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# Average Atomic Mass

Atoms are the basic building blocks of every materials or substances. It is very small in size and are not visible in naked eyes. Can we measure the size and mass of an atom? Yes we can. The  mass of the atom is termed as Atomic Mass. We know that most of this mass of the atom is concentrated in the nucleus as the electrons have negligible mass. The Nucleus is made up of protons and neutrons.Thus we can say the atomic mass is the mass of protons and neutrons present in the nucleus.

Consider a Carbon atom it contains 6 protons and 6 neutrons. Hence it has the atomic mass as 12.
Taking this element as standard we express the atomic mass of any in amu.
Where, 1 amu = $\frac{1}{12}$ th the mass of carbon atom. The atomic mass unit is represented as 'u'.

1 u = $\frac{Mass of ^{12}C_{6} atom}{12}$

The Average Atomic Mass is the average of all the atoms present in the element. Let us study the concepts lying behind the average atomic mass in detail.

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## What is Average Atomic Mass?

Atomic mass is the total matter present in one atom of an element. All atoms consist of three subatomic particles namely :
1. Protons
2. Electrons
3. Neutrons
Generally, the atomic mass of an element consists of protons and neutrons only, since electrons have negligible mass.
Thus, the Atomic Mass of an atom is defined as the total number of protons and neutrons present in the nucleus of that atom. The atomic mass is represented by the symbol A and is measured in amu (atomic mass unit).An atomic mass unit (amu) is equal to the mass of 1/12th the mass of a carbon atom of the isotope C-12.

The Average Atomic Mass is the average of the atomic mass of all the isotopes of the element. There are several elements which has number of naturally isotopes.

Difference between Atomic Mass and Average Atomic Mass:

 Atomic mass Average atomic mass It is the total mass of neutrons, Protons and electrons. It is average mixture of the isotopes of the element. It is obtained directly from the periodic table. It is the obtained by calculating percent abundance of each isotope and their atomic mass. It is an individual mass of the measure of each isotope. Average of the atomic mass when all the naturally occurring isotopes of the elements are mixed in a proportion.

## Average Atomic Mass Definition

The Average Atomic Mass is the weighted average of each isotope of an element.
Lets take an example of an element 'Y' which has isotopes Y1, Y2......Yn then its average atomic mass is given by :
Average Atomic Mass (Y) = (Atomic Mass of Y1) $\times$ $\frac{Y_{1}}{100}$ + (Atomic Mass Y2) $\frac{Y_{2}}{100}$ +.......+ (Atomic mass Yn) $\frac{Y_{n}}{100}$.It is expressed in Atomic Mass Unit (amu).

## Average Atomic Mass Formula

The Average atomic mass is the weighted sum of all the isotopes of the element. The Average atomic mass is given by formula:
Average Atomic Mass (Y) = (Atomic Mass Y1) $\times$ $\frac{Y_{1}}{100}$ + (Atomic Mass Y2) $\frac{Y_{2}}{100}$ +.........Where,
Y1 % = Percent Abundance of Y1
Y2 % = Percent Abundance of Y2.

For finding the average atomic mass of an element it is necessary to determine the number of the naturally occurring isotopes of that element and also their percent abundance in the nature.

## Percent Abundance

The Percent Abundance is the percentage amount of all the natural occurring isotopes of the element.
It is used to find the average atomic mass of the element.

As we know that the isotopes of each element occurs in different ratios and hence the percent abundance signifies the percentage of finding the isotope of the element while mining that element, since each element may contain a mixture of the isotopes.

## How to Calculate Average Atomic Mass?

Average Atomic Mass is given by the formula:
Average atomic mass= $\sum$ (Mass of isotope $\times$ Percentage isotope abundance)If the element Y has Y1, Y2, Y3 ......... Yn number of elements then for Calculating Average Atomic Mass, the percent abundance of each isotope and atomic mass of each isotope is obtained and then by using the following formula:
Average Atomic Mass (Y) = (Atomic mass Y1) $\frac{Y_{1}}{100}$ + (Atomic mass Y2) $\times$ $\frac{Y_{2}}{100}$ +....… (Atomic mass Yn) $\frac{Y_{n}}{100}$.Where
$\frac{Y_{1}}{100}$= Percent Abundance of Y1
$\frac{Y_{2}}{100}$ = Percent Abundance of Y2.

