# Mole Concept and Molar Mass

A mole is the amount of a substance that has $6.023\times10^{23}$ particles, and the molar mass is the mass of one mole of a substance.

The number of particles that makes up even a small amount of a substance is so large that chemists use a special number to describe them. A mole is the amount of a substance that contains as many particles as 12 grams of pure carbon-12, equal to $6.023\times10^{23}$ (Avogadro's number) particles. For example, one mole of neon (Ne) contains $6.023\times10^{23}$ neon atoms, and one mole of hydrogen gas (H2) contains $6.023\times10^{23}$ hydrogen molecules. Moles are useful in chemistry for describing the amounts of different substances that take part in a chemical reaction. However, along with the concept of the mole, it is necessary to understand the ratios of different types of atoms.

A molecular formula indicates the number of atoms of each type of element in a molecule. An empirical formula indicates the ratio of elements found in a compound in the lowest whole number possible. For example, the molecular formula of glucose (C6H12O6) shows that each glucose molecule has 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. The empirical formula of glucose (CH2O) shows that each molecule has twice the number of hydrogen atoms as carbon atoms and oxygen atoms. Similarly, the molecular formula of ethane (C2H6) shows that each molecule has 2 carbon atoms and 6 hydrogen atoms. Its empirical formula, CH3, shows that each molecule has three times as many hydrogen atoms as carbon atoms.

In a balanced chemical equation, the coefficients of the reactants (on the left side) and the coefficients of the products (on the right side) indicate the ratios of moles of the species participating in the reaction. This ratio of moles is called the mole ratio of reactants to products. Consider the reaction of copper oxide (CuO) and hydrochloric acid (HCl).
$\mathrm{CuO}+2\mathrm{HCl}\rightarrow{\mathrm{CuCl}}_2+{\mathrm H}_2\mathrm O$
The coefficients show that one mole of copper oxide reacts with two moles of hydrochloric acid to produce one mole of copper chloride, CuCl2, and one mole of water, H2O. Using these coefficients, the number of moles of any reactant and product involved in the reaction can be determined as long as the number of moles of any of the four is known.
Describing a reaction using the masses of the reactants and products requires a different unit. Molar mass is the mass of one mole of the atoms, molecules, or formula units that make up a substance. It is measured in grams per mole (g/mol). The molar mass of each element in its atomic state is equal to its atomic mass, which is found in the periodic table. For example, the periodic table shows that the atomic mass of sodium (Na) is 22.99 atomic mass units (amu). So, its molar mass is 22.99 g/mol. The atomic mass of chlorine is 35.45 amu. So, the molar mass of chlorine in its atomic state is 35.45 g/mol. The molar masses of individual elements can be used to calculate the molar mass of a compound.
Step-By-Step Example
Calculating the Molar Mass of a Compound
What is the molar mass of sodium chloride?
Step 1

Write the formula of the compound.

The formula of sodium chloride is NaCl.

Step 2

Find the number of each type of atom in the formula.

Number of atoms of Na is 1.

Number of atoms of Cl is 1.

Step 3
Multiply the number of each type of atom by the molar mass of the element, and add the results.
\begin{aligned}{\text{Molar mass of }{\rm{NaCl}}}&=(1)(22.99\rm{\; g/mol})+(1)(35.45\rm{\; g/mol})\\&=58.44\rm{\; g/mol}\end{aligned}
Solution
The molar mass of sodium chloride is 58.44 g/mol.
As an example, consider the calculation of the molar mass of water. The formula, H2O, shows that each molecule of water has 2 hydrogen atoms and 1 oxygen atom.
\begin{aligned}\text{Molar mass of }{\rm{H_2O}}&=(2)(1.01\rm{\; g/mol})+(1)(16.00\rm{\; g/mol})\\&=18.02\rm {\; g/mol}\end{aligned}

### Converting from Mass to Moles

Mass can be converted to moles by dividing the mass by the molar mass of the substance.
One of the most important stoichiometric calculations in chemistry is the conversion of moles of a substance to grams and vice versa. This relationship can be written as a conversion formula.
$\text{Amount of substance (moles)}=\frac{\text{Mass of substance}}{\text{Molar mass of substance}}$
This formula can be used to calculate either moles or mass of a substance, depending on the variables present. A diagram of the three factors involved in the formula is useful for remembering the formula.
Step-By-Step Example
Converting an Element from Mass to Moles
Calculate the number of moles present in 4.0 grams (g) of magnesium (Mg).
Step 1

Write the given and known information.

