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# The Mole and Avogadro's Constant

The mole, abbreviated mol, is an SI unit which measures the number of particles in a specific substance. One mole is equal to 6.02214179 x 1023 atoms, or other elementary units such as molecules.

### Introduction

The number of moles present can be accounted for using the atomic mass of an element, which can be found on the periodic table. The mass showed is commonly an average of the abundant forms of that element found on earth. An element's mass is listed as the average of all of its isotopes on earth.

Perodic Table Courtesy of UC Davis 2010.

 Example

if you have one mole of oxygen atoms then you have 6.02214179 x 1023  oxygen atoms. The number 6.02214179 x 1023 alone is called Avogadro's number (NA) or Avogadro's Constant, after the 19th century scientist Amedeo Avogadro.

Each Carbon-12 atom weighs about 1.99265 X 10-23 g, therefore (1.99265 X 10-23 g X 6.02214179 x 1023 atoms) = 12 g of Carbon-12

### Applications of the Mole

The mass of a mole of substance is called the molar mass. Molar mass is used to convert grams of a substance to moles and is used often in chemistry. The molar mass of an element is found on the periodic table and it is the element's atomic weight in grams/mole (g/mol); therefore the atomic weight of an element is the mass of one mole of that element.  We use the mole in various chemical equations and stoichiometry.

If we know the mass of a substance, we can then determine how many moles is in the substance.  Converting the mass, in grams, of a substance to moles requires a conversion factor of (one mole of substance/molar mass of substance). Simply put, we try to cancel units to obtain the desired unit.

The mole concept is also applicable to the composition of chemical compounds. For instance, let us take a look at methane which is CH4. This molecule and its molecular formula informs us that per mole of methane there is 1 mole of carbon and 4 moles of hydrogen. So, moles are also used as a common unit that can be used as a ratio shown here:

2 mol H +  1 mol O =  1 mol H2O

The moles of H and O describe the number of atoms of each element that react to form 1 mol of H2O.

To think about what a mole means, one should relate it to words like dozen or pair. Just as a pair can mean 2 of shoes, books, pencils, people, or anything as long as the numerical value is 2, a mole means 6.02214179 x 1023 of anything you may like to count. So take a look at the following.

1 mole = 6.02214179 x 1023

is the same as saying

1 Dozen    =    (12 eggs)

At last, it is very difficult to visualize a mole of something because Avogadro's Constant is far too great. For instance, consider the size of one single grain of wheat. If all the people who have existed in Earth's history did nothing but count individual wheat grains for their entire lives, the number of wheat grains would still be much less than Avogadro's Constant; the amount of wheat grains ever made in our history does not even come close to Avogadro's Number (Petrucci, 2007).

 Example

How many moles of Potassium (K) atoms are in a 3.04 grams of container of pure potassium metal?

In this example, we multiply the mass of K by the conversion factor:

1 mol K / 39.10 grams K

39.10 grams is the molar mass of one mole of K allowing us to cancel out grams, leaving us with the moles of K.

3.04 g K X (1 mol K/ 39.10 g K) =  0.0778 mol K

Similarly, if we know the moles of a substance, we can then determine how many grams is in the substance.  Converting moles of a substance to grams requires a conversion factor of molar mass of substance/one mole of substance. One simply needs to follow the same method but heading in a backwards direction.

 Example

Please convert to grams:  10.78 moles of Calcium (Ca)

10.78 mol Ca X (40.08 g Ca/ 1 mol Ca) =  432.1 g Ca

We multiply moles of Ca by the conversion factor 40.08 g Ca/ 1 mol Ca with 40.08 g being the molar mass of one mole of Ca, which then allows us to cancel out the moles, leaving us with grams of Ca.

The total number of atoms in a substance can also be determined by using the relationship between grams, moles, and atoms. If we are given the mass of a substance and we want to find the number of atoms in the substance, we would first have to convert the mass of the substance, in grams, to moles, as in example one. Then we would have to convert the moles of the substance to atoms. Converting moles of a substance to atoms requires a conversion factor of Avogadro's Constant (6.02214179 x 1023) / one mole of substance

A great way to make sure you are on the right track for calculating these problems is making sure your units cancel!

 Example

How many atoms are in a 3.5 g sample of sodium (Na)?

3.5 g Na X (1 mol Na/ 22.98 g Na) = .152 mol Na

0.152 mol Na X (6.02214179 x 1023 atoms Na/ 1 mol Na) = 9.15 X 1022 atoms of Na

In this example, we multiply the grams of Na by the conversion factor 1 mol Na/ 22.98 g Na, with 22.98g being the molar mass of one mole of Na, which then allows us to cancel out the grams, leaving us with moles of Na. Then we multiply the number of moles of Na by the conversion factor 6.02214179 x 1023 atoms Na/ 1 mol Na, with 6.02214179 x 1023 atoms being the number of atoms in one mole of Na (Avogadro's Constant), which then allows us to cancel out the moles, leaving us with the number of atoms of Na.

