18.2.2 Moles and Molar Mass

We often need to know the amount of substance in a macroscopic system: basically, how much “stuff” is present. The most familiar way of doing that is to specify the system’s mass $M$ . Another way is to measure the amount of substance in moles. By definition, 1 mole of matter—be it solid, liquid, or gas—is the amount of a substance that contains $6.02 \times 10^{23}$ basic particles.

The basic particle depends on the substance. Helium is a monatomic gas, meaning that the basic particle is the helium atom. Thus $6.02 \times 10^{23}$ helium atoms are 1 mol of helium. But oxygen gas is a diatomic gas because the basic particle is the two-atom diatomic molecule $O_{2}$ . 1 mol of oxygen gas contains $6.02 \times 10^{23}$ molecules of $O_{2}$ and thus $2 \times 6.02 \times 10^{23}$ oxygen atoms. TABLE 18.3 lists the monatomic and diatomic gases that we will use for examples and homework problems.

TABLE 18.3 Monatomic and diatomic gases

Monatomic		Diatomic
He	Helium	$H_{2}$	Hydrogen
Ne	Neon	$N_{2}$	Nitrogen
Ar	Argon	$O_{2}$	Oxygen

The number of basic particles per mole of substance is called Avogadro’s number, $N_{A}$ . The value of Avogadro’s number is

N_{A} = 6.02 \times 10^{23} {mol}^{- 1}

Despite its name, Avogadro’s number is not simply “a number”; it has units. Because there are $N_{A}$ particles per mole, the number of moles in a substance containing $N$ basic particles is

n = \frac{N}{N_{A}} (moles of substance)

(18.4)

Notice that we use uppercase $N$ for the number of atoms or molecules, lowercase $n$ for the number of moles.

The molar mass of a substance is the mass of 1 mol of substance. The molar mass, which we’ll designate $M_{mol}$ , has units $kg / mol$ . The definition of the mole is such that a substance with atomic or molecular mass $A$ u has a molar mass of almost exactly $A g / mol$ . (This is due to an older definition of the mole that was based on the mass of $^{12} C$ .) Thus the molar mass of $^{4} He,$ with $A = 4$ , is $4.0 g / mol$ . Chemists usually work with g/mol, but in physics we need molar masses to be in SI units of $kg / mol$ . Thus our rule is that the molar mass is the atomic or molecular mass in u divided by 1000. For example, the molar mass of He, with $m = 4 u,$ is $M_{mol} (He) = 0.004 kg / mol$ and the molar mass of diatomic $O_{2}$ is $M_{mol} (O_{2}) = 0.032 kg / mol$ .

Equation 18.3 uses the atomic mass to find the number of atoms in a system. Similarly, you can use the molar mass to determine the number of moles. For a system of mass $M$ consisting of atoms or molecules with molar mass $M_{mol}$ ,

n = \frac{M}{M_{mol}}

(18.5)

EXAMPLE 18.2 Moles of oxygen

100 g of oxygen gas is how many moles of oxygen?

SOLVE We can do the calculation two ways. First, let’s determine the number of molecules in 100 g of oxygen. The diatomic oxygen molecule $O_{2}$ has molecular mass $m = 32 u$ . Converting this to kg, we get the mass of one molecule:

m = 32 u \times \frac{1.66 \times 10^{- 27} kg}{1 u} = 5.31 \times 10^{- 26} kg

Thus the number of molecules in $100 g = 0.100 kg$ is

N = \frac{M}{m} = \frac{0.100 kg}{5.31 \times 10^{- 26} kg} = 1.88 \times 10^{24}

Knowing the number of molecules gives us the number of moles:

n = \frac{N}{N_{A}} = 3.13 mol

Alternatively, we can use Equation 18.5 to find

n = \frac{M}{M_{mol}} = \frac{0.100 kg}{0 .032 kg/mol} = 3.13 mol