boron Isotopes:
- Given Data:
- Isotope 1: ${}^{10}\text{B}$
  - Mass = $10.0129\,\text{amu}$
  - Natural Abundance = $19.80\%$
- Isotope 2: ${}^{11}\text{B}$
  - Mass = $11.0093\,\text{amu}$
  - Natural Abundance = $80.20\%$
- Calculation: $\text{Average Atomic Mass} = \frac{(19.80 \times 10.0129\,\text{amu}) + (80.20 \times 11.0093\,\text{amu})}{100}$ $\text{Average Atomic Mass} = \frac{198.25542\,\text{amu} + 882.94586\,\text{amu}}{100}$ $\text{Average Atomic Mass} = \frac{1081.20128\,\text{amu}}{100} = 10.8120128\,\text{amu}$
- Final Value: $\text{Average Atomic Mass of Boron} \approx 10.81\,\text{amu}$