Atomic Mass

15th Jun 2019 @ 22 min read

Atomic mass is a fundamental concept in chemistry. It relates the mass of an element to the number of atoms.

The atomic mass of an element is frequently used by chemists to determine the amount of substance required in a chemical reaction. The amount of product formed or the amount of reactant required is determined through the stoichiometry of a chemical reaction. A chemical reaction always deals with the number of atoms or moles of reactants and products. Practically, it is not easy to measure the number of atoms in a given sample of a substance; so, chemists measured the weight of a sample and converts it into the number of atoms or moles using atomic mass.

Definition and Formula of Atomic Mass

The Gold Book of the International Union of Pure and Applied Chemistry (IUPAC) defines atomic mass as “rest mass of an atom in its ground state”. In simple words, atomic mass is the mass of an atom of an element.

An Atom, as we know, consists of nucleons (protons and neutrons) and electrons, which revolve around the nucleus of an atom. Hydrogen atoms are the simplest of all atoms. They contain one electron and one proton.

Hence, we can say atomic mass or the mass of an atom is the sum of the mass of all electrons, protons, and neutrons contained in that atom.

$m_\text{a} =n_\text{e}m_\text{e} + n_\text{p}m_\text{p} + n_\text{n}m_\text{n}$

where:
m_a is the mass of an atom,
m_e is the mass of an electron,
m_p is the mass of a proton,
m_n is the mass of a neutron,
n_e is the number of electrons in an atom,
n_p is the number of protons in an atom,
n_n is the number of neutrons in an atom.

Since the mass of an electron is more than 1000 times less than the mass of a proton, we can ignore the mass of an electron.

$n_\text{e}m_\text{e} + n_\text{p}m_\text{p}+ n_\text{n}m_\text{n}\approx n_\text{p}m_\text{p} + n_\text{n}m_\text{n} \\ \because m_\text{e} << m_\text{p}$

So, the above statement can be simplified as atomic mass is the sum of the mass of all protons and neutrons in an atom.

$m_\text{a} \approx n_\text{p}m_\text{p} + n_\text{n}m_\text{n}$

lithium atom — Figure 1: Lithium-7 atom has 3 electrons, 3 protons, and 4 neutrons.

For example, consider the lithium-7 atom, which has 3 electrons, 3 protons, and 4 neutrons.

$m_\text{Li} \approx 3m_\text{p} + 4m_\text{n}$

Units of Atomic Mass

The standard unit of atomic mass is the unified mass unit. It is denoted as u. The unified mass unit is also known as the dalton (named after John Dalton, who known for his atomic theory). One unified mass unit or one dalton is equal to one-twelfth of the mass of a neutral and unbound atom of isotopic carbon-12 at the ground state. And its approximate value is 1.66 × 10⁻²⁷ kg.

$1\,\text{u} &=\frac{1}{12} \times m_{^{12}\text{C}} \\ &=1.660\,539\,066\,60(50) \times 10^{-27}\,\text{kg}$

The archaic version of the unified mass unit is the atomic mass unit and it is denoted as (amu).

From the above expression, we can say the mass of carbon-12 (¹²C) is exactly equal to 12 u or 12 amu.

Average Atomic Mass

Most of elements have two or more naturally occurring isotopes. Isotopes are atoms of the same element with different atomic mass. It is necessary to incorporate the existence of these isotopes with respect to their relative abundance (percentages). So, we average out the atomic masses. We, most of the time, use average atomic mass, not atomic mass for calculations. Masses of elements mention in the periodic table are also average atomic masses. The formula for calculating average atomic mass is described below.

$m_\text{a}=\sum_{i=1}^{k} m_i p_i$

where m_i is the atomic mass of an isotope with a relative abundance of p_i.

Consider an example of carbon. Carbon has three major naturally occurring isotopes, which are shown below with their relative abundance.

Table 1: Relative Abundance of Carbon Isotopes with their Atomic Masses
Isotope	Relative Abundance (%)	Unified Mass Unit (u)
¹²C	98.892	12
¹³C	1.108	13.003 35
¹⁴C	< 10⁻¹²	14.000 317

The average atomic mass of carbon can be calculated as:

$m_\text{a} &=\sum_{i=1}^{k} m_ip_i \\ &=12\times 0.988\,92+13.003\,35 \times 0.011\,08 \\ &\qquad \qquad+14.000\,317 \times 10^{-12} \\ &=12.011\,12\,\text{u}$

Relative Atomic Mass or Atomic Weight

Relative atomic mass or atomic weight is the average atomic mass divided by one unified atomic unit. So, average atomic weight of carbon is 12.011 12 u ÷ 1 u = 12.011 12.

