The Mole and Atomic Mass: _{Definitions, conversions, and Avogadro's number}

The mole and molar mass

The International Committee for Weights and Measures—a group that defines the metric system’s units of measurement (for more information, see our module on The Metric System)—defines one mole as the number of atoms in exactly 12 grams of carbon-12 (¹²C, Figure 2). Experiments counting the number of ¹²C atoms in a 12-gram sample have determined that this number is 6.02214076 x 10²³. Regardless of whether the substance is ¹²C, electrons, or gray squirrels, one mole represents the same number of each of these things.

Scientists have then defined the molar mass of a substance as the mass of 6.02214076 x 10²³ units of that substance. So, the molar mass of gray squirrels is 301,000,000,000,000,000,000,000,000 grams. With squirrels, this is not very useful. However, it is quite useful if we apply it to other substances, especially elements. By standardizing the number of atoms in a sample of an element, we also get a standardized mass for that element that can be used to compare different elements and compounds to one another. ¹²C’s molar mass is 12 grams, which represents the combined mass of 6.02 x 10^{23 12}C atoms. However, other elements have different molar masses; for example, 6.02 x 10²³ sulfur-32 (³²S) atoms have a mass together of 31.97 grams, which is ³²S’s molar mass.

Along with telling us the mass of one mole of an element, molar mass also acts as a conversion factor between the mass of a sample and the number moles in that sample. For example, 24 grams of ¹²C atoms would be equal to two moles since 24 grams divided by the mass of one mole (12) equals 2. Further, Avogadro’s number acts as the conversion factor for converting between the number of moles in a sample and the actual number of atoms or molecules in that sample. Extending our example, two moles of ¹²C atoms contains 2 times 6.02 x 10²³ atoms, which equals 12.04 x 10²³ atoms, which can be written as 1.204 x 10²⁴ atoms.

Example: Converting between mass and moles

Knowing a substance’s molar mass is useful, because the molar mass acts as a conversion factor between the mass of a sample and the number of moles in that sample (Equation 1). For converting between the number of moles in a sample and the number of molecules in the sample, Avogadro’s number acts as the conversion factor, as shown in Equation 2 below.

To understand how molar mass and Avogadro’s number act as conversion factors, we can turn to an example using a popular drink: How many CO₂ molecules are in a standard bottle of carbonated soda? (Figure 3 shows what happens when the CO₂ in soda is quickly converted to a gaseous form.)

Thanks to molar mass and Avogadro’s number, figuring this out doesn’t require counting each individual CO₂ molecule! Instead, we can start by determining the mass of CO₂ in this sample. In an experiment, a scientist compared the mass of a standard 16-ounce (454 milliliters) bottle of soda before it was opened, and then after it had been shaken and left open so that the CO₂ fizzed out of the liquid. The difference between the masses was 2.2 grams—the sample mass of CO₂ (for this example, we’re going to assume that all the CO₂ has fizzed out). Before we can calculate the number of CO₂ molecules in 2.2 grams, we first have to calculate the number of moles in 2.2 grams of CO₂ using molar mass as the conversion factor (see Equation 1 above):

Now that we’ve figured out that there are 0.050 moles in 2.2 grams of CO₂, we can use Avogadro’s number to calculate the number of CO₂ molecules (see Equation 2 above):

While scientists today commonly use the concept of the mole to interconvert number of particles and mass of elements and compounds, the concept started with 19th-century chemists who were puzzling out the nature of atoms, gas particles, and those particles’ relationship with gas volume.

Avogadro, Gay-Lussac, Dalton, and the history of the mole concept

In 1811, the Italian lawyer-turned-chemist Amedeo Avogadro published an article in an obscure French science journal that lay the foundation for the mole concept. However, as it turns out, that wasn’t his intention!

