Isotopes and atomic weight. How isotopes can be detected using mass spectrometry. 

Key points:

  • Atoms that have the same number of protons but different numbers of neutrons are known as isotopes.
  • Isotopes have different atomic masses.
  • The relative abundance of an isotope is the fraction of a single element that exists on Earth with a specific atomic mass.
  • Atomic weights are weighted averages calculated by multiplying the relative abundance of each isotope by its atomic mass and then summing up all the products.
  • The relative abundances of each isotope can be determined using mass spectrometry.
  • A mass spectrometer ionizes atoms and molecules with a high-energy electron beam and then deflects the ions through a magnetic field based on the mass-to-charge ratio of the ion, mz\dfrac{m}{z}.
  • The mass spectrum of a sample shows relative abundance of each ion on the y-axis and mz\dfrac{m}{z} along the x-axis.

Introduction: Dissecting an atom

Everything is made out of atoms. Your computer or phone screen, the chair you are sitting on, you, and I are all made out of atoms. If we were to zoom in further using our very technologically advanced and imaginary chemistry goggles, we'd be able to see that the atom itself is made out of subatomic particles with specific properties.
An atom is composed of protons, neutrons, and electrons. A proton carries a 11++ charge, an electron carries a 11- charge, and a neutron carries 00 charge. The nucleus is made up of protons and neutrons, and electrons are localized outside the nucleus. Since electrons have a negative charge, they are attracted to the positively charged protons in the nucleus. This gives an atom its stability, and we can think about this mathematically using Coulomb's law. On a very simplistic level, we can represent an atom using Figure 1:
A Bohr model of the helium atom.
Figure 1: a neutral helium atom. The protons and neutrons are located in the nucleus, while the electrons "orbit" outside the nucleus. Image from Wikispaces, CC BY-SA 3.0.
We count two protons in the nucleus of this atom, which means that our atom has an atomic number (Z\text Z) of 22. Since the identity of an element is determined by the atomic number, we can deduce that our atom must be an atom of helium. In order for the atom to be neutral, we have 22 electrons to balance out the positive charge from the protons. But what about the neutrons, which have no charge? This particular atom of helium happens to have two neutrons. Does that mean all helium atoms must have two neutrons?
Excellent question! Since like charges repel, the protons in the nucleus feel electrostatic repulsion from each other. This is balanced by the nuclear strong force, which is what holds the protons and the neutrons together in the nucleus. In order for the nucleus to be stable, the nuclear strong force must be greater than the electrostatic repulsion. It turns out that this depends on the ratio of the number of neutrons, N\text N, to the atomic number, Z\text Z.
If you would like to learn more about this, check out this video on the mass defect and binding energy and the follow-up video on nuclear stability and nuclear equations!
We know that if the nucleus had a different atomic number, then the atom would be a different element. However, the same is not true of the number of neutrons in the nucleus. It is possible that atoms of the same element may contain different numbers of neutrons; such atoms are known as isotopes. This comes from Greek: iso- meaning "same," and -tope meaning "place." Thus, isotopes—because they contain the same number of protons—occupy the same place on the periodic table. They differ, however, in the number of neutrons in their nuclei, which causes them to have different masses.

