Relative masses and mass spectrometry

In a nutshell

The relative mass of elements in The Periodic Table are often given to one decimal place rather than as whole numbers. This is because the relative atomic mass of an element is an average mass compared to Carbon $-12$ . You need to know the definitions of relative atomic mass and relative isotopic mass, and how to calculate them. You also need to know the processes which occur in a mass spectrometer and how to analyse data from a mass spectrometer to calculate relative atomic masses.

Relative atomic mass

As the mass of an atom is very small, relative mass is used by comparing the mass to carbon- $12$ . Most elements have more than one isotope, and therefore the relative atomic mass is one way of taking into account the abundance of each isotope.

Definition

Relative atomic mass, $A_r$ , is the weighted mean mass of an atom of an element, compared to $1/12^{th}$ the mass of an atom of carbon- $12$ .

Note: This is not usually a whole number, since it is an average of all the relative isotopic masses (more on this below).

Calculating $A_r$

Relative atomic mass can be calculated if the percentage abundance of each isotope is known.

procedure

1.	Multiply each relative isotopic mass by its percentage abundance, then add up the results.
2.	Divide the sum by $100$ .

Example

For $B$ , given that $B\,-\,10$ isotope has an abundance of $19.9\%$ and $B\,-\,11$ has an abundance of $80.1\%$ , calculate the $A_r$ .

First, add each isotopic mass multiplied by its percentage abundance:

$(10 \, \times\, 19.9) + (11 \, \times\, 80.1) = 199 \, + \, 881.1 = 1080.1$

Then, divide the sum by $100$ :

$\dfrac{1080.1}{100} \, = \, 10.8$

Therefore the $A_r$ of $B$ is $\underline{10.8}$ .

Relative isotopic mass

The mass of isotopes is also very tiny, so we use a number relative to the standard carbon- $12$ as a measure.

Definition

Relative isotopic masses and their percentage abundance in a sample of an element are used to calculate relative atomic masses.

Note: This is usually a whole number.

Calculating relative isotopic mass

PROCEDURE

1.	Find the abundance of the desired isotope by subtracting all the given percentages from $100$ .
2.	Using the procedure to calculate $A_r$ from above, substitute the amounts given and solve for isotopic mass.

Example

$Mg$ exists as three isotopes two of which are $Mg-24$ and $Mg-25$ in the abundances of $79\%$ and $10\%$ respectively. Given that the relative atomic mass of $Mg$  is $24.3$ , find the abundance and isotopic mass of the third isotope.

First, find the abundance of the third isotope by subtracting the given percentages from $100$ :

$100\%\,-\,79\%\,-\,10\%\,=\,11\%$

Then, use the calculation for $A_r$ with the known masses and abundances substituted in:

$\dfrac{(24\,\times\,79)\,+\,(25\,\times\,10)\,+\,(X\,\times\,11)}{100} \, = \,24.3$

Solve for $X$ :

$1896\,+\,250\,+\,11X\, = \,24.3\,\times\,100$

$X\,=\,\dfrac{2430\,-\,1896\,-\,250}{11}\,=\,26$

Therefore, the third isotope of $Mg$ is $\underline{26}$ .

The mass spectrometer

The masses of atoms, molecules and fragments of molecules can be measured using a time-of-flight (TOF) mass spectrometer. The following processes occur within a mass spectrometer:

1.	Ionisation - the sample is injected into the mass spectrometer in gaseous form and bombarded with high energy electrons. The high energy electrons strip the electrons from the atoms or molecules in the sample, forming positive ions.
2.	Acceleration - the positive ions are accelerated through a high energy electric potential. The ions pass through slits and emerge as parallel beams of ions.
3.	Deflection - the ions are then deflected by a magnetic field. Ions with a greater mass are deflected less than those with a smaller mass.
4.	Detection - the ions of lowest mass are detected first, as the magnetic field is increased. The detector is coupled to a computer to record the $m/z$ ratio and relative abundance. This is then used to generate mass spectra to be analysed.

Mass spectrometry data

Mass spectra is used to calculate the relative atomic masses of different elements.

The mass spectra for $Ga$ is given below:

Chemistry; Atomic structure and the periodic table; KS5 Year 12; Relative masses and mass spectrometry

The y-axis (A) gives the percentage abundance of isotopes.

The x-axis (B) gives the mass-to-charge ratio (m/z), which is assumed to be the relative isotopic mass as most ions have a charge of $+1$ .

Relative atomic mass can be calculated from the graph following the same steps as in the relative atomic mass section above:

$\dfrac{(69\,\times\,60.11) + (71\,\times\,39.89) }{100}\,=\,69.8$

Therefore the $A_r$ of $Ga$ is $\underline{69.8}$ .

However, sometimes a mass spectrum may contain relative abundances instead of % abundances. In this case the steps are slightly different.

procedure

1.	Multiply each m/z value by its percentage abundance, then add up the results.
2.	Divide by the sum of isotopic abundances.

Example

Chemistry; Atomic structure and the periodic table; KS5 Year 12; Relative masses and mass spectrometry

Using the mass spectra of $K$ above, calculate its relative atomic mass.

First, multiply each m/z value by its percentage abundance and add them together:

$(39\,\times\,111.91)\,+\,(40\,\times\,0.01)\,+\,(41\,\times\,8.08)\,=\,4696.17$

Then, divide by the sum of isotopic abundances:

$\dfrac{4696.17}{39\,+\,40\,+\,41}\,=\,39.1$

Therefore, the $A_r$ of $K$ is $\underline{39.1}$ .