Relative and Average Atomic Mass Notes

Initial Challenges in Measurement: Because an atom is extremely small, its absolute mass cannot be determined directly.
Definition: Early chemists used a method to measure the mass of an atom relative to a standard atom. This relative value is known as the Relative Atomic Mass (RAM).
Historical Method: Scientists established a scale by measuring the mass of a large number of atoms of two or more elements simultaneously to determine their relative masses.
Relative Atomic Mass of Common Elements (C-12 Scale): The following values represent the relative atomic masses as recorded in Table 7.1: * Hydrogen ( $H$ ): $1$ * Carbon ( $C$ ): $12$ * Nitrogen ( $N$ ): $14$ * Oxygen ( $O$ ): $16$ * Sodium ( $Na$ ): $23$ * Magnesium ( $Mg$ ): $24$ * Sulphur ( $S$ ): $32$

Hydrogen Standard: In the beginning, the mass of a hydrogen atom ( $^{1}H$ ) was chosen as the standard for comparison because it is the lightest atom.
Oxygen Standard: Later, the hydrogen standard was replaced by the oxygen atom as the reference point.
Carbon-12 Standard: Currently, the stable isotope of carbon with a mass of $12$ (Carbon-12) is used as the standard for measuring the relative atomic masses of other elements.
Arbitrary Value: Carbon-12 is assigned an arbitrary value of $12$ . The masses of all other elements are measured relative to this specific value.

Unitless Nature of RAM: Because Relative Atomic Mass is a ratio of masses, it has no unit.
Gram Atomic Mass: If the atomic mass of an element is expressed in grams, it is referred to as the Gram Atomic Mass.
Examples of Gram Atomic Mass: * Gram Atomic Mass of hydrogen: $1\,g$ * Gram Atomic Mass of carbon: $12\,g$ * Gram Atomic Mass of nitrogen: $14\,g$ * Gram Atomic Mass of oxygen: $16\,g$

The Complexity of Natural Elements: Measuring the standard atomic weight (denoted as $A$ ) of an element is complicated because most naturally occurring elements exist as a mixture of isotopes.
Isotopic Impact: Each isotope of a single element possesses its own unique mass. Therefore, calculations for the atomic mass must account for the isotopic mixture.
Definition of AAM: The average atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes.
Concept of Abundance: The abundance (percentage of occurrence in nature) of isotopes varies for each element. This abundance is a critical factor in determining the weighted average.

General Equation: The weighted average is calculated by multiplying the mass of each isotope by its fractional abundance and summing the results: * $\text{Average atomic mass} = (\text{Mass of 1st isotope} \times \%\text{ abundance of 1st isotope}) + (\text{Mass of 2nd isotope} \times \%\text{ abundance of 2nd isotope})$
Hypothetical Example: Consider an element existing as a mixture of $50\%$ of an isotope with a mass of $9\,amu$ and $50\%$ of another isotope with a mass of $10\,amu$ : * $\text{Average atomic mass} = (9 \times \frac{50}{100}) + (10 \times \frac{50}{100})$ * $\text{Average atomic mass} = 4.5 + 5 = 9.5\,amu$
Fractional Abundance Conversion: In calculations involving percentages, the percentage must be converted into fractional abundance (e.g., $50\%$ becomes $\frac{50}{100}$ or $0.50$ ).

Periodic Table Values: The atomic masses listed for elements in the periodic table are average atomic masses. The term "atomic weight" is often used interchangeably with average atomic mass.
Non-Whole Numbers: Most atomic masses in the periodic table are not whole numbers. This is a direct result of the weighted averaging of isotopes.
Case Study: Carbon: * In the periodic table, the atomic mass of carbon is listed as $12.01\,amu$ rather than exactly $12.00\,amu$ . * The calculation considers the two natural isotopes of carbon: Carbon-12 ( $C-12$ ) and Carbon-13 ( $C-13$ ). * Natural Abundance of Carbon Isotopes: * $C-12$ : $98.90\%$ * $C-13$ : $1.10\%$ * Calculation for Carbon: * $\text{Average atomic mass of carbon} = (12 \times \frac{98.9}{100}) + (13 \times \frac{1.1}{100})$ * $\text{Average atomic mass of carbon} = (12 \times 0.989) + (13 \times 0.011)$ * $\text{Average atomic mass of carbon} = 11.868 + 0.143 = 12.011\,amu$
Important Distinction: When it is stated that the atomic mass of carbon is $12\,amu$ , this generally refers to the average atomic mass of all carbon isotopes combined, not the mass of a single individual carbon atom.