Particle masses and unified atomic mass units

Atoms are extremely tiny, and the particles within atoms are even tinier. While we can talk about the masses of atoms and particles in terms of everyday units such as grams and kilograms, it is much more convenient to use a very tiny unit of mass to discuss such very tiny things. That unit is known as u\text{u}, which stands for unified atomic mass unit. By definition, 1 u1\text{ u} is equal to exactly 112\dfrac{1}{12} of the mass of a single neutral atom of carbon-12. The number after the hyphen, 12, is the sum of the protons and neutrons for that specific isotope of the element. The reason carbon-12 was chosen as the isotope to define the unit u\text{u} is because it is the most common isotope of carbon.
Historically, the units u\text{u} and amu\text{amu} were defined slightly differently. Today, however, they are often used interchangeably; depending upon your teacher and your textbook, you might come across amu\text{amu} (atomic mass unit) rather than u\text{u} (unified atomic mass unit). You should consider them to be one and the same.
Concept check: How many protons are in the nucleus of an atom of carbon-12?
If you look on the periodic table, you'll see that carbon has an atomic number of 6; thus, carbon-12, like any isotope of carbon, contains six protons in its nucleus.
As we will now examine in more detail, the bulk of an atom's mass is located in its nucleus. This is because protons and neutrons are much more massive than electrons. For example, a proton has a mass of 1.673×1027 kg1.673\times10^{-27}\text{ kg}, or 1.007 u1.007\text{ u}. A neutron is slightly heavier, with a mass of 1.675×1027 kg1.675\times10^{-27}\text{ kg}, or 1.009 u1.009\text{ u}. An electron, on the other hand, has a mass of only 9.109×1031 kg9.109\times10^{-31}\text{ kg}, or 5.486×104 u5.486\times10^{-4}\text{ u}. We can summarize this information in the following table:
NameChargeSymbolMass (kg)(\text{kg})Mass (u)(\text{u})Location
proton11++11p+_1^1\text{p}^+1.673×10271.673\times10^{-27}11Inside nucleus
neutron001n^1\text{n}1.675×10271.675\times10^{-27}11Inside nucleus
electron11-e\text{e}^-9.109×10319.109\times10^{-31}00Outside nucleus
There are a few things that we should conclude from this table. The first is that protons and neutrons have masses that are about 2,000 times greater than the mass of an electron. As such, electrons are considered to have a negligible effect upon the overall mass of an atom. This is a fancy way of saying that when we calculate the mass of atoms and molecules, we ignore the mass of electrons. This fact is illustrated further in the column listing the masses of these particles in u\text{u}. The masses have been rounded to the nearest integer; thus, protons and neutrons are considered to have identical masses of 1 u\text{ u}. While we know that neutrons are ever-so-slightly heavier than protons, this very small difference in mass is insignificant for many purposes, and we can simplify things greatly by assuming that protons and neutrons have equal mass.
We should also note that electrons are considered to have a mass of ~ 0 u\text{ u}. Again, this is not technically true. Keep in mind, however, that the masses of the electrons are so small in comparison to the masses of the protons and neutrons in the nucleus that we can simply ignore the electrons' masses altogether.

Mass number and isotope notation

Now that we have an understanding of the different masses and charges of protons, neutrons, and electrons, we can discuss the concept of mass number. By definition, the mass number is simply equal to the number of protons plus the number of neutrons in the nucleus.
Just as the atomic number defines an element, we can think of the mass number as defining the specific isotope of a particular element. In fact, a common way of specifying an isotope is to use the notation "element name-mass number" such as in carbon-12, which is a carbon atom with a mass number of 12. Using that information, we can calculate the number of neutrons in an atom of carbon-12 as follows.
We can rearrange the equation for mass number to solve for the number of neutrons:
Therefore, we see that an atom of carbon-12 has 6 neutrons in the nucleus. Let's try another example.
Concept check: Chromium-52 is the most stable isotope of chromium. How many neutrons are contained in a single atom of chromium-52?
From its name, we know that the mass number of chromium-52 is 52. The mass number is equal to the number of protons plus the number of neutrons. From the periodic table, we can see that chromium has an atomic number of 24, which means its nucleus contains 24 protons. To determine the number of neutrons, we simply subtract the number of protons, 24, from the mass number, 52:
Therefore, the chromium-52 isotope contains 28 neutrons in its nucleus.
We have now seen that isotopes are defined by their mass number, which is equal to the sum of the number of protons and neutrons. To show this most simply, chemists commonly use the following notation to indicate atomic number, mass number, and charge—all in one symbol:
Isotope notation for helium-4 and magnesium-24.
Figure 2: isotopic notation for neutral hydrogen-3 (left) and the magnesium-24 cation with a 2+2+ charge (right).
In Figure 2, we have the isotopic notation for neutral hydrogen-3 and the magnesium-24 cation. In the center of each is the chemical symbol for each element. On the lower left is the atomic number, which corresponds to the number of protons in the atom's nucleus. Above the atomic number is the mass number, which is equal to the sum of the number of protons and the number of neutrons in the atom. To the upper right of the chemical symbol is the net charge on the species, if any. Keep in mind that for neutral atoms, the net charge is zero, and nothing will be written in this space.