Note: the average atomic weight is dimensionless quantity while atomic mass has the dimension of unified mass unit (u), But both has the same numerical value.

$\text{Relative Mass} =\frac{\text{Atomic Mass}}{1\,\text{u}}$

1 u equals the one-twelfth mass of carbon-12 atom; so, we can define the atomic weight in terms of carbon-12 as the ratio of average atomic mass to the one-twelfth mass of carbon-12 atom.

Mass Defect

Mass defect is the difference between the sum of the masses of all constituents and observed the atomic mass of an atom of an element. The observed atomic mass is always less than the sum of the masses of all constituent particles. This was first discovered by Einstein. When an atom is formed from all its constituent particles (electrons, protons, and neutrons), some of the mass is transformed into binding energy, which is calculated using famous Einstein equation, E = mc². Because of this, the observed mass of an atom is less than the sum mass of its constituent particles. The graph below represents the nuclear binding energy curve.

Figure 2: Graph of Binding Energy per Nucleon vs Number of Nucleons
[Data Source: Brookhaven National Laboratory]

As we can obverse, initially binding energy per nucleon increases with a rise in the number of nucleons. But for heavier atoms, the binding energy decreases with an increase in nucleons.

Note: The mass loss due to binding energy is very small in comparison to the overall mass of an atom.

Measurement of Atomic Mass

Since atoms are extremely small, their mass is also very small. It is not possible to measure the mass of an atom with normal weighing machines. With the advent of technology, we have developed sophisticated techniques, which can measure the mass of an atom with considerable accuracy. One of them is mass spectrometry. A mass spectrometer is a very powerful device which can identify elements, isotopes, and compounds.

Figure 3: Schematics of Simple Mass Spectrometer

A brief working principle of a mass spectrometer is described below.

A mass spectrometer works with a gaseous sample, and it uses mass to charge ratio to determine the mass of an atom.
We can divide a mass spectrometer into three components: an ion source, a mass analyser, and a detector (or collector). An ioniser ionises a given sample. When a sample flows through an ioniser, it is subjected to a highly energised condition, which rips electrons from atoms of a given sample. This causes the ionisation of a sample. An analyser separates these generator ions based on mass to charge ratio. This separation is nothing but the deflection of the sample from the normal path (see above figure). This is achieved by subjecting the sample to a magnetic field. Finally, these deflections are collected and analysed to give the output on a monitor. This is done using a detector. There are various types of ionisers, analysers, and detectors used based on requirements, but we will not discuss it in this article.
The sequence of mass spectrometry starts with the injection of a sample. Initially, the injected sample passes through positively charged plates. Then they are subjected to an ioniser, which ionises the sample.
After the ionisation of the sample, it passes through an accelerator, which is positively charged.
These ionised particles pass through an analyser. Here, they under the influence of a magnetic field are deflected. This deflection depends on mass to charge ratio, for example, heavier particles may deflect less in comparison to lighter particles.
Collected is the final component, which received these deflected particles. Based on the amount of deflection of particles, we can determine various properties like mass, composition, isotopes.

Unified Mass Units and Grams

The unified mass unit is a very small unit. It is comfortable to use at the atomic scale. But is not very efficient when practical life calculation, for example, in chemical laboratories, we use weighting machine, which measured the amount of sample in grams, not in atomic mass units. So, it is necessary to establish a relationship between both the units. This is where atomic mass constant comes in the picture. The atomic mass constant (M_u) relates atomic mass units (u) by Avogadro’s constant (N_A). The relationship between them is as follows.

$1 \text{u} =\frac{M_\text{u}}{N_\text{A}}$

The value of M_u is 0.999 999 999 65(30) g mol⁻¹. It can be approximated to 1 g mol⁻¹.

$1 \text{u} =\frac{1\,\text{g}\,\text{mol}^{-1}}{N_\text{A}}$

History of Atomic Mass

The history of atomic mass goes back to the beginning of the 19^th century when John Dalton purposed the atomic theory and he was the first chemist to determine the relative atomic mass. He used hydrogen, which is the lightest of all, as a reference element. The relative atomic mass assigned to hydrogen was one. Around 1900, hydrogen was replaced by oxygen as a reference element. Therefore, the atomic mass unit at that time was defined as the one-sixteenth mass of an oxygen atom. Later, it was discovered that oxygen has two heavier isotopes (¹⁷O and ¹⁸O), and presence of these was not incorporated in the definition. So, oxygen was also replaced by carbon-12. As of today, carbon-12 remains a reference element.

