#### Learning Objective

- Convert between the mass and the number of moles, and the number of atoms, in a given sample of compound

#### Key Points

- The molar mass of a compound is equal to the sum of the atomic masses of its constituent atoms in g/mol.
- Although there is no physical way of measuring the number of moles of a compound, we can relate its mass to the number of moles by using the compound’s molar mass as a direct conversion factor.
- To convert between mass and number of moles, you can use the molar mass of the substance. Then, you can use Avogadro’s number to convert the number of moles to number of atoms.

#### Terms

- moleThe amount of substance that contains as many elementary entities as there are atoms in 12 g of carbon-12.
- dimensional analysisThe analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
- molar massThe mass of a given substance (chemical element or chemical compound) divided by its amount of substance (mol), in g/mol.

Chemists generally use the mole as the unit for the number of atoms or molecules of a material. One mole (abbreviated mol) is equal to 6.022×10^{23} molecular entities (Avogadro’s number), and each element has a different molar mass depending on the weight of 6.022×10^{23} of its atoms (1 mole). The molar mass of any element can be determined by finding the atomic mass of the element on the periodic table. For example, if the atomic mass of sulfer (S) is 32.066 amu, then its molar mass is 32.066 g/mol.

By recognizing the relationship between the molar mass (g/mol), moles (mol), and particles, scientists can use dimensional analysis convert between mass, number of moles and number of atoms very easily.

## Determining the Molar Mass of a Compound

In a compound of NaOH, the molar mass of Na alone is 23 g/mol, the molar mass of O is 16 g/mol, and H is 1 g/mol. What is the molar mass of NaOH?

[latex]Na+O+H=NaOH[/latex]

[latex]23 \space \text{g/mol} +16 \space \text{g/mol}+ 1 \space \text{g/mol} = 40 \space \text{g/mol}[/latex]

The molar mass of the compound NaOH is 40 g/mol.

## Converting Mass to Number of Moles

How many moles of NaOH are present in 90 g of NaOH?

Since the molar mass of NaOH is 40 g/mol, we can divide the 90 g of NaOH by the molar mass (40 g/mol) to find the moles of NaOH. This the same as multiplying by the reciprocal of 40 g/mol.

If the equation is arranged correctly, the mass units (g) cancel out and leave moles as the unit.

[latex]90 g \space \text{NaOH} \times \frac{1 \space mol}{40 g} = 2.25 \space \text{mol NaOH}[/latex]

There are 2.25 moles of NaOH in 90g of NaOH.

## Converting Between Mass, Number of Moles, and Number of Atoms

How many moles and how many atoms are contained in 10.0 g of nickel?

According to the periodic table, the atomic mass of nickel (Ni) is 58.69 amu, which means that the molar mass of nickel is 58.69 g/mol. Therefore, we can divide 10.0 g of Ni by the molar mass of Ni to find the number of moles present.

Using dimensional analysis, it is possible to determine that:

[latex]10\:g\:Ni \times \frac{1\:mol \ Ni}{58.69g \ Ni} = 0.170\: mol\:Ni[/latex]

To determine the number of atoms, convert the moles of Ni to atoms using Avogadro’s number:

[latex]0.170\:moles\:Ni \times \frac {6.022×10^{23} atoms \ Ni}{1\:mol \ Ni} = 1.02\times10^{23}\:atoms\:Ni[/latex]

Given a sample’s mass and number of moles in that sample, it is also possible to calculate the sample’s molecular mass by dividing the mass by the number of moles to calculate g/mol.

What is the molar mass of methane (CH_{4}) if there are 0.623 moles in a 10.0g sample?

[latex]\frac{10.0 \ g\ CH_4}{0.623 \ mol\ CH_4} = 16.05 \ g/mol \ CH_4 [/latex]

The molar mass of CH_{4} is 16.05 g/mol.