The mass value for any element listed on the Periodic Table is not a simple whole number. This value represents a calculated average of all the naturally occurring forms of that element, known as isotopes. Atoms of the same element exist with slightly different masses. Calculating this representative mass, called the average atomic mass, requires knowing the mass and natural frequency of each isotope. This calculation provides the single, consistent mass value used in chemical equations.
Defining Isotopes and Relative Abundance
An isotope refers to atoms of the same element that contain an identical number of protons but a varying number of neutrons. Since the number of protons defines the element’s atomic number, isotopes maintain the same chemical identity but possess different physical masses. For instance, a carbon atom always has six protons. It can have six, seven, or eight neutrons, leading to isotopes Carbon-12, Carbon-13, and Carbon-14, respectively.
The mass of a single, specific isotope is called the isotopic mass, measured in atomic mass units (amu). This mass is distinct from the element’s overall atomic mass, which is a weighted figure. Scientists must account for the natural distribution of isotopes because a sample of any element contains a mixture of these forms.
Relative abundance describes the percentage or fraction of a specific isotope that exists naturally in a typical sample of that element. For example, if 98.9% of all naturally occurring carbon atoms are Carbon-12, its relative abundance is 98.9%. This percentage is a stable, measurable quantity determined through techniques like mass spectrometry. The higher the relative abundance, the greater its influence on the element’s calculated average atomic mass.
The Formula for Calculating Average Atomic Mass
The average atomic mass is calculated by determining a weighted average of the masses of all natural isotopes. This method ensures that the mass of more common isotopes contributes more significantly to the final average. A simple arithmetic mean would not accurately reflect the element’s mass in nature, as it would treat all isotopes as equally abundant.
The mathematical relationship is a summation of products for every isotope of the element. The formula is expressed as: (Isotopic Mass\(_{1}\) \(\times\) Fractional Abundance\(_{1}\)) \(+\) (Isotopic Mass\(_{2}\) \(\times\) Fractional Abundance\(_{2}\)) \(+\) \(\dots\). The final result of this calculation is the element’s average atomic mass, which is the value listed on the Periodic Table.
Before beginning the calculation, any relative abundance given as a percentage must be converted into a fractional abundance. This conversion requires dividing the percentage value by 100. For example, an isotope with a 75.77% abundance is converted to a fractional abundance of 0.7577 for use in the formula.
Applying the Calculation Step-by-Step
To illustrate this process, consider the element Chlorine, which has two main naturally occurring isotopes. Chlorine-35 has an isotopic mass of 34.969 atomic mass units (amu) and a relative abundance of 75.77%. Chlorine-37 has an isotopic mass of 36.966 amu and a relative abundance of 24.23%.
The first step is to convert the percentage abundances into fractional abundances by dividing by 100. Chlorine-35’s fractional abundance becomes 0.7577, and Chlorine-37’s fractional abundance is 0.2423. The sum of these fractional abundances must always equal 1.0.
Next, the isotopic mass of each isotope is multiplied by its corresponding fractional abundance. For Chlorine-35, this product is \(34.969 \text{ amu} \times 0.7577\), which equals \(26.50 \text{ amu}\). For Chlorine-37, the product is \(36.966 \text{ amu} \times 0.2423\), which equals \(8.96 \text{ amu}\).
The final step involves summing these products to find the average atomic mass. Adding the two partial masses together (\(26.50 \text{ amu} + 8.96 \text{ amu}\)) yields a final average atomic mass of \(35.46 \text{ amu}\). This result is closer to \(35 \text{ amu}\) because the lighter isotope, Chlorine-35, is significantly more abundant.