Atomic mass vs. mass number

Atomic mass is a concept that is very closely related to mass number. The atomic mass is the mass of a specific isotope of an element expressed in units of u\,\text{u} or amu\text{amu}. Since the mass of a neutron and the mass of a proton are both very, very close to 1 amu\text{amu}, the atomic mass of an isotope is often very close to the mass number. However, they are different! The mass number is an integer, since we always count whole numbers of protons and neutrons (that is, you can't have 1.05 protons, or 0.27 neutrons). The mass number is also, usually, written as being unitless. In comparison, the atomic mass is only a whole number if you round to the nearest integer, and atomic mass has units of mass (u\text{u}).
Another term that students often might find confusingly similar to atomic mass and mass number is atomic weight, which is also a related but different term. Don't worry, though, we will discuss atomic weight in the following section!

Relative abundance and atomic weight

There are two stable isotopes of chlorine: chlorine-35, and and chlorine-37.
Chlorine-36 can be found in trace amounts, relative to chlorine-35 and chlorine-37, but it is radioactive and therefore unstable. The half-life of chlorine-36 is about 301,000 years, which makes it useful for radioactive dating of geological events.
And yet, if you look on the periodic table, you'll find that the mass of an atom of chlorine is given as 35.45 u\text{ u}. Where do the numbers after the decimal come from? If you guessed that this number might be the mass of an average atom of chlorine, you would be correct. In fact, all of the masses that you see on the periodic table are averages that are based on the masses and abundances of all the stable isotopes of each element. These average masses are referred to as atomic weights. In comparison, atomic mass refers to the mass of a specific isotope. We use atomic masses to calculate the atomic weight of a given element.
Let's now further consider the atomic weight of chlorine. As we said before, chlorine has two stable isotopes: chlorine-35 and chlorine-37. The atomic weight of chlorine given on the periodic table is 35.45 u\text{ u}. This begs the question, why isn't the atomic weight of chlorine simply the average of 35 and 37, which would be 36u\,\text u?
The answer has to do with the fact that different isotopes have different relative abundances, meaning that some isotopes are more naturally abundant on Earth than others. In the case of chlorine, chlorine-35 has a relative abundance of 75.76%, whereas chlorine-37 has a relative abundance of 24.24%. Note that relative abundances are percentages, and thus the relative abundances of all the different stable isotopes of an element will add up to 100%. The atomic weight that you find on the periodic table is actually a weighted average calculated from these values. To better illustrate this, let's calculate the atomic weight of chlorine.
Chemists almost always use the atomic weight from the periodic table to calculate molecular weight, which is also sometimes called the relative molecular mass.
If you needed to know the different masses resulting from specific isotopes in the molecule, you would calculate the mass of the molecule using the atomic masses of those isotopes. The resulting mass would be called the exact mass of the molecule. You might need to calculate exact masses when extreme precision is required, such as when doing experiments with isotopically labeled chemicals.

Example: Calculating the atomic weight of chlorine

When we want to calculate a weighted average, we multiply the value of every item in our set—in this case, the atomic mass of each isotope of chlorine—by its relative abundance expressed as a fraction, and then sum up all of our products. This can be written as follows:
atomic weight=i=1n(relative abundance×atomic mass)i\text{atomic weight}=\sum\limits_{i=1}^n(\text{relative abundance}\times\text{atomic mass})_i
If we apply this formula for chlorine, we get the following:
atomic weight of chlorine=(0.7576×34.969 u)+(0.2424×36.966 u)=26.49u+8.960u=35.45 u\begin{aligned}\text{atomic weight of chlorine} &= (0.7576\times34.969\text{ u})+(0.2424\times36.966\text{ u}) \\ \\ &= 26.49 \,\text u+8.960\,\text u \\ \\ &= 35.45\text { u} \end{aligned}
We can see that because chlorine-35 is about 3 times more abundant than chlorine-37, the weighted average is closer to 35 than to 37.
Concept check: Bromine has two stable isotopes—bromine-79 and bromine-81. The relative abundance of each isotope is 50.70% and 49.30%, respectively. Is the atomic weight of bromine closest to 79, 80, or 81 u\text{ u}?
We can see that the two isotopes of bromine have relative abundances that are just about 50/50, or 1:1. Therefore, the weighted average will be almost exactly in between the atomic mass of each isotope, and the answer is 80. (The actual atomic weight of bromine is 79.90 u\text{ u}.)

