# 2.9: The Avogadro Constant

- Page ID
- 49413

Although chemists usually work with moles as units, occasionally it is helpful to refer to the actual number of atoms or molecules involved. When this is done, the symbol N is used. For example, in referring to 1 mol of mercury atoms, we could write

\[n_{\text{Hg}} = 1 \text{mol}\]

and

\[ N_{\text{Hg}} = 6.022\times 10^{23}\]

Notice that *N*_{Hg} is a pure number, rather than a quantity. To obtain such a pure number, we need a conversion factor which involves the number of particles per unit amount of substance. The appropriate factor is given the symbol* N*_{A} and is called the **Avogadro constant**. It is defined by the equation

\[N_{\text{A}} = \dfrac{N}{n} \label{1}\]

Since for any substance there are 6.022 × 10^{23} particles per mole,\[\textit{N}_\text{A}=\tfrac{6.022\cdot10^{23}} {1\text{ mol}}=6.022\cdot10^{23}\text{ mol}^{\text{–1}}\]

Example \(\PageIndex{1}\): Moles to Molecules

Calculate the number of O_{2} molecules in 0.189 mol O_{2}.

**Solution**

Rearranging Equation \(\ref{1}\), we obtain

\[N = n \times N_{\text{A}} = 0.189 \text{ mol} \times 6.022 \times 10^{23} \tfrac{1}{\text{ mol}} \ = 1.14 \times 10^{23} \nonumber \]

Alternatively, we might include the identity of the particles involved:

\begin{align}\text{N}&=&\text{0.189 mol O}_{\text{2}}\cdot \tfrac{6.022\cdot 10^{23}\text{ O}_2\text{ molecules}} {\text{1 mol O}_2} \\ &=& 1.14\cdot10^{23}\text{ O}_{\text{2}}\text{ molecules}\end{align}

Notice that Equation \(\ref{1}\), which defines the Avogadro constant, has the same form as the equation which defined density. The preceding example used the Avogadro constant as a conversion factor in the same way that density was used. As in previous examples, all that is necessary is to remember that number of particles and amount of substance are related by a conversion factor, the Avogadro constant.

\[\large\text{Number of particles } \large\overset{\text{Avogadro constant}}{\longleftrightarrow} \large\text{amount of substance} \\ \quad \\ \large N \large\overset{\text{N}_{\text{A}}}{\longleftrightarrow} { \space} \large n\label{2}\]

As long as the units *mole* cancel, *N*_{A} is being used correctly.

### Contributors

Ed Vitz (Kutztown University), John W. Moore (UW-Madison), Justin Shorb (Hope College), Xavier Prat-Resina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.