Atoms of the same element have an identical number of protons but can differ in the number of neutrons; these variations are known as isotopes. For example, carbon always has six protons but can exist as carbon-12, carbon-13, or carbon-14, each having a different mass due to the varying neutron count. Isotopic abundance refers to the relative amount of each specific isotope found naturally in a sample. Understanding this natural distribution is necessary because it directly determines the average atomic mass of the element, which is the value listed on the periodic table. This weighted average reflects the collective mass of all naturally occurring isotopes in the proportions they are typically found.
Required Data for Isotope Calculations
Calculating relative isotope abundance requires two sets of mass data. First, the calculation relies on the element’s relative atomic mass, which is the average mass value displayed on the periodic table. This number represents the known weighted average of all the element’s isotopes. The second necessary input is the precise, individual mass of each specific isotope involved. These masses are generally very close to, but not exactly equal to, the isotope’s mass number, and are used along with the average atomic mass to solve for the unknown percentage of each isotope.
Calculating Relative Isotopic Abundance
The method for calculating relative isotopic abundance uses a weighted average model. This model links the known atomic masses of the isotopes to the element’s overall average atomic mass. The overall mass is the sum of each isotope’s fractional abundance multiplied by its specific isotopic mass. This approach is most straightforward for elements with only two naturally occurring isotopes.
In a two-isotope system, the fractional abundances must total 1.0. If the fractional abundance of the first isotope is \(x\), the fractional abundance of the second isotope must be \((1-x)\). The average atomic mass, \(M(E)\), is defined by the equation: \((M_1)(x) + (M_2)(1-x) = M(E)\), where \(M_1\) and \(M_2\) are the precise masses of the two isotopes.
Consider chlorine, which exists primarily as chlorine-35 and chlorine-37. The average atomic mass of chlorine is \(35.45\) atomic mass units (amu). The precise mass of chlorine-35 (\(M_1\)) is \(34.969\) amu, and the precise mass of chlorine-37 (\(M_2\)) is \(36.966\) amu. Setting up the equation to solve for the fractional abundance (\(x\)) of chlorine-35 yields: \((34.969)(x) + (36.966)(1-x) = 35.45\).
To isolate the unknown variable \(x\), distribute the mass of the second isotope to get the expanded equation: \(34.969x + 36.966 – 36.966x = 35.45\). Combining the terms containing \(x\) results in the simplified expression: \(-1.997x = 35.45 – 36.966\), which simplifies further to \(-1.997x = -1.516\).
Solving for \(x\) yields \(x \approx 0.759\). This decimal value represents the fractional abundance of chlorine-35, meaning its natural abundance is approximately \(75.9\%\).
The fractional abundance of chlorine-37 is found by calculating \((1-x)\), which is \(1 – 0.759 = 0.241\). Therefore, the natural abundance of chlorine-37 is approximately \(24.1\%\). This algebraic method determines the natural proportions of each isotope from the known mass values.
Determining Abundance Through Mass Spectrometry
The actual abundances and precise isotopic masses are determined experimentally using a technique called mass spectrometry. This analytical technique provides a direct physical measurement of the relative amounts of each isotope in a sample. The process begins by vaporizing the sample and then ionizing the atoms by bombarding them with a stream of high-energy electrons, which creates positively charged ions.
These ions are accelerated through a vacuum chamber and passed into a magnetic field. Ions of the same charge but different masses are deflected by the magnetic field to different degrees, separating them. Lighter ions are deflected more significantly than heavier ions, causing them to follow distinct paths.
A detector records where each stream of ions lands, and the instrument generates a mass spectrum. This spectrum plots the mass-to-charge ratio (x-axis) against the signal intensity (y-axis). Since the ions typically have a single positive charge, the x-axis represents the mass of the isotopes.
The height of each peak on the mass spectrum is directly proportional to the relative number of ions that reached the detector. Therefore, the height of a peak indicates the relative abundance of that particular isotope in the original sample. This experimental data provides the precise isotopic masses and their natural abundances.