How to find the most abundant isotope

To calculate percentage abundance, you must recall the atomic mass of an element is calculated by using the formula:

In the above formula you see fractional abundance. How do you get that? To get fractional abundance, you usually divide the percentage abundance of each isotope by 100. And when you add all the fractional abundance values of all the isotopes, you will notice they all add up to 1. To calculate the percent abundance of each isotope in a sample of an element, chemists usually divide the number of atoms of a particular isotope by the total number of atoms of all isotopes of that element and then multiply the result by 100. Now, let’s apply our understanding to solve the following question:

Silver (Ag) has two stable isotopes: silver-107(107Ag) and silver-109 (109Ag). Silver-107 has a mass of 106.90509 amu and silver-109 has a mass of 108.90476 amu. Calculate the percentage abundance of each isotope.

Strategy

To calculate percentage abundance, we must first know the fractional abundance of each isotope. But from the question, we are not given these values, which means we must think of a way of finding them. One way we can find them is to remember that the:

 fractional abundance of isotope 1 plus the fractional abundance of isotope 2 = 1

We know this because the sum of the percentage abundance values always equal 100. And since the fractional abundance is obtained by dividing the percentage abundance by 100, then, it follows that the sum of the fractional abundance must equal 1. Once we understand this, we can then let X represent the fractional abundance for isotope 1 and (1-X) represent the fractional abundance for isotope 2. Once we substitute these variables into the formula above, we will generate an algebraic equation from which we can solve to find X and then (1-X).  Once we get the values for fractional abundance (X and 1-X), we can then multiply each fractional abundance by 100 to get percentage abundance.

Also, notice the question gave only the isotopic masses and not the atomic mass of silver. How do we get the atomic mass of silver to plug into the above equation? We get the atomic mass of silver by reading its value from the periodic table. And If you do, you will notice the atomic mass of silver is 107.8682. After gathering all the information necessary to solve the question, here is how the setup will appear based on the above information and formula:

Next, we then apply our algebra skills to solve for X. To do this, we will expand the right side of the equation by multiplying X and 1-X by the numbers in front of them. If we do, here is what we will get:

107.8682 amu = 106.90509X amu   +  108.90476 amu  –   108.90476X amu

Next, we bring like terms together by moving 108.90476 amu from the right over to the left side of the equation. Since it’s a positive number on the right, it will become a negative number when it moves across the equal sign to the left. When we move it, here is how the rearranged equation will appear:

107.8682 amu – 108.90476 amu = 106.90509 X amu – 108.90476 X amu

Next, we subtract like terms

-1.03656 amu = -1.99967 X amu

Next, we divide by -1.99967 amu to isolate X. 

-1.03656 amu/-1.99967 amu        = -1.99967 amu  X/-1.99967 amu

X = 0.518

Here is a more clear representation of the division step

Notice that the negative signs and units cancel each other when we divide by -1.99967 amu, hence, fractional abundance has no units

X= 0.518     and 1 – X = 1-.518 = 0.482

Therefore, the fractional abundance of isotope 1 (Silver-107) is 0.518 and isotope 2 (Silver-109) is 0.482.

How to find percentage abundance

To get the percentage abundance, we will simply multiply each fractional abundance by 100. Recall that fractional abundance is calculated by dividing the percentage abundance by 100. Therefore, to get back percentage abundance, we multiply fractional abundance by 100. If we do, the percentage abundance for silver-107 is 0.518 x 100 = 51.8%. And percentage abundance for silver-109 is 0.482 x 100 = 48.2%

To learn how to calculate atomic mass using percentage abundance and isotopic masses click here.

When looking at the periodic table, each element has a value displayed for the atomic mass. If you look closely, it is clear that these values are almost never whole numbers. This is due to isotope abundance. In this tutorial, we will learn what isotope abundance is and how to use it to calculate the atomic weight of an element.

Topics Covered in Other Articles

Vocabulary

• Isotope: when an element has a different form in which it contains the same number of protons, but differs in the number of neutrons.
• Proton: Positively charged subatomic particle located in the nucleus of an atom.
• Neutron: Neutrally charged subatomic particle located in the nucleus of an atom.

What is an isotope?

