The mass value listed for every element on the periodic table is not simply the mass of a single atom. This number, often displayed beneath the element symbol, represents the average atomic mass, a calculated value used by chemists and physicists. Deriving this representative number requires understanding the natural composition of the element. This calculation involves accounting for the different forms an element can take and how frequently each form is found.
Why Atomic Mass is a Weighted Average
The concept of a weighted average is necessary because most elements naturally exist as a mixture of different atomic variations called isotopes. An isotope is an atom of an element that has the same number of protons but a varying number of neutrons in its nucleus. This difference in the number of neutrons results in a difference in mass for each isotope.
The natural percentage likelihood of finding a specific isotope is referred to as its percent abundance or relative abundance. This figure is determined experimentally, often using a technique called mass spectrometry. Because certain isotopes are far more common than others, their masses must influence the overall average mass more heavily, requiring a weighted average to accurately reflect the element’s mass in nature.
The Formula for Calculating Average Atomic Mass
The calculation of average atomic mass synthesizes the mass of each isotope with its presence in the natural world. This method is mathematically identical to calculating a weighted average, where the mass of the isotope acts as the value and the fractional abundance acts as the weight. The final value is reported in atomic mass units, abbreviated as amu or u.
Step 1: Convert Percent Abundance
The first step is to convert the percent abundance of each isotope into a decimal fraction by dividing the percentage value by 100. For example, an isotope with a natural abundance of 75.77% converts to the fractional abundance of 0.7577.
Step 2: Calculate Mass Contribution
Next, multiply the mass of that specific isotope by its corresponding fractional abundance. This product represents the mass contribution of that single isotope to the element’s overall average atomic mass. This step must be repeated for every known naturally occurring isotope.
Step 3: Sum the Contributions
The final step is to sum all the resulting products from each isotope to determine the average atomic mass. The formula can be generalized as the sum of (Isotope Mass \(\times\) Fractional Abundance) for all isotopes.
Applying the Calculation: A Worked Example
To demonstrate this process, we use the element chlorine, which has two main naturally occurring isotopes: Chlorine-35 and Chlorine-37. Chlorine-35 has an isotopic mass of 34.969 amu and an abundance of 75.77%. Chlorine-37 has an isotopic mass of 36.966 amu and a natural abundance of 24.23%.
First, convert the percentages to their decimal equivalents: 75.77% becomes 0.7577, and 24.23% becomes 0.2423.
Next, calculate the mass contribution of each isotope.
For Chlorine-35: \(34.969 \text{ amu} \times 0.7577 = 26.50 \text{ amu}\).
For Chlorine-37: \(36.966 \text{ amu} \times 0.2423 = 8.957 \text{ amu}\).
The final step is to sum these mass contributions to find the average atomic mass for chlorine: \(26.50 \text{ amu} + 8.957 \text{ amu} = 35.457 \text{ amu}\). This calculated value closely matches the mass listed for chlorine on the periodic table.