### Applications

The mass of a mole of substance is called the molar mass. Molar mass is used to convert grams of a substance to moles and is used often in chemistry. The molar mass of an element is found on the periodic table and it is the element's atomic weight in grams/mole (g/mol). The atomic mass is the mass of one mole of that element.

If we know the mass of a substance, we can then determine how many moles are in the substance.  Converting the mass, in grams, of a substance to moles requires a conversion factor of (one mole of substance/molar mass of substance). Using Avogadro's constant, it is also easy to calculate the number of atoms or molecules present in a substance. By multiplying the number of moles by Avogadro's constant, the mol units cancel out, leaving the number of atoms.

Known Information             Multiply By                                                                   End Up With

Mass of substance (g)         Molar mass (g/mol)                                                      Moles of substance

Moles of substance (mol)     Avogadro's constant (atoms/mol)                                  Atoms (or molecules)

Mass of substance (g)        Molar mass (mol/g) x Avogadro's constant (atoms/mol)    Atoms (or molecules)

 Example

1.    Please convert to moles:   3.00 grams of Potassium (K)  =   3.00 g K X (1 mol K/ 39.10 g K) =  0.076726 mol K

In this example, we multiply the mass of K by the conversion factor:

1 mol K / 39.10 grams K

39.10 grams is the molar mass of one mole of K allowing us to cancel out grams, leaving us with the moles of K.

You can also use this knowledge to do the opposite and calculate the mass from the number of moles:

 Example

2. Please convert to grams:  10.00 moles of Calcium (Ca)

We multiply moles of Ca by the conversion factor 40.08 g Ca/ 1 mol Ca with 40.08 g being the molar mass of one mole of Ca, which then allows us to cancel out the moles, leaving us with grams of Ca.

10.00 mol Ca X (40.08 g Ca/ 1 mol Ca) =  400.8 grams of Ca

We can also find the number of atoms from this method, using Avogadro's Constant (6.02214179 x 1023) / one mole of substance

 Example

3.     How many atoms are in a 3.0 g sample of sodium (Na)?

3.0 g Na X (1 mol Na/ 22.98 g Na) = .130548 mol Na

0.130548 mol Na X (6.02214179 x 1023 atoms Na/ 1 mol Na) = 7.86179 x 1022 atoms of Na

### Practice Problems

Using a periodic table, What is the Molar Mass of

1)

a. H

b. Se

c. Ne

d. Cs

e. Fe

For problems 2-4, convert to moles and find the total number of atoms

2)    5.06 grams of Oxygen            3)    2.14  grams of K            4)    0.134 kg of Li

Convert the following to grams

5)   4.5 mols of C             6) 7.1 mols of Al            7)  2.2 mols of Mg

How many moles are in the product of the reaction

8)    6 mol H + 3 mol  O → ? mol H20

9)    1 mol Cl + 1 mol Cl → ? mol Cl2

10)    5 mol Na + 4 mol Cl → ? mol NaCl

1a.  1.008 g/mol

1b. 78.96 g/mol

1c. 20.18 g/mol

1d.  132.91g/mol

1e.  55.85 g/mol

2. 5.06g O (1mol/16.00g)=  0.316 mol of O

0.316 mols (6.022x1023 atoms/ 1mol) = 1.904x1023 atoms of O

3. 2.14g K (1mol/39.10g)=  0.055 mol of K

0.055 mols (6.022x1023 atoms/ 1mol) = 3.312x1022 atoms of K

4. 0.134kg Li (1000g/1kg)= 134g Li (1mol/6.941g)= 19.3 mols Li

19.3 (6.022x1023 atoms/ 1mol) = 1.16x1025 atoms of Li

5. 4.5 mols of C (12.011g/1mol) = 54.05 g of C

6. 7.1 mols of Al (26.98g/1mol) = 191.56 g of Al

7. 2.2 mols of Mg (24.31g/1mol) = 53.48 g of MG

8. 6 mol H + 3 mol  O → 3 mol H20

9. 1 mol Cl + 1 mol Cl → 1 mol Cl2

10. 5 mol Na + 4 mol Cl → 4 mol NaCl + 1 mol Na (excess)

### References

1. Keenan, Charles W. and Wood, Jesse H. . General College Chemistry. 4th ed. New York: Haper and Row, 1971.
2. Mortimer, Charles E. Chemistry a Conceptual Approach. 2nd ed. New York: Van Nostrand Reinhold, 1971.
3. Jones, Loretta and Atkins, Peter. Chemistry : Molecules, Matter, and Change. 4th ed. New York: W.H. Freeman, 2000.
4. Petrucci, Ralph H., Herring, Goeffrey F., Madura, Jeffrey D., and Bissonnette, Carey. General Chemistry: Principles and Modern Applications. 10th ed. New Jersey: Pearson Canada, 2011.

### Contributors

• Ryan Benoit (UCD), Michael Thai (UCD), Charlie Wang (UCD), Jacob Gomez (UCD)

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