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Atomic Mass

15th Jun 2019 @ 22 min read

Physical Chemistry

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}

ma is the mass of an atom,
me is the mass of an electron,
mp is the mass of a proton,
mn is the mass of a neutron,
ne is the number of electrons in an atom,
np is the number of protons in an atom,
nn 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 mi is the atomic mass of an isotope with a relative abundance of pi.

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
IsotopeRelative Abundance (%)Unified Mass Unit (u)
13C1.10813.003 35
14C< 10−1214.000 317

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.

Graph of Binding Energy per Nucleon vs Number of Nucleons
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.

Schematics of Simple Mass Spectrometer
Figure 3: Schematics of Simple Mass Spectrometer

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

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 (Mu) relates atomic mass units (u) by Avogadro’s constant (NA). The relationship between them is as follows.

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

The value of Mu 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 19th 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
ElementAtomic Mass, mu in uMass Number, AmuA
15.995160.999 69
55.935560.998 84
233.0482331.045 06
238.0512381.000 21

Atomic Mass Calculations and Examples

Example 1

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

isotopes of hydrogen
Figure 4: Three isotopes of Hydrogen: Protium (1
), Deuterium (2
), and Tritium (3

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

Table 3: Isotopes of Hydrogen with their Atomic Mass and Relative Abundance
IsotopeAtomic Mass (u)Abundance (%)
Protium1.007 82599.988 5
Deuterium2.014 1010.011 5
Tritium3.016 049trace

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
and 37
. Their respective abundance with atomic mass is mentioned in the below table.

Table 4: Isotopes of Chlorine with their Atomic Mass and Relative Abundance
IsotopeAtomic Mass (u)Abundance (%)
Chlorine-3534.968 85376
Chlorine-3736.965 90324

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
), oxygen-17 (17
), and oxygen-18 (18

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

Table 3: Isotopes of Oxygen with their Atomic Mass and Relative Abundance
IsotopeAtomic Mass (u)Abundance (%)
Oxygen-1615.994 91599.76
Oxygen-1716.999 1310.04
Oxygen-1817.999 1600.20

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}

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