Mass spectrometry

We now know how to find atomic weights by calculating weighted averages using relative abundances. But, where do these relative abundances come from? For example, how do we know that 75.76% of all chlorine on Earth is chlorine-35?
The answer is that these relative abundances can be determined experimentally using a technique called mass spectrometry.
A diagram of a mass spectrometer.
Figure 3: A diagram of a mass spectrometer. A sample is injected into the machine, vaporized by a heater, and then ionized by a stream of high-energy electrons. The resulting Ions are accelerated through parallel electric plates and then deflected in a magnetic field before they reach a detector. Image from Openstax, CC BY 4.0.
In a mass spectrometer, a sample containing the atoms or molecules of interest is injected into the instrument. The sample—typically in an aqueous or organic solution—is immediately vaporized by a heater, and the vaporized sample is then bombarded by high-energy electrons. These electrons are powerful enough to knock electrons off atoms in the sample, which creates cationic versions of the sample. These cations are then accelerated through electric plates and subsequently deflected by a magnetic field.
Once the ions reach the magnetic field, they are deflected different amounts depending on their speed and charge. Ions that are moving more slowly, the heavier ions, are deflected less, and ions that are moving more quickly, the lighter ions, are deflected more. Think of the force you need to apply to accelerate a bowling ball versus the force necessary to accelerate a tennis ball. It's much easier to accelerate the tennis ball! Also, the higher the charge on the ion, the more it will be deflected.
The amount that the ions are deflected is inversely proportional to their mass-to-charge ratio, mz\dfrac{m}{z}, where mm is equal to the mass of the ion and zz is equal to the charge. The detector records the mz\dfrac{m}{z} values for each ion, as well how many of each ion it sees. The relative abundance for a specific ion within the sample can be calculated by dividing by the number of ions of that type by the total number of ions detected. The instrument then generates a mass spectrum for the sample, which plots relative abundance against the mass-to-charge ratio, mz\dfrac{m}{z}.
Concept check: A sample of copper is injected into a mass spectrometer. After the sample is vaporized and ionized, the ions 63Cu2+^{63}\text{Cu}^{2+} and 65Cu2+^{65}\text{Cu}^{2+} are detected. Which ion is deflected more inside the spectrometer?
The ion 63Cu2+^{63}\text{Cu}^{2+} has a lower mass, mm, with the same charge, zz, compared to 65Cu2+^{65}\text{Cu}^{2+}. This means that 63Cu2+^{63}\text{Cu}^{2+} has a lower mass-to-charge ratio, mz\dfrac{m}{z}. Since the amount of deflection is inversely proportional to the mass-to-charge ratio, the ion with the lower mz\dfrac{m}{z} is deflected more.
Therefore, 63Cu2+^{63}\text{Cu}^{2+} will be deflected more by the magnetic field in the spectrometer.

