Mole and Number of Particles Calculations (n = N/N_{A}) Chemistry Tutorial
Key Concepts
 1 mole of any substance contains 6.022 × 10^{23} particles.
 6.022 × 10^{23} is known as the Avogadro Number or Avogadro Constant and is given the symbol N_{A}^{(1)}
 N = n × N_{A}
N = number of particles in the substance
n = amount of substance in moles (mol)
N_{A} = Avogardro Number = 6.022 × 10^{23} particles mol^{1}
N = n × (6.022 × 10^{23})
 To find the number of particles, N, in a substance:
N = n × N_{A}
N = n × (6.022 × 10^{23})
 To find the amount of substance in moles, n :
n = N ÷ N_{A}
n = N ÷ (6.022 × 10^{23})
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Deriving and Applying the Equation N = n × N_{A}
1 mole of a pure substance contains N_{A} particles, or 6.022 × 10^{23} particles.
Imagine a box containing 1 mole of helium gas, He(g), represented in the diagram on the right as an X
This box contains:
 1 mole of He atoms
 = Avogadro number of He atoms
 = N_{A} He atoms
 = 6.022 × 10^{23} He atoms


Imagine we now add another mole of helium gas, He(g), also represented in the diagram on the right as an X
This box now contains:
 1 + 1 = 2 moles of He atoms
 = Avogadro number + Avogadro number = 2 × Avogadro number of He atoms
 = N_{A} + N_{A} = 2 × N_{A} He atoms
 = (6.022 × 10^{23}) + (6.022 × 10^{23}) = 2 × (6.022 × 10^{23}) He atoms


