Isotopes and Relative Atomic Mass

Defining Isotopes

Atoms of the same element always contain the same number of protons in their nuclei, meaning they share the same atomic number. However, the number of neutrons can vary between atoms of the same element. These variations are known as isotopes. Because the number of neutrons differs, isotopes of the same element have different mass numbers.

The term isotope comes from the Greek words isos, meaning same, and topos, meaning place. This refers to the fact that isotopes occupy the same position in the Periodic Table. A specific isotope is identified by its atomic and mass numbers using standard notation.

Consider the three isotopes of hydrogen as an example:

$^{1}_{1}H$ (Protium), $^{2}_{1}H$ (Deuterium), and $^{3}_{1}H$ (Tritium).

In each case, the number of protons remains 1. The number of neutrons for Protium is 0, for Deuterium is 1, and for Tritium is 2. This results in mass numbers of 1, 2, and 3 respectively.

Mass Spectrometry and Abundance

The number and relative abundances of an element's isotopes are determined using a mass spectrometer. In this device, atoms or molecules are ionised, accelerated, and then separated based on their mass to charge ratio ($m/z$) as they drift through the machine toward a detector.

A mass spectrum is a plot showing the number of ions detected (the abundance) against their $m/z$ values. The x-axis represents the $m/z$ value, while the y-axis indicates the relative abundance or intensity. Often, the peak for the most abundant ion is set to 100 percent, with others measured relative to it.

Interpreting Mass Spectra

By looking at a spectrum, you can identify the number of isotopes and their relative quantities. For instance, the mass spectrum of neon shows three distinct peaks.

This indicates neon has three isotopes with mass numbers 20, 21, and 22. In contrast, the spectrum for boron shows two peaks.

These peaks are at $m/z$ 10 and 11. The ratio of the heights for $m/z$ 10 to 11 is 1 to 4. This means that in a sample of boron, there are four times as many atoms with a mass of 11 as there are with a mass of 10. Consequently, 80 percent of the atoms are boron-11 and 20 percent are boron-10.

Calculating Relative Atomic Mass

The relative atomic mass, $A_r$, is not a simple average. It is a weighted mean that takes the abundance of each isotope into account. The values obtained from mass spectra are relative to one twelfth the mass of an atom of carbon-12. Because it is a ratio, $A_r$ has no units.

Calculation Methods

1. Using Percentage Data

If you are given percentages, such as $a\%$ of isotope $q$ and $b\%$ of isotope $r$, the formula is:

$A_r (X) = \frac{(a \times q) + (b \times r) + \dots}{100}$

For chlorine, 75 percent of atoms are $^{35}Cl$ and 25 percent are $^{37}Cl$:

$A_r (Cl) = (\frac{75}{100} \times 35) + (\frac{25}{100} \times 37) = 26.25 + 9.25 = 35.5$

2. Using Mass Spectrum Peak Heights

If the data is presented as arbitrary peak heights $a$ for mass $q$ and $b$ for mass $r$, the formula is:

$A_r (X) = \frac{(a \times q) + (b \times r) + \dots}{(a + b + \dots)}$

Using the chlorine mass spectrum below:

The peaks are at 50 units for $m/z$ 35 and 20 units for $m/z$ 37. The total height is $50 + 20 = 80$.

$A_r (Cl) = \frac{(50 \times 35) + (20 \times 37)}{80} = \frac{1750 + 740}{80} = \frac{2490}{80} = 35.5$

Worked Examples from the Guide

Example: Copper Isotopes

In this spectrum, there are 35 units at mass 63 and 15 units at mass 65. The total number of atoms represented is $35 + 15 = 50$.

$A_r (Cu) = (\frac{35}{50} \times 63) + (\frac{15}{50} \times 65) = \frac{2205}{50} + \frac{975}{50} = \frac{3180}{50} = 63.6$

Example: Diatomic Bromine Gas

Bromine exists as $Br_2$ molecules. The peaks at $m/z$ 79 and 81 are $Br^+$ ions. Since they are equal height, the abundance of $^{79}Br$ and $^{81}Br$ is 50 percent each. The peaks at $m/z$ 158, 160, and 162 are $Br_2^+$ molecular ions. These correspond to molecules made of 79 plus 79, 79 plus 81 (or 81 plus 79), and 81 plus 81. Because there are two ways to form the 160 mass ion, it is twice as abundant as the 158 or 162 peaks.