Now we will solve some problems of finding the average atomic mass to understand it better.

### Solved Examples

Question 1: There are two isotopes of Rubidium, 85Rb and 87Rb. The abundance of 85Rb is calculated to be 72.2 % and the abundance of 87Rb is calculated to be 27.8 %, find the average atomic mass of rubidium?
Solution:

By using the following formula we can find the average atomic mass of rubidium,

Average atomic mass= $\sum$ (Mass of Isotope $\times$ Percent abundance)

Average atomic mass = 85 $\times$ $\frac{72.2}{100}$ + 87 $\times$ $\frac{27.8}{100}$.

Average atomic mass = 61.37 amu + 24.186 amu.

Average atomic mass = 85.556 amu

The Average atomic mass of the rubidium is 85.56 amu.

Question 2: Find the average atomic mass of the titanium which has five naturally occurring isotopes:
1. 46 Ti (8.0%)
2. 47 Ti (7.8%)
3. 48 Ti (73.4%)
4. 49 Ti (5.5%)
5. 50 Ti (5.3%).

Solution:

The Average atomic mass of titanium is obtained by using the following formula:

Average atomic mass= $\sum$ (mass of isotope $\times$ Percent abundance).

Average atomic mass=46 $\times$ $\frac{8}{100}$ + 47 $\times$ $\frac{7.8}{100}$ + 48 $\times$ $\frac{73.4}{100}$ + 49 $\times$ $\frac{5.5}{100}$ + 50 $\times$ $\frac{5.3}{100}$.

Average atomic mass = (3.68 + 3.666 + 35.232 + 2.695 + 2.65) amu.

Average atomic mass = 47.932 amu.

The Average atomic mass of the titanium is 47.93 amu.

Question 3: Calculate the average atomic mass of the chlorine which has two naturally occurring isotopes. The Percent abundance of 35 Cl (mass = 34.969 amu) is 75.53% and the percent abundance of 37 Cl (mass = 36.966 amu) is 24.47%.
Solution:

The Average atomic mass of chlorine is obtained by using the following formula:

Average atomic mass= $\sum$ (Mass of isotope $\times$ Percent abundance)

Average atomic mass=34.969 $\times$  $\frac{75.53}{100}$ + 36.966 $\times$ $\frac{24.47}{100}$.

Average atomic mass = 26.412 amu + 9.045 amu

Average atomic mass = 35.457 amu.

The Average atomic mass of the chlorine is 35.46 amu.

Question 4: The Magnesium has three naturally occurring isotopes 24Mg, 25Mg and 26Mg. The percent abundance of 24Mg is 78.7%, 25Mg is 10.13% and 26Mg is 11.7%. If the average atomic mass of magnesium is 24.3050 and the atomic mass of 25Mg is 24.98584 and 26Mg is 25.98259. Find the atomic mass of 24Mg?
Solution:

The atomic mass of 24Mg is obtained by using the following formula:

Average atomic mass = $\sum$ (Mass of isotope $\times$ Percent abundance) 24.3050
= X $\frac{78.7}{100}$ + 24.98584 $\times$ $\frac{10.13}{100}$ + 25.98259 $\times$ $\frac{11.7}{100}$.

X = 23.98504 amu.

The atomic mass of 24Mg is 23.98504 amu.

Question 5: Find the Average atomic mass of uranium. Uranium has three common isotopes. The abundance of 234 U is 0.01%, the abundance of 235 U is 0.71% and the abundance of 238U is 99.28%?
Solution:

The Average atomic mass of uranium is obtained by using the following formula:

Average atomic mass = $\sum$ mass of isotope $\times$ Percent abundance.

Average atomic mass = 234 $\times$ $\frac{0.01}{100}$ + 235 $\times$ $\frac{0.71}{100}$ + 238 $\times$ $\frac{99.28}{100}$.

Average atomic mass = 0.0234 amu + 1.6685 amu + 236.2864 amu.

Average atomic mass = 237.9783 amu.

The Average atomic mass of the uranium is 237.9783 amu.

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