The number of helium atoms (N) in the box is equal to the moles of helium atoms (n) multiplied by the Avogadro number (N_{A}):
N = n × N_{A}
We can use this mathematical equation (mathematical formula or mathematical expression) to find the number of particles (N) in any amount of substance (n) just by multiplying the amount in moles (n) by the Avogadro number (N_{A}) as shown in the table below:
n (amount of substance in moles) 
× 
N_{A} (Avogadro's number) 
= 
N (number of particles) 
1 mol 
× 
6.022 × 10^{23} 
=
 6.022 × 10^{23} particles 
2 mol 
× 
6.022 × 10^{23} 
=
 1.204 × 10^{24} particles 
10 mol 
× 
6.022 × 10^{23} 
=
 6.022 × 10^{24} particles 
0.5 mol 
× 
6.022 × 10^{23} 
=
 3.011 × 10^{23} particles 
The mathematical equation, N = n × N_{A}, can be used to find the number of atoms, ions or molecules in any amount (in moles) of atoms, ions or molecules:
 10 moles of helium atoms = 10 × (6.022 × 10^{23}) = 6.022 × 10^{24} helium atoms
 10 moles of sodium ions = 10 × (6.022 × 10^{23}) = 6.022 × 10^{24} sodium ions
 10 moles of water molecules = 10 × (6.022 × 10^{23}) = 6.022 × 10^{24} water molecules
The mathematical equation, N = n × N_{A}, can also be used to find the number of atoms of each element in a known amount (in moles) of a compound.
For a compound with the molecular formula X_{a}Y_{b}:
 1 molecule of compound X_{a}Y_{b} contains
a atoms of element X
b atoms of element Y
 1 mole of compound X_{a}Y_{b} contains
a moles of atoms of element X
b moles of atoms of element Y
 n moles of compound X_{a}Y_{b} contains
(n × a) moles of atoms of element X
(n × b) moles of atoms of element Y
 n moles of compound X_{a}Y_{b} contains
(n × a) × N_{A} atoms of element X
(n × b) × N_{A} atoms of element Y
 n moles of compound X_{a}Y_{b} contains
(n × a) × 6.022 × 10^{23} atoms of element X
(n × b) × 6.022 × 10^{23} atoms of element Y
Consider n moles of each of these compounds with the general formula XY_{2}.
The table below gives the moles of each element present in the compound, and also shows us how to calculate the number of atoms of each element present:
XY_{2} formula 
n(XY_{2}) moles of XY_{2} 
n(X) moles of atoms of element X 
N(X) number of X atoms 
n(Y) moles of atoms of element Y 
N(Y) number of Y atoms 
CO_{2} 
n 
n × 1 = n mol of C atoms 
n × N_{A} atoms of C 
n × 2 = 2n mol of O atoms 
2n × N_{A} atoms of O 
NO_{2} 
n 
n × 1 = n mol of N atoms 
n × N_{A} atoms of N 
n × 2 = 2n mol of O atoms 
2n × N_{A} atoms of O 
SCl_{2} 
n 
n × 1 = n mol of S atoms 
n × N_{A} atoms of S 
n × 2 = 2n mol of Cl atoms 
2n × N_{A} atoms of Cl 
If we have 5 moles of each the compounds above, for example, then we can calculate the moles of each element, and the number of atoms of each element as shown in the table below:
XY_{2} formula 
n(XY_{2}) moles of XY_{2} 
n(X) moles of atoms of element X 
N(X) number of X atoms 
n(Y) moles of atoms of element Y 
N(Y) number of Y atoms 
CO_{2} 
5 
5 × 1 = 5 mol of C 
5 × N_{A} C atoms 
5 × 2 = 10 mol of O 
10 × N_{A} O atoms 
NO_{2} 
5 
5 × 1 = 5 mol of N 
5 × N_{A} N atoms 
5 × 2 = 10 mol of O 
10 × N_{A} O atoms 
SCl_{2} 
5 
5 × 1 = 5 mol of S 
5 × N_{A} S atoms 
5 × 2 = 10 mol of Cl 
10 × N_{A} Cl atoms 
Deriving and Applying the Equation n = N ÷ N_{A} (n=N/N_{A})
In the previous section we derived the mathematical equation:
N = n × N_{A}
where:
N = the number of particles present in the substance
n = the amount of particles in the substance in moles (mol)
N_{A} = Avogadro number = 6.022 × 10^{23} particles mol^{1}
If we divide both sides of this equation by N_{A} as shown below:
We arrive at the equation shown below:
which we can use to find the moles of substance if we know how many particles of the substance are present.
The equation n=N/N_{A} can be used to calculate:
 moles of atoms (n) if you know the number of atoms present (N)
 moles of ions (n) if you know the number of ions present (N)
 moles of molecules (n) if you know the number of molecules present (N)
The table below shows the calculation of moles (n) given then number of particles (N):
N (number of particles) 
÷ 
N_{A} (Avogadro's number) 
= 
n (moles of particles) 
(3.011 × 10^{23}) 
÷ 
(6.022 × 10^{23}) 
= 
0.5 mol 
(1.204 × 10^{24}) 
÷ 
(6.022 × 10^{23}) 
= 
2 mol 
(6.022 × 10^{24}) 
÷ 
(6.022 × 10^{23}) 
= 
10 mol 
If you know a substance contains 3.011 × 10^{23} particles of the substance, then the moles of substance will be (3.011 × 10^{23}) ÷ (6.022 × 10^{23}) = 0.5 mol
3.011 × 10^{23} helium atoms = 0.5 mol of helium atoms
3.011 × 10^{23} sodium ions = 0.5 mol of sodium ions
3.011 × 10^{23} water molecules = 0.5 mol of water molecules
The equation n = N ÷ N_{A} can also be used to find the amount in moles of atoms or ions in a compound if you know both the molecular formula for the compound and the number of molecules of the compound that are present.
For N molecules of a compound with the general formula X_{a}Y_{b}:
Consider the following examples in which 1.927 × 10^{24} molecules of a compound with the general formula X_{2}Y are present
X_{2}Y formula 
N(X_{2}Y) (number of X_{2}Y molecules) 
N(X) (number of atoms of element X) 
n(X) (moles of X atoms) 
N(Y) (number of atoms of element Y) 
n(Y) (moles of Y atoms) 
H_{2}S 
1.927 × 10^{24} 
2 × (1.927 × 10^{24}) H atoms = 3.854 × 10^{24} H atoms 
(3.854 × 10^{24}) ÷ (6.022 × 10^{23}) = 6.4 mol H atoms 
1 × (1.927 × 10^{24}) S atoms = 1.927 × 10^{24} S atoms 
(1.927 × 10^{24}) ÷ (6.022 × 10^{23}) = 3.2 mol S atoms 
H_{2}O 
1.927 × 10^{24} 
2 × (1.927 × 10^{24}) H atoms = 3.854 × 10^{24} H atoms 
(3.854 × 10^{24}) ÷ (6.022 × 10^{23}) = 6.4 mol H atoms 
1 × (1.927 × 10^{24}) O atoms = 1.927 × 10^{24} O atoms 
(1.927 × 10^{24}) ÷ (6.022 × 10^{23}) = 3.2 mol O atoms 
Worked Examples of MolesAvogadro Number Calculations
Calculating the number of particles (N = n × N_{A})
Question 1: Calculate the number of ammonia, NH_{3}, molecules in 3.5 moles of ammonia.
Solution:
 What is the question asking you to do?
Calculate the number of ammonia molecules.
N(ammonia) = number of ammonia molecules = ?
 What information (data) has been given in the question?
molecular formula for ammonia: NH_{3}
n = amount of ammonia molecules in moles = 3.5 mol
 What is the relationship between the moles of particles and the number of particles?
N = n × N_{A} 
where  
 N = number of particles 
 n = moles of particles 
 N_{A} = Avogadro number = 6.022 × 10^{23} 
 Write the equation for the relationship between between moles of ammonia molecules and number of ammonia molecules:
N(NH_{3}) = n(NH_{3}) × N_{A}
= n(NH_{3}) × (6.022 x 10^{23})
 Substitute in the vales and solve for N:
N(NH_{3}) = 3.5 × (6.022 × 10^{23})
= 2.1 × 10^{24} ammonia (NH_{3}) molecules
Question 2. Determine the number of hydrogen atoms in 1.5 moles of water, H_{2}O, molecules.
Solution:
 What is the question asking you to do?
Calculate the number of hydrogen atoms.
N(H atoms) = number of hydrogen atoms = ?
 What information (data) has been given in the question?
molecular formula for water: H_{2}O
n(H_{2}O molecules) = moles of water molecules = 1.5 mol
 What is the relationship between moles of particles and number of particles?
N = n × N_{A} 
where  
 N = number of particles 
 n = moles of particles 
 N_{A} = Avogadro number = 6.022 × 10^{23} 
 What is the relationship between moles of water molecules and number of hydrogen atoms?
(i) relationship between moles of water molecules and number of water molecules is:
N(H_{2}O molecules) = n(H_{2}O molecules) × N_{A} 
where  
 N(H_{2}O molecules) = number of water molecules 
 n(H_{2}O molecules) = moles of water molecules = 1.5 mol 
 N_{A} = Avogadro number = 6.022 × 10^{23} 
N(H_{2}O molecules) = 1.5 × (6.022 × 10^{23}) = 9.033 × 10^{23} 
(ii) relationship between number of hydrogen atoms and number of water molecules:
From the molecular formula we see that 1 molecule of water is made up of 2 atoms of hydrogen and 1 atom of oxygen.
N(H atoms) = 2 × N(H_{2}O molecules)
 Substitute in the values and solve the equation:
N(H atoms) = 2 × N(H_{2}O molecules)
= 2 × (9.033 × 10^{23})
= 1.8 × 10^{24} hydrogen atoms
Calculating the moles of substance (n=N/N_{A})
Question 1. A sample of gas contains 4.4 × 10^{24} carbon dioxide molecules.
How many moles of carbon dioxide molecules are present in the sample?
Solution:
 What is the question asking you to do?
Calculate the moles of carbon dioxide molecules.
n(carbon dioxide molecules) = moles of carbon dioxide molecules = ?
 What information (data) has been given in the question?
 N(carbon dioxide molecules) 
= number of carbon dioxide molecules 
 