Isotopes are very similar versions of the same element, only having one difference: the number of neutrons. Though these two versions of the same element differ in the number of neutrons, it is important to note that they do not differ in the number of protons and electrons. In some instances, isotopes can have different reactivity, but in most cases, the defining difference is the number of neutrons.

A common example of an isotope having reactivity that differs from what the element is known for is carbon. Carbon is known to be a very stable element, often being involved in predictable reactions. One isotope of carbon, carbon-14, defies the normal reactivity of the stable element. Carbon-14 is a naturally occurring carbon isotope that radioactively decays. Read more about carbon here.

How does isotope abundance impact atomic weight?

Atomic mass depends on the composition of protons and neutrons in an element, with each weighing 1 atomic mass unit (amu). Electrons are an important part of elements as well, but they have such a small mass that they are considered negligible when calculating atomic mass. As both protons and neutrons make up an atom’s mass, when an element differs in its number of neutrons, it is impactful on the mass.

Though they sound like synonyms, atomic mass and atomic weight are different. Isotopes impact the value of both. Atomic mass is defined as the mass of an individual atom of an element. This is solely the calculation of the weight of protons and neutrons in amu. Atomic weight on the other hand is the weighted average of all of the isotopes of an element that exist. This is where isotope abundance comes in. Though there may be many naturally occurring isotopes of an element, they do not exist in equal amounts. There are many isotopes that occur much more commonly than others, and therefore have a greater impact on the atomic weight. If given the atomic mass of the isotopes of an element as well as their relative abundances, we can follow simple steps to calculate the atomic weight.

Using isotope abundance to calculate atomic weight

As stated previously, the number of isotopes and their percent abundance are all that are needed to calculate the atomic weight of an element. We can start by using magnesium as an example. Magnesium has three naturally occurring isotopes: 24Mg, 25Mg, and 26Mg. Each isotope has an abundance of 78.70 %, 10.13%, and 11.17%, respectively. The atomic mass of each isotope is usually very close to each isotope value. In this example, the mass of each isotope is 23.985 amu, 24.985 amu, and 25.982 amu respectively.

Now that we have all of the information about mass and abundance, we can calculate the atomic weight of magnesium. If you have trouble visualizing all of the values, you can organize them in a table to make your information more clear.

We start by multiplying each isotopes’ mass by its abundance. This can be done in two ways. First, we can directly multiply the mass by the percent:

23.985 amu (78.70)= 1887.6 amu

On the other hand, we can change the percent to a decimal out of one and then multiply by the mass. This can be done by dividing the percent by 100.

23.985 amu (0.7870)= 18.876 amu

With both of these methods, the next step is to repeat for the other isotopes and add the values together.

Method 1:

23.985 amu (78.70) + 24.985 amu (10.13) + 25.982 amu (11.17) =

1887.6 amu + 253.09 amu + 290.21 amu = 2430.90 amu

Method 2:

23.985 amu (0.7870) + 24.985 amu (0.1013) + 25.982 amu (0.1117) = 24.3090 amu

If you follow method one, the final step is to divide the product by 100 to compensate for the percentages being whole numbers.

2430.90 amu/100= 24.3090 amu

Now that we have obtained the same value for both methods, the last step is to make sure your answer has the correct number of significant figures. In this case, the final value is 24.31 amu.

Other Examples

Now that you have seen a general example of how to calculate atomic mass from isotope abundances, we can understand other problems involving isotope abundance.

Calulating abundance frorm atomic mass and atomic weight

Lithium has two isotopes, 6Li and 7Li, with masses of 6 amu and 7 amu respectively. If Lithium has an atomic weight of 6.94 amu, we can determine the abundance of each isotope. First, we define one. of the abundances as x. In this case, the abundance of 6Li will be x. This means that the abundance of 7Li= 1-x. Using what we learned above, we can set up an equation.

6 amu (x) + 7 amu (1-x)= 6.94 amu

There is only one variable, so we can easily solve for x.

6x + 7 – 7x= 6.94

6x-7x= -0.06

-x = -0.06

x= 0.06

Now that we have found the abundance of 6Li, we can use 1-x to find 7Li.

1-0.06= 0.94

So we have determined that the abundances of 6Li and 7Li are 6 % and 94 % respectively.

Further Reading

• How to find Molar Mass
• How to Calculate Molarity
• Balancing Chemical Equations