Atomic Mass and Mass Number

Atomic mass and mass number are two different quantities. Atomic mass, as mentioned above, is the mass of an atom while mass number is the number of nucleons (protons and neutrons) in an atom. Mass number is always a whole number while atomic mass is not (except in the case of carbon-12 when expressed in u. The ratio of atomic mass to mass number is always closed to one. This is shown in the table below.

Table 2: Ratio of Atomic Mass to Mass Number of Various Elements
Element	Atomic Mass, m_u in u	Mass Number, A	^m_u∕_A
¹ ₁H	1.008	1	1.008
⁴ ₂He	4.003	4	1.001
¹² ₆C	12	12	1
¹⁶ ₈O	15.995	16	0.999 69
⁵⁶ ₂₆Fe	55.935	56	0.998 84
²³³ ₈₈Ra	233.048	233	1.045 06
²³⁸ ₉₂U	238.051	238	1.000 21

Atomic Mass Calculations and Examples

Example 1

Consider three isotopes of hydrogen: protium (¹
₁H), deuterium (²
₁H), and tritium (³
₁H).

Their respective abundance with atomic mass is mentioned in the below table.

Table 3: Isotopes of Hydrogen with their Atomic Mass and Relative Abundance
Isotope	Atomic Mass (u)	Abundance (%)
Protium	1.007 825	99.988 5
Deuterium	2.014 101	0.011 5
Tritium	3.016 049	trace

We can calculate average atomic mass as:

$m_\text{a} &=\sum_{i=1}^{k} m_ip_i \\ &=1.007\,825\times 0.999\,885\\ &\qquad \qquad+2.014\,101 \times 0.000\,115 \\ &=1.008\,\text{u}$

Example 2

Consider two isotopes of chlorine: ³⁵
₁₇Cl and ³⁷
₁₇C. Their respective abundance with atomic mass is mentioned in the below table.

Table 4: Isotopes of Chlorine with their Atomic Mass and Relative Abundance
Isotope	Atomic Mass (u)	Abundance (%)
Chlorine-35	34.968 853	76
Chlorine-37	36.965 903	24

We can calculate the average atomic mass as:

$m_\text{a} &=\sum_{i=1}^{k} m_ip_i \\ &=34.968\,825\times 0.76\\ &\qquad \qquad+36.965\,903 \times 0.24 \\ &=35.45\,\text{u}$

Example 3

The mass of a hydrogen atom is 1.673 6 × 10⁻²⁴ g. Determine the mass in the unified mass unit.

We know that 1 u = 1.660 539 × 10⁻²⁴ g.

$m_\text{u} &=\frac{1.673\,6 \times 10^{-24} \,\text{g}}{1.660\,539 \times 10^{-24}\,\text{g}}\\ &=1.007\,8\,\text{u}$

Also, we can write the value in terms of g mol⁻¹.

$m_\text{u} &=1.007\,8\,\text{u}\\ &=1.007\,8\,\text{g}\,\text{mol}^{-1}$

Example 4

Consider three isotopes of oxygen: oxygen-16 (¹⁶
₈O), oxygen-17 (¹⁷
₈O), and oxygen-18 (¹⁸
₈O).

Their respective abundance with atomic mass is mentioned in the below table.

Table 3: Isotopes of Oxygen with their Atomic Mass and Relative Abundance
Isotope	Atomic Mass (u)	Abundance (%)
Oxygen-16	15.994 915	99.76
Oxygen-17	16.999 131	0.04
Oxygen-18	17.999 160	0.20

We can calculate the average atomic mass as:

$m_\text{a} &=\sum_{i=1}^{k} m_ip_i \\ &=15.994\,915\times 0.997\,6\\ &\qquad \qquad +16.999\,131 \times 0.000\,4\\ &\qquad \qquad +17.999\,160 \times 0.002\,0\\ &=15.999\,\text{u}$

Associated Articles

If you appreciate our work, consider supporting us on ❤️ patreon.

Atom

5
cite
response
facebook
twitter

5
twitter
facebook
cite

Copy Article Cite

Thanks for your response!