1 u = 1.66 × 10-27 kg let x = %abundance of isotope-a Please do not block ads on this website. Abundance of Naturally Occurring IsotopesMost elements occur in nature as a mixture of different isotopes. The element carbon, for example, exists in nature as a mixture of different isotopes: stable1 carbon-12 atoms and carbon-13 atoms. If you were to take a sample of carbon atoms, for example the soot from a chimney or a lump of coal, you would find that most of the carbon atoms are the carbon-12 isotope and only a few would be the carbon-13 isotope. We call this amount of each isotope found in the naturally occurring element its abundance, or its isotopic abundance to be more precise. The abundance of the carbon-12 isotope in naturally occurring bulk carbon is 98.90% while the abundance of the carbon-13 isotope in nature is 1.10% This means that if you take a lump of coal from nature, 98.90% of the carbon in the coal will be atoms of the carbon-12 isotope, while only 1.10% of it will be atoms of the carbon-13 isotope. The table below gives the isotopic abundances for some elements on Earth:
You can find a more complete list of isotopic abundances at the bottom of this page.
Do you know this? Join AUS-e-TUTE! Play the game now! The relative atomic mass of a carbon-12 atom is defined as 12.00 We can estimate the mass of any isotope of an element, its isotopic mass, using its mass number (A). The mass number (A) of an isotope tells us how many protons and neutrons are in the nucleus of an atom of this isotope. Nucleon is the term used to describe both protons and neutrons. So, the mass number (A) tells us the number of nucleons in the nucleus of an atom of the isotope of the element. For example:
The mass of a proton is almost exactly the same as the mass of a neutron. The mass of a proton is about 1 u (1 atomic mass unit), so the mass of a neutron is also about 1 u. The mass of an electron is so small compared to the mass of a proton or neutron that it can be ignored when estimating the mass of an isotope of an element.2 We can estimate the mass of an atom of an isotope of an element by adding together the mass of its nucleons:3 isotopic mass = number of nucleons × mass of nucleon For example:
Isotopic masses can be measured using mass spectroscopy. You will find a discussion of calculating relative atomic mass (atomic weight) using these measured isotopic masses in the Mass Spectroscopy tutorial.
Do you understand this? Join AUS-e-TUTE! Take the test now! The relative atomic mass of an element is the weighted average of the masses of the isotopes in the naturally occurring element relative to the mass of an atom of the carbon-12 isotope which is taken to be exactly 12. What is a "weighted average" ? First lets look at what the "average weight" of carbon would be: mass of carbon-12 isotope is 12 u mass of carbon-13 isotope is 13 u So we can calculate the average mass (average weight) of carbon as:
If we look up the atomic weight of carbon in the Periodic Table we find that it is 12.01 NOT 12.5 This is because most of the atoms found in naturally occurring carbon are atoms of the carbon-12 isotope while very few of the atoms will be of the carbon-13 isotope. We need to know the abundance of each isotope, that is we need to know how much of the mass (weight) of the naturally occuring bulk carbon is due to each of the isotopes (carbon-12 and carbon-13). This means that if I had 100 atoms of bulk carbon (like in coal or soot), then:
So, the total mass of 100 naturally occurring carbon atoms is:
So the mass of 100 naturally occurring carbon atoms is 1201.1 u Therefore the "weighted average mass" of 1 carbon atom is 1201.1 ÷ 100 = 12.011 u This value for the atomic weight of carbon agrees with the value in the Periodic Table. Note that this does NOT mean that the mass of 1 atom of carbon is 12.011 u Let's review how we calculated the "weighted average" mass of bulk carbon atoms:
In general, to calculate the "weighted average" mass of an element that occurs naturally as two different isotopes, isotope 1 and isotope 2, then:
If we estimate the mass of each isotope by using its mass number (A), then we can re-write the expression as:
Or, put another way:
Can you apply this? Join AUS-e-TUTE! Take the exam now! The Periodic Table gives us the weighted average for the mass of an element, referred to as the element's atomic weight (or relative atomic mass). If we know the mass of each isotope making up this naturally occurring element (estimated by its mass number), then we can calculate the abundance of each isotope in nature. We wrote a general mathematical expression (mathematical equation) above for calculating the "weighted average" mass, also known as the relative atomic mass or atomic weight, of an element, which was:
Let's say I wanted to find the abundance (%) of each isotope of nitrogen. Nitrogen has two naturally occurring stable isotopes: nitrogen-14 and nitrogen-15. So, substituting these into the mathematical equation above I get:
I can look up the atomic weight (relative atomic mass, or, "weighted average" mass) of nitrogen in the Periodic Table: atomic weight of nitrogen is 14.01 I can estimate the mass of an atom of each isotope by using its mass number (A): The mass of an atom of nitrogen-14 = its mass number = 14 u The mass of an atom of nitrogen-15 = its mass number = 15 u I can substitute these values into the mathematical equation above:
I can multiply both sides of the mathematical equation by 100:
But how can I solve this equation when there are 2 unknowns, the abundance of nitrogen-14 is unknown and the abundance of nitrogen-15 is unknown. The trick is to remember that we are talking about percentage abundance! Which means that: %abundance of nitrogen-14 + %abundance of nitrogen-15 = 100 Or, put a different way: %abundance of nitrogen-14 = 100 - %abundance of nitrogen-15 So, if I let the %abundance of nitrogen-14 be equal to x then: %abundance of nitrogen-14 = x %abundance of nitrogen-15 = 100 - x If I substitute these into the mathematical equation above I will have only 1 unknown value, x:
To solve for x I will clear the brackets first:
Next I collect like terms, starting with x:
Then subtract 1500 from both sides of the equation:
Note that the abundance of an isotope must be a positive number (not a negative number), so I divide both sides of the equation by -1 to find the value of x : Then substitute this value for x back into the expressions we wrote for the abundance of each isotope: %abundance of nitrogen-14 = x = 99 % %abundance of nitrogen-15 = 100 - x = 100 - 99 = 1% In general, if an element occurs in nature in 2 isotopic forms, isotope 1 and isotope 2, then we can estimate the percentage abundance of each isotope using the mass number (A) of each isotope because the %abundance of isotope 2 equals 100 - %abundance of isotope 1:
let abundance isotope 1 = x
So,
Can you apply this? Join AUS-e-TUTE! Take the exam now! Question: Naturally occurring silver is 51.84% silver-107 and 48.16% silver-109. Solution: (Based on the StoPGoPS approach to problem solving.)
Worked Example 2: Calculating an Element's Isotopic AbundanceQuestion: Copper consists of two isotopes, copper-63 and copper-65. Solution: (Based on the StoPGoPS approach to problem solving.)
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Footnotes 1. A stable isotope is one that does not undergo radioactive decay (nuclear decay). An unstable isotope is one that does undergo radioactive decay, and therefore, the abundance of a naturally occurring unstable isotope will decrease over time .... Isotopic abundances can change over time, if a radioactive isotope decays to produce a stable isotope of a different element then the isotopic abundance of this stable isotope will increase over time. Man-made nuclear reactions will also change the isotopic abundance. There is a discussion of the variation in isotopic abundance in the Carbon-14 Dating tutorial. 2. Mass of a proton = 1.673 × 10-27 kg = 1.01 u 3. In effect we are estimating the mass of the nucleus rather than the atom (since we are ignoring the mass of the electrons which contribute very little to the mass of an atom). Isotopic Abundance Data
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