Analyzing the mass spectrum of zirconium

Let's suppose we analyzed a sample of elemental zirconium, atomic number 40, using mass spectrometry. After putting the sample through the instrument, we would get a mass spectrum that looks like Figure 4, where the height of a peak is roughly proportional to the relative abundance for a given value of mz\dfrac{m}{z}:
Concept check: Based on this spectrum, what is the most common isotope of zirconium in our sample?
The simulated mass spectrum for zirconium.
Figure 4: The simulated mass spectrum for a sample of elemental zirconium. Image from Openstax, CC BY 4.0.
The height of a peak in the mass spectrum is proportional to the relative abundance. Zr-90 is the most common isotope in this sample since it has the most intense, or tallest, peak.
Based on the relative height or intensity of the peaks at a given mass-to-charge ratio, we can find the relative abundances of the isotopes. The peaks are labeled in the simulated mass spectrum assuming a charge of +1+1 for the ions pictured, which allows us to calculate the atomic mass of each isotope. Using this information, we can calculate the atomic weight of zirconium by finding a weighted average of the atomic masses for each isotope. To try this calculation yourself, see the practice problem below!
Nowadays, we already know the atomic weights for all the most common elements, and so it is not often necessary to analyze individual elements using mass spectrometry—except to teach students! Most of the time, working chemists use mass spectrometry in the lab in order to help them determine the structure or identity of unknown molecules and compounds. In today's world, mass spectrometry is an invaluable analytical tool, not only in chemistry, but also in medicine, forensics, environmental science, and other important fields.

Summary

  • Atoms that have the same number of protons and electrons but different numbers of neutrons are known as isotopes.
  • Isotopes of a given element have different atomic masses.
  • The relative abundance of an isotope is the fraction of a single element that exists on Earth with a specific atomic mass.
  • Atomic weights are weighted averages calculated by multiplying the relative abundance of each isotope by its atomic mass and then summing up all the products.
  • The relative abundances of each isotope can be determined using mass spectrometry.
  • A mass spectrometer ionizes atoms and molecules with a high-energy electron beam and then deflects the ions through a magnetic field based on the mass-to-charge ratio of the ion, mz\dfrac{m}{z}.
  • The mass spectrum of a sample shows relative abundance of each ion on the y-axis and mz\dfrac{m}{z} along the x-axis.

Attributions

  1. “OpenStax College. "Atomic Structure and Symbolism." Openstax. October 2, 2014. http://cnx.org/contents/havxkyvS@9.58:ZV-IsnqQ@8/Atomic-Structure-and-Symbolism. CC BY 4.0.
The modified article is licensed under a CC-BY-NC-SA 4.0 license.

Additional References

Zumdahl, S. S., and S. A. Zumdahl. "Atomic Structure and Periodicity." In Chemistry, 290-94. 6th ed. Boston, MA: Houghton Mifflin Company.

Try it!

Based on the simulated mass spectrum of zirconium above, we get the following atomic masses and relative abundances for the isotopes of zirconium:
IsotopeZr-90\text{Zr-}90Zr-91\text{Zr-}91Zr-92\text{Zr-}92Zr-94\text{Zr-}94Zr-96\text{Zr-}96
Atomic mass (u\text u)89.90589.90590.90690.90691.90591.90593.90693.90695.90895.908
Relative abundance (%\%)51.4551.4511.2211.2217.1517.1517.3817.382.802.80
What is the atomic weight of zirconium based on our mass spectrometry sample?
Express your answer as a decimal rounded to the nearest hundreth.
u\text u
The equation for atomic weight is:
atomic weight=i=1n(relative abundance×atomic mass)i\text{atomic weight}=\sum\limits_{i=1}^n(\text{relative abundance}\times\text{atomic mass})_i
We can use this equation, plus the relative abundances and atomic masses from our mass spectrum, to calculate the atomic weight of zirconium!
Plugging in the relative abundances and atomic masses into the above equation, we get:
atomic weight=i=1n(relative abundance×atomic mass)i=(0.5145×89.905u)+(0.1122×90.906u)+(0.1715×91.905u)+(0.1738×93.906u)+(0.0280×95.908u)=46.26u+10.20u+15.76u+16.32u+2.68u=91.22 u\begin{aligned}\text{atomic weight} &= \sum\limits_{i=1}^n(\text{relative abundance}\times\text{atomic mass})_i \\ &= (0.5145\times89.905\,\text{u})+(0.1122\times90.906\,\text{u}) +(0.1715\times91.905\,\text u)+(0.1738\times93.906\,\text{u})+(0.0280\times95.908\,\text u) \\ &= 46.26 \,\text u+10.20\,\text u +15.76\,\text u +16.32\,\text u +2.68\,\text u \\ &= 91.22\text { u} \end{aligned}
The atomic weight of zirconium from our sample is 91.22u91.22\,\text u.
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