6.5: Chemical Formulas as Conversion Factors
 Page ID
 98021
Learning Objectives
 Use chemical formulas as conversion factors.
Figure \(\PageIndex{1}\) shows that we need 2 hydrogen atoms and 1 oxygen atom to make one water molecule. If we want to make two water molecules, we will need 4 hydrogen atoms and 2 oxygen atoms. If we want to make five molecules of water, we need 10 hydrogen atoms and 5 oxygen atoms. The ratio of atoms we will need to make any number of water molecules is the same: 2 hydrogen atoms to 1 oxygen atom.
Using formulas to indicate how many atoms of each element we have in a substance, we can relate the number of moles of molecules to the number of moles of atoms. For example, in 1 mol of water (H_{2}O) we can construct the relationships given in (Table \(\PageIndex{1}\)).
1 Molecule of \(H_2O\) Has  1 Mol of \(H_2O\) Has  Molecular Relationships 

2 H atoms  2 mol of H atoms  \(\mathrm{\dfrac{2\: mol\: H\: atoms}{1\: mol\: H_2O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: H_2O\: molecules}{2\: mol\: H\: atoms}}\) 
1 O atom  1 mol of O atoms  \(\mathrm{\dfrac{1\: mol\: O\: atoms}{1\: mol\: H_2O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: H_2O\: molecules}{1\: mol\: O\: atoms}}\) 
The Mole is big
A mole represents a very large number! The number 602,214,129,000,000,000,000,000 looks about twice as long as a trillion, which means it’s about a trillion trillion.
(CC BYSA NC; https://whatif.xkcd.com/4/).
A trillion trillion kilograms is how much a planet weighs. If 1 mol of quarters were stacked in a column, it could stretch back and forth between Earth and the sun 6.8 billion times.
1 Molecule of \(C_2H_6O\) Has  1 Mol of \(C_2H_6O\) Has  Molecular and Mass Relationships 

2 C atoms  2 mol of C atoms  \(\mathrm{\dfrac{2\: mol\: C\: atoms}{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{2\: mol\: C\: atoms}}\) 
6 H atoms  6 mol of H atoms  \(\mathrm{\dfrac{6\: mol\: H\: atoms}{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{6\: mol\: H\: atoms}}\) 
1 O atom  1 mol of O atoms  \(\mathrm{\dfrac{1\: mol\: O\: atoms}{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{1\: mol\: O\: atoms}}\) 
2 (12.01 amu) C 24.02 amu C 
2 (12.01 g) C 24.02 g C 
\(\mathrm{\dfrac{24.02\: g\: C\: }{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{24.02\: g\: C\: }}\) 
6 (1.008 amu) H 6.048 amu H 
6 (1.008 g) H 6.048 g H 
\(\mathrm{\dfrac{6.048\: g\: H\: }{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{6.048\: g\: H\: }}\) 
1 (16.00 amu) O 16.00 amu O 
1 (16.00 g) O 16.00 g O 
\(\mathrm{\dfrac{16.00\: g\: O\: }{1\: mol\: C_2H_6O\: molecules}}\) or \(\mathrm{\dfrac{1\: mol\: C_2H_6O\: molecules}{16.00\: g\: O\: }}\) 
The following example illustrates how we can use the relationships in Table \(\PageIndex{2}\) as conversion factors.
Example \(\PageIndex{1}\): Ethanol
If a sample consists of 2.5 mol of ethanol (C_{2}H_{6}O), how many moles of carbon atoms does it have?
Solution
Steps for Problem Solving 
If a sample consists of 2.5 mol of ethanol (C_{2}H_{6}O), how many moles of carbon atoms does it have? 

Identify the "given" information and what the problem is asking you to "find." 
Given: 2.5 mol C_{2}H_{6}O 
List other known quantities. 
1 mol C_{2}H_{6}O = 2 mol C 
Prepare a concept map and use the proper conversion factor. 

Cancel units and calculate. 
Note how the unit mol C_{2}H_{6}O molecules cancels algebraically. \(\mathrm{2.5\: \cancel{mol\: C_2H_6O\: molecules}\times\dfrac{2\: mol\: C\: atoms}{1\: \cancel{mol\: C_2H_6O\: molecules}}=5.0\: mol\: C\: atoms}\) 
Think about your result.  There are twice as many C atoms in one C_{2}H_{6}O molecule, so the final amount should be double. 
Exercise \(\PageIndex{1}\)
If a sample contains 6.75 mol of Na_{2}SO_{4}, how many moles of sodium atoms, sulfur atoms, and oxygen atoms does it have?
 Answer
 13.5 mol Na atoms, 6.75 mol S atoms, and 27.0 mol O atoms
The fact that 1 mol equals 6.022 × 10^{23} items can also be used as a conversion factor.
Example \(\PageIndex{2}\): Oxygen Mass
Determine the mass of Oxygen in 75.0g of C_{2}H_{6}O.
Solution
Steps for Problem Solving 
Determine the mass of Oxygen in 75.0g of C_{2}H_{6}O 

Identify the "given" information and what the problem is asking you to "find." 
Given: 75.0g C_{2}H_{6}O 
List other known quantities. 
1 mol O = 16.0g O 1 mol C_{2}H_{6}O = 1 mol O 1 mol C_{2}H_{6}O = 46.07g C_{2}H_{6}O 
Prepare a concept map and use the proper conversion factor. 

Cancel units and calculate. 
\(\require{cancel}\mathrm{75.0\: \cancel{g\: C_2H_6O}\times\dfrac{1\: \cancel{mol\: C_2H_6O}}{46.07\:\cancel{g\: C_2H_6O}}\times\dfrac{1\: \cancel{mol\:O}}{1\: \cancel{mol\:C_2H_6O}}\times\dfrac{16.00\: g\: O}{1\: \cancel{mol\:O}}=26.0\: g\: O}\) 
Think about your result. 
Exercise \(\PageIndex{2}\)
 How many molecules are present in 16.02 mol of C_{4}H_{10}? How many C atoms are in 16.02 mol?
 How many moles of each type of atom are in 2.58 mol of Na_{2}SO_{4}?
 Answer a:
 9.647 x 10^{24} C_{4}H_{10 }molecules and 3.859 x 10^{25} C atoms
 Answer b:
 5.16 mol Na atoms, 2.58 mol S atoms, and 10.3 mol O atoms
Summary
In any given formula, the ratio of the number of moles of molecules (or formula units) to the number of moles of atoms can be used as a conversion factor.