 ## Relationship Between Atomic Mass, Average Atomic Mass, & Relative Isotopic Mass

Here are some similar Terms for Different Quantities

Atomic Mass (ma or m): mass of one atom ⇒ Expressed in Kg, or gram, or a.m.u, or u
Relative Isotopic Mass: relative mass of one isotope of an element ⇒ Dimensionless
Average Atomic Mass or Relative Atomic Mass (Ar): average mass of all the isotopes of a given element sample ⇒ Dimensionless
Standard Relative Atomic Mass (Ar, standard(E)): a standard value for average or relative atomic mass ⇒ Dimensionless

## Atomic Mass

The atomic mass is the mass of a single atom of an element. Mass of an atom (atomic mass) can be expressed in Kilogram or Gram or a.m.u or u.

The SI unit of mass is the kilogram (kg), but the atomic mass is most often expressed in the non-SI unit a.m.u (atomic mass unit), also known as u (unified mass).

Where 1 a.m.u (or u) is defined as 112 absolute mass of a single Carbon-12 atom at rest. This is the standard unit to calculate the mass of one atom of any element in a.m.u.

1 a.m.u = Absolute mass of one Carbon atom ÷ 12

Absolute mass of one Carbon-12 atom (in kg) = 1.99264687992 × 10−26 kg = 1.99264687992 × 10−23 g. (We will see how to calculate it.)

So, 1 a.m.u = (1.99264687992 × 10−23 g) ÷ (12)
1 a.m.u = 1.66053907 x 10-24 g = (1 Dalton)
⇒ 1 a.m.u. = 1 u = 1 Da (dalton) = 112 of the mass of a single Carbon-12 atom = 1.66053907 x 10-24 g

1.66053907 x 10-24 g known as Atomic Mass Constant (mu)

• When the Mass of an atom is expressed in grams or kilograms, it is called absolute mass of the atom.
• Mass of Carbon-12 atom is 12 a.m.u, which is also the mass number (no. of protons + no. of neutrons) of Carbon-12.
• That means 1 a.m.u is the average mass of one proton or one neutron of Carbon-12.

### Gram to A.M.U. Conversion Factor

We know,
1 a.m.u = 1.66053907 x 10-24 g (Atomic Mass Constant)
So,
1g = (1 a.m.u) ÷ (1.66053907 x 10-24)
= 6.02214075 x 1023 a.m.u.

Now, If we know the mass of an object in Kilogram or Gram then we can easily calculate its mass in a.m.u.

6.02214075 x 1023 known as Avogadro’s Number (NA)
So,
1 a.m.u. = (1g) ÷ 6.02214075 x 1023
1 a.m.u. = (1g) ÷ NA

### Calculate Atomic Masses in A.M.U.

#### Mass of Proton in a.m.u.

The mass of a Proton = 1.67262779 x 10-24 g, its mass in a.m.u.
1g = 6.02214075 x 1023 a.m.u.
1.67262779 x 10-24 g = (1.67262779 x 10-24) x (6.02214075 x 1023 a.m.u.) = 1.00728 a.m.u.

#### Mass of Neutron in a.m.u.

The mass of a Neutron = 1.67493594 x 10-24 g, its mass in a.m.u.
1g = 6.02214075 x 1023 a.m.u.
1.67493594 x 10-24 g = (1.67493594 x 10-24) x (6.02214075 x 1023 a.m.u.) = 1.00867 a.m.u.

#### Mass of Electron in a.m.u.

The mass of an Electron = 9.1 x 10^-28 g, its mass in a.m.u.
1g = 6.02214075 x 1023 a.m.u.
9.1 x 10-28 g = (9.1 x 10-28) x (6.02214075 x 1023 a.m.u.) = 0.00055 a.m.u. = 5.5 x 10-4 a.m.u

### Calculate Atomic Mass of an Element

• The protons and neutrons of the nucleus account for nearly all of the total mass of an atom.
• The electrons make negligible contributions to mass of the atom.
• Thus, the numeric value of the atomic mass when expressed in a.m.u has nearly the same value as the mass number.