Avogadro was trying to explain a strangely simple observation made by one of his contemporaries. This contemporary was the French chemist and hot air balloonist Joseph-Louis Gay-Lussac, who was fascinated by the gases that lifted his balloons and performed studies on gas behavior (for more about gas behavior, see the module Properties of Gases). In 1809, Gay-Lussac published his observation that volumes of gases react with each other in ratios of small, whole numbers. For example, Gay-Lussac observed that 2 volumes of carbon monoxide reacted with 1 volume of oxygen to yield 2 volumes of carbon dioxide. Modern scientists would immediately recognize this reaction as: 2CO + 1O₂ → 2CO₂ (Figure 4). But how could early 19th century scientists explain this tidy observation of small, whole numbers?

In his 1811 paper, Avogadro drew from British scientist John Dalton’s atomic theory—the idea that all matter, whether gas or liquid or solid, is made of extremely tiny particles (to learn more about Dalton’s idea, see our module on Early Ideas about Matter). Avogadro assumed that for substances in a gas state, the gas particles maintained fixed distances from one another. These fixed distances varied with temperature and pressure, but were the same for all gases at the same temperature and pressure.

Avogadro’s assumption meant that a defined volume of one gas, such as CO₂, would have the same number of particles as the same volume of a totally different gas, such as O₂. Avogadro’s assumption also meant that when the gases reacted together, the whole number ratios of their volumes ratios reflected how the gas reacted on the level of individual molecules. Thus, 2 volumes of CO reacted with 1 volume of O₂, because on the molecular level, 2 CO molecules were reacting with 1 molecule of O₂.

Avogadro’s paper was largely ignored for many decades: the existence of atoms was still being debated, and Avogadro didn’t offer experimental evidence to support his suggestions. However, his former student, the Italian chemist Stanislao Cannizzaro, used Avogadro’s ideas to figure out a vital connection between gas volume, the number of particles, and weight. Cannizzaro reasoned that if he had samples of two different gases with the same number of atoms, such as nitrogen and oxygen, then he could weigh each sample and compare the weight ratios. By comparing the ratios, he could figure out the relative weight of one nitrogen atom compared to one oxygen atom. This approach is still used by scientists today to figure out the relative mass of an atom from one element compared to the mass of an atom from a different element.

Avogadro’s number combines with the mole

The idea that equal gas volumes had the same number of molecules challenged scientists to figure out how many gas molecules there were in a defined volume. In 1865, an Austrian chemist named Josef Loschmidt came up with a way of estimating that number. Working with the assumptions of kinetic-molecular theory on the size of gas molecules and the distance between them (see our module on Kinetic-Molecular Theory), Loschmidt estimated the number of particles in one cubic centimeter of a gas to be 1.83 x 10¹⁸ atoms/cm³.

By the start of the 20th century, this enormous number was given a new name. In a 1908 paper, the French physical chemist Jean Perrin proposed renaming this constant Avogadro’s number. The name took off among chemists because Perrin used it in his popular chemistry books. Around the same time, the concept of having a standard number of particles correspond to the mass of a solid was being developed by German chemists. The scientists called this concept the Kilogrammemolekuel, which was soon shortened to "mole." However, it was mid-20th century physicists who linked Avogadro’s number to this concept of the mole.

Since the early 20th century, physicists had been developing a unified atomic mass scale—basically, a way for comparing the mass of an atom from one element to the mass of an atom from another element. Because this scale was relative (just like Cannizzaro’s ideas about comparing the weight ratio of gas molecules), the scientists had reached a consensus where a single oxygen-16 (¹⁶O) atom was assigned an arbitrary value of 16. The atoms of all other elements were then compared to this ¹⁶O reference.

However, in 1961, a new reference point was chosen by scientists with the International Committee for Weights and Measures, the group that defines the metric system’s units of measurement. In place of ¹⁶O, the group decided to use the most common isotope of carbon, carbon-12 (¹²C), as the reference. The group decided that the mass of one ¹²C atom would be set as 12 atomic mass units (amu), and the atomic mass of the atoms of all other elements would be determined relative to ¹²C—the standard we still use today.

A decade later, the same group added the mole into the metric system as a unit for the “amount of a substance.” To define the exact amount that is in one mole unit, scientists again turned to ¹²C. To link the relative atomic mass scale to both absolute mass and moles, the group defined one mole as equal to the number of ¹²C atoms in 12 grams of ¹²C. The number of ¹²C atoms in 12 grams was experimentally determined to be 6.022 x 10²³. This value was named Avogadro’s number (N_A), thereby replacing its earlier definition as the number of gas atoms in a cubic centimeter.

