# Molar Mass

Molar mass is the mass of a substance in a mole of that substance. It can be used to calculate the number of moles and molecules of a substance in a given amount of that substance.

When considering compounds and chemical reactions, scientists need a method to understand the ways in which atoms arrange and rearrange. Atoms are far too small to easily measure their number and size directly, but the mass of the materials they make up when they combine can be easy to measure, such as 351 grams of sodium chloride (351 g NaCl) or 12 kilograms of aluminum phosphate (12 kg AlPO4).

Italian scientist Amadeo Avogadro first proposed in 1811 that the number of atoms of a substance determines the amount of the substance that exists. Today, the number scientists use to compare quantities of atoms is named after Avogadro. Avogadro's number is the number of particles of a substance per mole of that substance, equal to $6.022\times{10}^{23}$ atoms, ions, or molecules. A mole is the amount of a substance that contains as many particles as 12 grams of pure carbon-12. (Carbon-12 was chosen as a reference because it is abundant and easily measured.) Both Avogadro's number and a mole can be used to describe any substance, such as molecules, atoms, and compounds.

A mole of any element in pure form is $6.022\times{10}^{23}$ atoms of that element. When atoms combine to make compounds, a mole of the compound consists of quantities of each element in proportion to their abundance in the compound. For example, a mole of sodium chloride (NaCl) consists of one mole of sodium atoms and one mole of chlorine atoms. A mole of water (H2O) consists of two moles of hydrogen atoms and one mole of oxygen atoms.

Molecular mass is the sum of the atomic weights of all atoms in a molecule. Thus, the molecular mass of water is the atomic weight of hydrogen multiplied by 2 plus the atomic weight of oxygen.
\begin{aligned}{\text{molecular mass of }}{\rm{H_2O}}&=(2)(1.01\;\rm{amu})+(1)(16.00\,\rm{amu})\\&=18.02\;\rm{amu}\end{aligned}
A related term is molecular weight, which is the weighted average of the molecular masses of all isotopes (atoms that have the same number of protons but different numbers of neutrons) of an element.

The atomic mass unit (amu) used in these calculations is the mass of a single proton or neutron, equal to 1 gram per mole (g/mol), defined as one-twelfth the mass of a carbon-12 atom. For water, the molecular mass is also the formula mass, the sum of the atomic weights of all atoms in a compound.

Molecular mass and formula mass may be given in varying units, such as daltons or atomic mass units, or without units. To state the mass of a substance in a way that is meaningful, scientists can use molar mass, the mass of one mole of the atoms, molecules, or formula units (the lowest whole number ratio of ions in an ionic compound) that make up a substance. The standard units for molar mass are kilograms per mole (kg/mol) but are more commonly reported in g/mol. The molar mass of an element is the same value as its atomic weight listed on the periodic table except it has the units g/mol. For example, the atomic weight of sodium is 22.99 amu, and its molar mass is 22.99 g/mol.
To calculate the number of moles in 408.8 g of NaCl, set up a ratio to convert grams to moles. Then divide 408.8 by the molar mass of NaCl, 58.44 g/mol.
$\frac{408.8\;\rm g}1\times\frac{1\;\rm{mol}}{58.44\;\rm g}=6.995\;\rm{mol}\;\rm{NaCl}$
Similarly, the number of moles of atoms in a certain mass of an element can be calculated. For example, to calculate the number of moles in 253 g of aluminum (Al), set up a ratio to convert grams to moles. Then divide by its molar mass, 26.98 g/mol.
$\frac{253\;\rm g}1\times\frac{1\;\rm{mol}}{26.98\;\rm g}=9.38\;\rm{mol}\;\rm{Al}$
Molar mass can also be used to calculate the mass of a substance given the number of moles of that substance. For example, given 3.80 moles of beryllium (Be) with molar mass 9.01 g/mol, multiply moles by molar mass.
$\frac{3.80\;\rm{moles}}1\times\frac{9.01\;\rm g}{1\;\rm{mol}}=34.2\;\rm g\;\rm{Be}$
The same is true if the substance is a compound. For example, to calculate the mass of 18.60 moles of carbon dioxide (CO2), which has a molar mass of 44.01 g/mol, multiply moles by molar mass.
$\frac{18.60\;\rm{mol}}1\times\frac{44.01\;\rm g}{1\;\rm{mol}}=818.6\;\rm g\;{\rm{CO}}_2$
Avogadro's number, $6.022\times{10}^{23}$, can be used to calculate the number of atoms in a given mass of a substance. For example, we can calculate the number of atoms of neon (Ne), which has a molar mass of 20.18 g/mol, in 15.9 grams.
$\frac{15.9\;{\rm{g}}}1\times\frac{1\;{\rm{mol}}}{20.18\;\rm{g}}\times\frac{6.022\times10^{23}\;{\text{atoms}}}{1\;{\rm{mol}}}=4.74\times10^{23}\;{\text{atoms}}\;{\rm{Ne}}$
The same is true for a compound. How many molecules of hydrogen peroxide (H2O2), which has a molar mass of 34.02 g/mol, are in 242.6 grams?
$\frac{242.6\;{\rm{g}}}1\times\frac{1\;{\rm{mol}}}{34.02\;{\rm{g}}}\times\frac{6.022\times10^{23}\;{\text{molecules}}}{1\;{\rm{mol}}}=4.294\times10^{24}\;{\text{molecules}}\;{\rm{H_2O_2}}$