#### Atomic Mass of Oxygen (O16)

Oxygen (O-16) has 8 protons, 8 neutrons, and 8 electrons
and above we have calculated the mass one proton, one neutron, and one electron

Therefore,
= (8 x 1.00728 a.m.u) + (8 x 1.00867 a.m.u) + (8 x 0.00055 a.m.u)
= 16.1319886 a.m.u, should be the atomic mass of oxygen (O-16) but

• The mass of nucleus of an atom is less than the sum of masses of all nucleons making it = Mass Defect.
• That is, some energy releases when the nucleons (protons and neutrons) fuse to form a nucleus, thus the mass decreases (E = MC2).
• The amount of energy released during the fusion of protons and neutrons will be the same as the amount of energy required to separate the nucleons from the nucleus = Binding Energy.
• Binding Energy of elements determined experimentally.
• So, Now we have to calculate mass defect of Oxygen-16 Nucleus

You might think
It should be, More binding energy == More stable nucleus but
Iron (56Fe26) has a binding energy = 492 M.e.v (Mega electron volt) and
Uranium (238U92) has a binding energy = 1800 M.e.v

but Iron (56Fe26) is the most stable nucleus, that is because
for iron, the binding energy per nucleon = 8.8 M.e.v while
for Uranium, the binding energy per nucleon = 7.6 M.e.v

More binding energy per nucleon == More stable nucleus

Now, If we first calculate the energy for 1 a.m.u mass, and then if we are given the binding energy of a nucleus, it will be easy to calculate the mass defect for that atomic nucleus.

We know,
1 a.m.u. = 1.66053907 x 10-27 Kg
1 = 1.66053907 x 10-27 Kg/a.m.u.
E = MC2
= (1.66053907 x 10-27 Kg/a.m.u.) x (3 x 108 m/s)2
= (1.66053907 x 10-27) x (3 x 108)2 Joule/a.m.u.
Converting Joule to electron volt (1 J = 1.6 x 1019 e.v.)
= (1.66053907 x 10-27) x (3 x 108)2 x (1.6 x 1019) e.v./a.m.u.
Converting electron volt to Mega electron volt (e.v to M.e.v)
= (1.66053907 x 10-27) x (3 x 108)2 x (1.6 x 1019) x 10-6
≈ 931.5 M.e.v/a.m.u.

1 a.m.u ≈ 931.5 M.e.v

##### Actual Atomic Mass of Oxygen (O-16)

Above we have calculated the atomic mass of oxygen = 16.1319886 a.m.u but during the fusion of protons and neutrons, the nucleus of the oxygen atom releases some energy hence will lose some mass.

Mass defect of Oxygen (O-16)
For Oxygen, the binding energy per nucleon = 8 M.e.v. (Determined experimentally)
Oxygen has 16 nucleons (Protons + Neutrons)
Mass defect = (8 M.e.v x 16) ÷ 931.5 M.e.v/a.m.u. = 0.1374 a.m.u

So the atomic mass of one oxygen atom (O16)
= 16.1319886 a.m.u Mass defect
= 16.1319886 a.m.u 0.1374 a.m.u = 15.9946 a.m.u

And, the Absolute mass of one oxygen atom (O16)
= 15.9946 a.m.u x (Atomic Mass Constant/a.m.u.)
= 15.9946 a.m.u x (1.66053907 x 10-24 gram/a.m.u)
= 2.655965 x 10-23 gram
= 2.655965 x 10-26 Kg

#### Atomic Mass of Carbon (C-12)

Carbon has 6 protons, 6 neutrons, and 6 electrons
= [(6 × 1.00728 amu) + (6 × 1.00867 amu) + (6 × 0.00055 amu)] – Mass defect
= 12.09900 amu – Mass defect

Mass defect of Carbon-12 atom
For Carbon the binding energy per nucleon = 7.68 M.e.v. (Determined experimentally)
Carbon-12 has 12 nucleons (Protons + Neutrons)
Mass defect = (7.68 M.e.v x 12) ÷ 931.5 M.e.v/a.m.u = 0.0989371981 a.m.u ≈ 0.099 a.m.u

= 12.09900 amu – 0.099 amu
= 12 a.m.u.

And, the Absolute mass of one Carbon-12 atom
12 a.m.u × (Atomic Mass Constant/a.m.u.)
12 a.m.u. x (1.66053907 x 10-24 gram/a.m.u.)
= 1.99264688 × 10−23 g

## Relative Isotopic Mass

Relative Isotopic Mass simply means HOW MANY TIMES the absolute mass of one atom (in grams or Kilograms) is heavier than the 112 absolute mass of one carbon-12 atom.