Atomic weight, molecular weight, and molar mass

By setting the mass of one mole of ¹²C equal to 12 grams and one ¹²C atom to 12 amu, the scientists made it possible to easily convert between an element’s atomic mass and its molar mass—the mass of one mole of molecules. In the case of ¹²C, we can see that the value for its molar mass and atomic mass both equal 12, although the units are different. While atomic mass is measured in amu, molar mass is measured in grams per mole.

This shared value between molar mass and atomic mass applies to all elements. Going back to ³²S, we know that because its molar mass is 31.97 grams per mole, its atomic mass must be 31.97 amu.

However, as you can see on the periodic table, sulfur is listed as 32.07 amu—not 31.97! This is because the periodic table lists atomic weights—the averages of the atomic mass for each one of the element’s stable isotopes. Specifically, the periodic table lists the average atomic weight calculated based on the relative abundance of an element’s different isotopes, as shown in the example with sulfur below.

Example: Calculating sulfur’s atomic weight

To calculate sulfur’s atomic weight, we first need to know its isotopes and the atomic mass and relative abundance (proportion) each isotope. Sulfur has four stable isotopes, as shown in the chart below with their relative proportion in nature:

Next, we calculate sulfur’s atomic weight by multiplying each isotope’s atomic mass by its proportion:

Molecular weight and atomic weight

Atomic weight is useful for determining a substance’s molecular weight—the average combined weight of a molecule’s individual atoms. Like atomic weight, molecular weight is an averaged weight, based on the relative abundance of each atom’s isotopes.

Let’s go back to carbon dioxide (CO₂) for an example of how this works. Each CO₂ molecule has one carbon atom and two oxygen atoms. Therefore, CO₂’s molecular weight (MW) can be calculated by adding the atomic weights of one carbon atom and two oxygen atoms:

Because molecular weight and molar mass are linked, we’ve also just determined the mass of one mole of CO₂: one mole of CO₂ has a molar mass of 44.0098 grams per mole.

Limitations of the mole

While the mole is very useful for scientists, it is not without its problems and controversies. One major challenge with the mole is that it is defined as the number of ¹²C atoms in 12 grams of ¹²C. Because the number of atoms must be determined through experiments, this number changes as experimental techniques evolve and improve. Furthermore, there is also experimental uncertainty about the exact number of atoms in 12 grams of ¹²C (to learn more about uncertainty, see our module on Uncertainty, Error, and Confidence). We can see these problems even in very recent examples: in 2011, two papers were published on experiments that determined Avogadro’s number to be 6.02214082(18) x 10²³, with a relative uncertainty of 3.0 x 10^-8. However, a 2015 paper determined Avogadro’s number to be 6.02214076(12) x 10²³, with a relative uncertainty of 2.0 x 10^-8.

Because of these problems, the General Conference on Weights and Measures, which meets periodically to coordinate and set international standards for the metric system, proposed in the early 21st century to set a permanent definition for the number contained in one mole. According to this proposal, one mole contains a number equal to Avogadro’s constant, which would be set (regardless of further experiment results) as 6.02214129 x 10²³.

Some scientists support this permanent definition of the mole. Others object because this would transfer the uncertainty in the mole to experiments determining mass. Specifically, scientists would still have to experimentally determine the mass for a mole of each element. This would introduce uncertainty into the mass, and possibly change the atomic weights listed on the periodic table (Figure 5).

This controversy over defining the mole can seem a little strange. After all, the mole is not a term we need to refer to most things in daily life (such as gray squirrels!). But as we’ve seen in this module, the mole is very useful for scientists when talking about very large numbers. Furthermore, the mole provides a vital link that allows scientists to connect the number of particles in a substance to both the combined mass of all those particles and the mass of each individual particle. As we’ve seen, the mole concept is useful for so much more than just referring to the amount of a substance.