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The atomic weight of an element is related to its isotopes. One way to calculate it is to use the values of the masses of the isotopes and their relative abundance. To perform this calculation easily it is necessary to first understand each of these different concepts.
atomic weight
The atomic weight is also known as the “average atomic mass” of an element. It is an average calculated by multiplying the relative abundance of isotopes of a given element by their atomic masses, and then adding their products.
Therefore, the atomic weight can be expressed in this way:
Atomic weight = ∑ (atomic mass x relative abundance)
Each element has a unique number of positively charged protons in its nucleus. However, the number of neutrons can vary. Atoms of an element with different numbers of neutrons are the isotopes of that element.
In the periodic table there are 20 elements that have only one natural isotope. Others have more than one and some items have many. For example, tin (Sn) has 10 naturally occurring isotopes.
Neutrons have the same mass as protons, and some isotopes have different atomic masses. Thus, the atomic weight of an element on the periodic table is a weighted average (according to relative abundance) of the atomic masses of each isotope. To express the atomic weight, atomic mass units are used: u , Da , amu .
How to calculate the atomic weight of an element: carbon example
Review the periodic table
To calculate the atomic weight of carbon (C), we must first identify its symbol on the periodic table. The atomic weight is the number (usually with decimal places) below the element symbol. In this case it is approximately 12.01. As mentioned before, the atomic weight is an average of the atomic masses of the different isotopes of carbon, therefore, the figures may vary.
Obtain the atomic weight of the isotope
The next step in calculating the atomic weight of a single atom or an isotope of an element is to add the masses of the protons and neutrons that make up its nucleus. The value obtained is known as the atomic mass.
Continuing with the example of carbon, we know that its isotope has 7 neutrons. The atomic number of carbon is 6, and it is equal to the number of protons in its nucleus. Therefore, the atomic weight of this carbon isotope will be the sum of the masses of protons and neutrons: 6 + 7 = 13.
Calculate atomic weight
The third step is to obtain the atomic weight, that is, the weighted average of the atomic masses of the isotopes of the element. The average weighting factor is the natural abundance of each isotope, in this case, the carbon isotope.
Typically, when performing these types of calculations, a list of the element’s isotopes is provided with their atomic mass and isotopic abundance, expressed as a fraction or percentage.
The calculation of the atomic weight consists of multiplying the mass of each isotope by its abundance and adding the results of these operations. If the isotopic abundance is expressed as a percentage, the final result must be divided by 100, or the percentage value of each isotope must be converted to the corresponding decimal expression.
Example:
For example, if we have a sample of carbon atoms with a composition of 98% 12 C and 2% 13 C, we must perform the following steps:
First step: convert the isotopic abundance from percentage to fraction by dividing each value by 100:
Isotopic abundance of 12 C = 0.98
13 C isotopic abundance = 0.02
Since the total isotopic abundance must be 1 (ie 100%), the calculation can be verified by adding the isotopic abundances of each isotope: 0.98 + 0.02 = 1.00.
Second step: multiply the atomic mass of each isotope by its isotopic abundance:
0.98 x 12 = 11.76
0.02 x 13 = 0.26
Third step: add the values obtained to obtain the atomic weight.
11.76 + 0.26 = 12.02 g/mol
What is relative abundance
Isotopes are atoms that have the same number of protons but a different number of neutrons. They also have different atomic masses. The relative abundance of an isotope or isotopic abundance is the percentage of atoms that have a given atomic mass.
To know the relative abundance, the fractional abundance must be calculated. The sum of the fractional values of abundance must be equal to 1.
Suppose we have an element with two isotopes of masses m1 and m2 . Since the sum of the fractional abundances must give a total equal to 1, if the abundance of the first mass is “x” and of the second is “y”, then x + y = 1. That is, the relative abundance of the second is y = 1 – x. This can be expressed as follows:
Atomic weight = m1 . x + m2 . and
Atomic weight = m1 . x + m2 . (1–x)
Atomic weight = m1 . x + m2 – m2 . x
Atomic weight – m2 = (m1 – m2) . x
x = (Atomic Weight – m2) ÷ (m1 – m2)
In this way, we obtain that the quantity x is the relative abundance of the isotope with mass m1. From this value, we determine the relative abundance of the isotope with mass m2 knowing that y = 1 – x.
Example to calculate the abundance of an isotope
For example, suppose we have an element whose atomic weight is 5.2. This element also has two isotopes with atomic masses of 6 and 5 respectively.
If we introduce these values in the above formula, we get:
m1 . x + m2 . y = atomic weight
6 . x + (1 – x) . 5 = 5.2.
6 . x + (1 – x) . 5 = 5.2
6x + 5 – 5x = 5.2
x + 5 = 5.2
x = 5.2 – 5
x = 0.2
Then we find and
y = 1 – x
y = 1 – 0.2
y = 0.8
To know the percentage abundance of the first isotope, multiply “x” by 100. The result is: 0.2. 100 = 20%.
Finally, to obtain the percentage abundance of the second isotope, we must multiply “y” by 100. Thus we obtain: 0.8 . 100 = 80%.
Example to calculate the atomic weight and abundance of an isotope
To better understand how to calculate the atomic weight of an element, let’s look at the case of chlorine (Cl), which has two natural isotopes:
35 Cl: which has a mass of 34.9689 amu.
37 Cl: with a mass of 36.9659 amu.
So, knowing the atomic weight of chlorine (Cl), which is 35.453 amu, we can also calculate the relative abundances of each isotope. To do this, we apply the previous equation:
Atomic weight = m1 . x + m2 . (1–x)
If we assume that x is the fractional abundance of 35 Cl, we identify its mass as m1 and that of 37 Cl as m2, the calculation would be as follows:
x = (35.453 – 36.9659) ÷ (34.9689 – 36.9659)
x = -1.5129 / -1.9970
x = 0.7575
In this way, we obtain that the fractional abundance of the 35 Cl isotope is 0.7575 (ie, 75.75%) and that of the 37 Cl isotope is 0.2425 (ie, 24.25%).
Relative abundances can be calculated for elements that have two isotopes, based on the atomic masses of their isotopes. Elements with more than two isotopes require more complex calculations.
Bibliography
- Llansana, J. Basic Atlas of Physics and Chemistry. (2010). Spain. Parramon.
- Delgado Ortiz, SE; Solíz Trinta, LN Manual of General Chemistry. (2015). Spain. CreateSpace.
- Patiño, A. Introduction to chemical engineering: mass and energy balances. Volume II. (2000). Mexico. UIA.
