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# Isotopic Mass

One of the most common physical properties of matter is mass. Atomic mass can be defined as the mass of an atom or a molecule. It is a quantity which can be used to determine the average mass of elements and molecules. Atomic mass also involves in stoichiometry problems. An atom is composed of electrons, protons and neutrons. The total number of protons in an atom is called as atomic number. The sum of number of neutrons and protons is known as mass number. In an atom, the number of protons is always equal to the total number of electrons which makes it neutral due to equal and opposite charges of electrons and protons. The atomic mass is expressed in unified atomic mass units (u). Some of the atoms contain same number of protons but different mass number due to different number of neutrons. They are called as isotopes. For example; three isotopes of hydrogen are; hydrogen (H), deuterium (D) and tritum (T).

They have same atomic number but different mass number; 1, 2 and 3 respectively. Isotopes are found in different percentage in nature. Some of the isotopes are found in abundance whereas some of them are radioactive and decay continually in nature. For example; C-12 is the most abundant isotope of carbon whereas C-14 is a radioactive isotope of it with half life of 5500 years. Some of the isotopes are very useful and widely used in various fields like medical and chemical industries. Isotopes have same chemical properties as they have same number of electrons and their arrangement in shell. They have different number of neutrons which effects their mass and mass number. Their physical properties are different and also depend on their masses.

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## Isotopic Mass Definition

On macroscopic level most mass measurements of pure substances gives the mass of a mixture of isotopes. In other words we can say that mixtures are not pure but the mixture of all known mixture such as the macroscopic mass of oxygen molecule does not correspond to the microscopic mass. The macroscopic mass implies a certain isotopic distribution while microscopic refers to the mass most common isotope of oxygen that is O-16. Remember the macroscopic mass is also called as molecular weight or atomic weight. The isotopic mass is referred as mass of most abundant isotope of an element.

## How to Find Isotopic Mass?

We know that isotopes are atoms with the same atomic number and different mass numbers due to different number of neutrons. The mass number of an element is a whole number whereas actual mass of an atom is not a whole numbers except for carbon-12. For example the atomic mass of Lithium is 6.941 Da. On the basis of abundance of isotopes, we can calculate the isotopic mass and average atomic mass of element.  The average mass of the element E can be expressed as:

m(E) = $\sum_{n=1}m(I_{n}) \times p(I_{n})$.

For example the mass and abundance of isotopes of Boron is given below.

 N isotope $I_{n}$ mass m (Da) isotopic abundance p 1 10B 10.013 0.199 2 11B 11.009 0.801

The average mass of Boron can be calculated as:
m(B)=(10.013)(0.199)+(11.009)(0.801)
=1.99 +8.82
= 10.81 Da

## Relative Isotopic Masses

It’s difficult to express the mass of an element, relative mass is one of the best methods to express the mass of known element. It can be defined as ‘Ar’;
Ar = $\frac{m}{mu}$.

The relative isotopic mass is a unit less quantity with respect to some standard mass quantity. The relative atomic mass can be taken as the weighted mean mass of an atom of an element compared to the mass of 1/12 of the mass of an atom in C-12. Similarly relative isotopic mass referred as the mass of an atom of an isotope with respect to mass of 1/12 of the mass of an atom in C-12. The isotopic abundances are used to calculate the average atomic weight and the isotopic weights.

## Average Isotopic Mass

The percentage abundance and isotopic mass is used to calculate the average isotopic mass. For example two isotopes of Nitrogen are N-14 and N-15 and average isotopic mass of Nitrogen is 14.007. The percentage abundance of both isotopes can be calculated as given below.

(14.003074) (x) + (15.000108) (1 - x) = 14.007
(14) (x) + (15) (1 - x) = 14.007
x = 15 - 14.007 = 0.993
1 - x = 0.007

So percentage abundance of N-14 would be 99.3 % and for N-15 it would be 0.7 %. Similarly of we have the average isotopic mass of copper is 63.546 and the atomic mass of Cu-63 is 62.929 amu and Cu-65 is 64.927 amu, the percentage abundance would be;

(62.9296) (x) + (64.9278) (1 - x) = 63.546
x = 0.6915

Hence the percentage abundance of Cu-63 would be 69.15 % and rest of would be Cu-65.