The given information is:
$\text{Mass of magnesium} = 4.0 \rm{\; g}$
The periodic table shows that the atomic mass of magnesium is 24.31 atomic mass units (amu). That means that the known information is:
$\text{Molar mass of magnesium} = 24.31 \rm{\; g/mol}$
Step 2
Use the molar mass equation to convert from mass to moles.
\begin{aligned}\text{Amount of magnesium (moles)} &= \frac{\text {Mass of magnesium}}{\text{Molar mass of magnesium}}\\ &=\frac{4.0\rm{\; g}}{24.31\rm{\; g/mol}}\\ &=0.16\rm{\; mol}\end{aligned}
Solution
A mass of 4.0 g of magnesium has 0.16 mole of magnesium.
Step-By-Step Example
Converting Compounds from Mass to Moles
Calculate the number of moles in 50.5 grams of calcium carbonate, CaCO3, and the number of moles in 20.3 grams of sodium chloride (NaCl).
Step 1
The formula indicates one calcium atom, one carbon atom, and three oxygen atoms. The molar mass of CaCO3 can be calculated using these numbers as well as the atomic mass of each element.
\begin{aligned}{\text{Molar mass of }{\rm CaCO_3}} &= (1)(40.08{\rm{\; g/mol}}) + (1)(12.01{\rm{\; g/mol}}) +(3)(16.00{\rm{\; g/mol}}) \\&= 100.09{\rm{\; g/mol}}\end{aligned}
Step 2
The number of moles is the mass of CaCO3 divided by its molar mass.
\begin{aligned}\text{Amount of }{\rm CaCO}_3&=\frac{\text{Mass of }{\rm CaCO_3}}{\text{Molar mass of }{\rm CaCO_3}}\\&=\frac{50.5\rm{\; g}}{100.09\rm {\; g/mol}}\\&=0.505\rm{\; mol}\end{aligned}
Step 3
Next, consider the number of moles in 20.3 grams of NaCl. The formula indicates one sodium atom and one chlorine atom. Use these numbers and the atomic mass of each element to calculate the molar mass of NaCl.
\begin{aligned}\text{Molar mass of }{\rm NaCl}&= (1)(22.99{\rm{\; g/mol}}) + (1)(35.45{\rm{\; g/mol}})\\&= 58.44{\rm{\; g/mol}}\end{aligned}
Step 4
The number of moles is the mass of NaCl divided by its molar mass.
\begin{aligned}\text{Amount of }{\rm NaCl}&=\frac{\text{Mass of }{\rm NaCl}}{\text{Molar mass of }{\rm NaCl}}\\&=\frac{20.3\rm{\; g}}{58.44\rm {\; g/mol}}\\&=0.347\rm{\; mol}\end{aligned}
Solution

A 50.5-gram sample of CaCO3 has 0.505 mole of CaCO3.

A 20.3-gram sample of NaCl has 0.347 mole of NaCl.

### Converting from Moles to Mass

Moles can be converted to mass by multiplying the number of moles of the substance by the molar mass of the substance.
The molar mass formula can be rearranged to find the mass of a substance if the number of moles is given.
\begin{aligned}\text {Moles of substance}&=\frac{\text{Mass of substance}}{\text{Molar mass of substance}}\\\\ \text{Mass of substance}&=(\text{Moles of substance})(\text{Molar mass of substance})\end{aligned}
Step-By-Step Example
Converting an Element from Moles to Mass
Calculate the mass of 2.45 moles of oxygen (O).
Step 1

Write the given and known information.

The given information is:
$\text{Moles of oxygen}=2.45\;\rm{mol}$
The periodic table shows that the atomic mass of oxygen is 16.00 atomic mass units (amu). That means that the known information is:
$\text{Molar mass of oxygen}= 16.00\;\rm{g/mol}$
Step 2
Use the molar mass equation to convert from mass to moles.
\begin{aligned}{\text{Mass of oxygen}}&= ({\text{Moles of oxygen}})({\text{Molar mass of oxygen}})\\&= (2.45{\rm{\; mol}})(16.00{\rm{\; g/mol}})\\&= 39.2{\rm{\; g}}\end{aligned}
Solution
The mass of 2.45 mol of oxygen is 39.2 g.
Step-By-Step Example
Converting Compounds from Moles to Mass
Calculate the mass of 1.25 mol of water (H2O) and the mass of 3.4 mol of hydrochloric acid (HCl).
Step 1
Notice that the formula H2O shows that each water molecule has two hydrogen atoms and one oxygen atom. The periodic table shows that the atomic weight of hydrogen is 1.01 g/mol and the atomic weight of oxygen is 16.00 g/mol. The molar mass can be calculated using these values.
\begin{aligned}{\text{Molar mass of }{\rm H_{2}O}}&= (2)(1.01{\rm{\; g/mol}})+(1)(16.00{\rm{\; g/mol}})\\&=18.02{\rm{\; g/mol}}\end{aligned}
Step 2
The mass of water is the product of the number of moles and the molar mass.
\begin{aligned}{\text{Mass of }{\rm H_{2}O}}&= ({\text{Moles of }{\rm H_{2}O}})({\text{Molar mass of }{\rm H_{2}O}})\\&= (1.25{\rm{\; mol}})(18.02{\rm{\; g/mol}})\\&= 22.5{\rm{\; g}}\end{aligned}
Step 3
The formula for hydrochloric acid shows that a molecule of HCl is composed of one hydrogen atom and one chlorine atom. Use these numbers and the atomic weight of each element to calculate the molar mass of HCl.
\begin{aligned}{\text{Molar mass of }{\rm HCl}}&=(1)(1.01{\rm{\; g/mol}})+(1)(35.45{\rm{\; g/mol}})\\&= 36.46{\rm{\; g/mol}}\end{aligned}
Step 4
The mass of HCl is the product of the number of moles and the molar mass.
\begin{aligned}{\text{Mass of }{\rm HCl}}&= ({\text{Moles of }{\rm HCl}})({\text{Molar mass of }{\rm HCl}})\\&=(3.48{\rm{\; mol}})(36.46{\rm{\; g/mol}})\\&= 127{\rm{\; g}}\end{aligned}
Solution
The calculations show that with appropriate significant figures, 1.25 mol of H2O has a mass of 22.5 g, and 3.48 mol HCl has a mass of 127 g.