= 4.4 × 10^{24} carbon dioxide molecules 
 What is the relationship between moles (n) of particles and number (N) of particles?

n 
= 
N N_{A} 
where N_{A} = Avogadro number = 6.022 × 10^{23} 
 What is the relationship between moles (n) of carbon dioxide molecules and number (N) of carbon dioxide molecules?

n(carbon dioxide molecules) 
= 
N(carbon dioxide molecules) N_{A} 


= 
N(carbon dioxide molecules) 6.022 × 10^{23} 
 Substitute the values into the equation and solve:

n(carbon dioxide molecules) 
= 
N(carbon dioxide molecules) 6.022 × 10^{23} 


= 
4.4 × 10^{24} 6.022 × 10^{23} 


= 
7.3 carbon dioxide molecules 
Question 2. A sample contains 2.4 × 10^{22} molecules of oxygen gas (O_{2}).
How many moles of oxygen atoms are present in the sample?
Solution:
 What is the question asking you to do?
Calculate the moles of oxygen atoms.
n(O atoms) = moles of oxygen atoms = ?
 What information (data) has been given in the question?
molecular formula for oxygen gas: O_{2}
N(O_{2} molecules) = number of oxygen molecules (O_{2}) = 2.4 × 10^{22}
 What is the relationship between moles (n) of particles and number (N) of particles?