### Relative Isotopic Mass of Carbon-12

To determine the Relative Isotopic Mass of Carbon-12, we divide the mass of one C-12 atom by the 112 mass of C-12 (Atomic Mass Constant).

Relative Isotopic Mass of Carbon-12
= (Mass of C-12 isotope) ÷ (Atomic mass constant)
= (1.99264688 × 10−23 g) ÷ (1.66053907 x 10-24 g)
= 12

So, the Atomic Mass of Carbon-12 is 12 a.m.u but the Relative Isotopic Mass of one Carbon-12 atom is simply 12 with no dimensions.

• Relative Isotopic Mass represents HOW MANY TIMES the absolute mass of one atom of an element is heavier then 112 mass of one Carbon-12 atom.
• Relative Isotopic Mass obtained by dividing the Atomic Mass (ma)of an isotope by the Atomic Mass Constant (mu) yielding a dimensionless value.
• The word “relative” in the term “relative isotopic mass” refers to the scaling relative to 112 mass of one Carbon-12 atom.

### Relative Isotopic Mass of Oxygen-16

Relative Isotopic Mass of Oxygen-16
= (Mass of O-16 isotope) ÷ (Atomic mass constant)
= (2.655965 x 10-23 gram) ÷ (1.66053907 x 10-24 gram)
= 15.9946

The Atomic Mass and the Relative Isotopic Mass refer to a certain specific isotope of an element. Because substances are usually not isotopically pure, it is convenient to use the Standard Elemental Atomic Mass

## Standard Relative Atomic Mass (Ar, standard(E))

The Standard Relative Atomic Mass is the elemental atomic mass which is the average (mean) relative isotopic mass of an element, weighted by the abundance of its isotopes on earth.

On the Periodic Table, the atomic mass of carbon is reported as 12.011. This is the Standard Relative Atomic Mass (Standard Average Relative Atomic Mass or Standard Atomic Weight or Standard Average Atomic Weight) of carbon.

No single carbon atom has a relative mass of 12.011, but a handful of Carbon atoms (including its isotopes: C-13 & C-14) have an average mass of 12.011.

### Calculate Relative Atomic Mass

Ar, standard = [Relative isotopic mass x (% abundance / 100)] + [Relative isotopic mass x (% abundance / 100)] + ……

#### Standard Relative Atomic Mass of Chlorin

isotope 35Cl constitutes 75.77% of the copper on Earth and the rest 24.23% being 37Cl, so
Ar, standard(Cl) = [35Cl relative isotopic mass x (% abundance / 100)] + [37Cl relative isotopic mass x (% abundance / 100)]
Ar, standard(Cl) = [34.96885269 x (75.77 / 100)] + [36.96590258 x (24.23 / 100)]
Ar, standard(Cl) = [34.96885269 x 0.7577] + [36.96590258 x 0.2423]
Ar, standard(Cl) = 35.4527379 ≈ 35.5

Because relative isotopic masses are dimensionless quantities, this weighted mean is also dimensionless. It can be converted into a measure of mass by multiplying it with the atomic mass constant.

## Relative Atomic Mass (Ar)

Relative Atomic Mass or Atomic Weight or Average Atomic Weight or Average Atomic Mass: It is the same as Standard Relative Atomic Mass, this term is used for a single given sample from a particular location.

The relative atomic mass of a given element sample is the weighted arithmetic mean of the masses of the individual atoms (including their isotopes) that are present in the sample.

This quantity can vary substantially between samples because the sample’s origin may have produced unique combinations of isotopic abundances.

Standard Relative Atomic Mass is a standardized value obtained with the various element samples being taken from Earth and is a more common, and more specific quantity.

References:
https://goldbook.iupac.org/terms/view/A00496
https://goldbook.iupac.org/terms/view/A00497
https://goldbook.iupac.org/terms/view/U06554
https://goldbook.iupac.org/terms/view/R05258
https://goldbook.iupac.org/terms/view/BT07001
https://en.wikipedia.org/wiki/Atomic_mass
https://en.wikipedia.org/wiki/Relative_atomic_mass
https://courses.lumenlearning.com/introchem/chapter/nuclear-binding-energy-and-mass-defect/
https://www.westfield.ma.edu/PersonalPages/cmasi/gen_chem1/Atomic%20and%20molar%20mass/atomic_and_molar_mass.htm