2.2.3: Moles

A mole of an element (or of a compound) is defined as containing 6.022 × 10²³ atoms (or molecules). The number, 6.022 × 10²³, is known as Avogadro’s number. So, one mole of carbon is equivalent to 6.022 × 10²³ carbon atoms. The scale used to measure atomic mass has changed slightly over time. Today, it is standardized relative to carbon so that the mass of one mole of ¹²C is exactly 12.0000. Consequently, all atomic masses are given in atomic mass units (amus), defined as one-twelfth the mass of ¹²C. Both protons and neutrons have equivalent mass, about one amu, and electrons have almost no mass (less than 1/1000^th the mass of protons and neutrons). So, we might expect the mass of an atom to be equal to the mass number (A, the total number of protons plus neutrons). But for several reasons, not worth going into here, the mass of an atom is generally close to, but not the same as, the mass number.

Elements are different from atoms. The atomic mass, also called the atomic weight, of an element is the sum of the masses of its naturally occurring isotopes weighted in accordance with their abundances. Atomic masses/weights of elements are molal quantities (and often given in units of grams/mole) but they are really dimensionless numbers because they are all calculated relative to the atomic mass/weight of a mole of carbon. Although isotope mass numbers are always integers, atomic weights of elements are not. For example, many tables and charts give the atomic weight of oxygen as 15.999 and that of iron as 55.847. When elements combine to produce a compound, the atomic weight of the compound is the sum of the weights of the elements in the compound. FeO, for example, has atomic weight 71.846 (15.999 + 55.847) grams/mole.

Most elements have very small isotopic variation in nature, no matter where they are found. Thus, most quartz (SiO₂) contains about the same relative amounts of the three natural oxygen isotopes (mostly ¹⁶O) depicted in Figure 2.5 above. Furthermore, isotopic variations have extremely small effects on the properties of minerals. So, mineralogists generally do not worry much about isotopes. Small isotopic variations, however, may be significant to a geochemist trying to determine the genesis of a particular mineral or rock.

Example \(\PageIndex{1}\): What Is a Mole of Quartz?

Quartz (SiO₂) is one of the most common and well-known minerals. A mole of quartz is 6.022 × 10²³ SiO₂ molecules, and that sounds like a lot. How much quartz is that?

Solution

To answer this question, we use the atomic weights of silicon and oxygen as well as some crystallographic data. Silicon and oxygen have atomic weights of 28.0855 and 15.9994, respectively. The atomic weight of quartz, SiO₂, is therefore 60.0843 (= 28.0855 + 15.9994 + 15.9994). This means that a mole of quartz, 6.022 × 10²³ SiO₂ molecules, weighs 60.0843 gm.

Crystallographers have determined that quartz crystals are made of fundamental unit cells shaped like trigonal prisms (discussed in detail in later chapters) containing three (\(Z = 3\)) SiO₂ molecules. Each unit cell has a volume of 112.985 ų. So we may calculate the volume of a mole of quartz as:

\[V = N_A \times v/Z\]

where \(V, N_A, v\), and \(Z\) are the molar volume, Avogadro’s number, the unit cell volume, and the number of molecules per unit cell, respectively.

So, \(V\) = 6.022 × 10²³ SiO₂/mole × 112.986 ų/unit cell \(\div\) 3 SiO₂/unit cell

= 2.268 × 10²⁵ ų = 22.68 cm³

which is slightly smaller than a golf-ball.

We can, if we wish, then calculate the density (\(\rho\)) of quartz from the molar data:

\(\rho\) = molar weight \(\div\) molar volume = 60.0843 gm/mole \(\div\) 22.68 cm³/mole = 2.649 gm/cm³