n 
= 
N N_{A} 
where N_{A} = Avogadro number = 6.022 × 10^{23} 
 What is the relationship between moles of oxygen atoms, n(O atoms), and number of oxygen molecules, N(O_{2} molecules)?
(i) relationship between moles of oxygen atoms, n(O atoms), and number of oxygen atoms, N(O atoms)

n(O atoms) 
= 
N(O atoms) N_{A} 


where N_{A} 
= 
Avogadro number
 = 6.022 × 10^{23} 
(ii) relationship between number of oxygen molecules, N(O_{2} molecules), and number of oxygen atoms, N(O atoms):
One O_{2} molecule is made up of 2 oxygen atoms
number of oxygen atoms = 2 × number of oxygen molecules
N(O atoms) = 2 × N(O_{2} molecules)
(iii) relationship between moles of oxygen atoms, n(O atoms), and number of oxygen molecules, N(O_{2} molecules):

n(O atoms) 
= 
N(O atoms) N_{A} 



= 
2 × N(O_{2} molecules) N_{A} 



= 
2 × N(O_{2} molecules) 6.022 × 10^{23} 

 Substitute in the values and solve the equation:

n(O atoms) 
= 
2 × N(O_{2} molecules) 6.022 × 10^{23} 



= 
2 × (2.4 × 10^{22}) 6.022 × 10^{23} 



= 
0.080 moles of oxygen atoms 

Problem Solving Using Moles, Number of Particles, and Avogadro Number
The Problem: Bo the Biologist has been studying the effect of chloride ions, Cl^{}, on plant cells.
Bo has asked Chris the Chemist to make 1 litre of a solution containing 0.50 moles of chloride ions, Cl^{}, dissolved in water.
The solution must also contain equal numbers of sodium ions, Na^{+}, and potassium ions, K^{+}.
Chris makes the solution by dissolving some sodium chloride, NaCl, and potassium chloride, KCl, in 1 litre of deionised water.
How many sodium ions are present in the solution?
Solving the Problem using the StoPGoPS model for problem solving:
STOP!
 State the question. 
What is the question asking you to do?
Determine the number of sodium ions in the solution.
N(sodium ions) = number of sodium ions = ?

PAUSE!
 Plan. 
What chemical principle will you need to apply?
Apply stoichoimetry (N = n × N_{A})
What information (data) have you been given?
Step 1: Calculate the number of Cl^{} ions in the solution
Assume the water used to make the solution does NOT contain any chloride ions, Cl^{}.
N(Cl^{} in solution) = n(Cl^{}) × N_{A}
Use a data sheet to find the value of Avogadro Number, N_{A}:
N_{A} =
Step 2: Write an equation for the relationship between number of chloride ions, sodium ions and potassium ions
Assume the water used to make the solution does NOT contain any sodium ions, Na^{+}, or potassium ions, K^{+}
Assume the sodium chloride and potassium chloride used to make the solution are 100% pure, that is, contain no impurities.
Step 3: Solve the equation to determine the number of sodium ions

GO!
 Go with the Plan. 
Step 1: Calculate the number of Cl^{} ions in the solution
Assume the water used to make the solution does NOT contain any chloride ions, Cl^{}.
N(Cl^{} in solution) = n(Cl^{}) × N_{A}
Use a data sheet to find the value of Avogadro Number, N_{A}:
N_{A} = 6.022 × 10^{23}
N(Cl^{} in solution) = 0.50 × (6.022 × 10^{23})
= 3.011 × 10^{23} chloride ions
Step 2: Write an equation for the relationship between number of chloride ions, sodium ions and potassium ions
Assume the water used to make the solution does NOT contain any sodium ions, Na^{+}, or potassium ions, K^{+}
Assume the sodium chloride and potassium chloride used to make the solution are 100% pure, that is, contain no impurities.
N(Cl^{} in solution) = N(Cl^{} from NaCl) + N(Cl^{} from KCl)
Consider just the NaCl:
1 "molecule" of NaCl contains 1 Na^{+} and 1 Cl^{}, so
N(Cl^{} from NaCl) = N(Na^{+}) = N(NaCl)
Consider just the KCl:
1 "molecule" of KCl contains 1 K^{+} and 1 Cl^{}, so
N(Cl^{} from KCl) = N(K^{+}) = N(KCl)
Therefore:
N(Cl^{} in solution) = N(Na^{+}) + N(K^{+})
Since N(Na^{+}) = N(K^{+})
N(Cl^{} in solution) = 2 × N(Na^{+})
Step 3: Solve the equation to determine the number of sodium ions
N(Cl^{} in solution) = 2 × N(Na^{+})
3.011 × 10^{23} = 2 × N(Na^{+})
3.011 × 10^{23} 2 
= 
2 × N(Na^{+})
2 
1.5 × 10^{23} 
= 
N(Na^{+}) 

PAUSE!
 Ponder Plausability. 
Have you answered the question that was asked?
Yes, we have determined the number of sodium ions in the solution.
Is your solution to the question reasonable?
Let's work backwards to see if the number of sodium ions we have calculated will give us the correct moles of chloride ions in solution.
N(Na^{+} calculated) = 1.5 × 10^{23}
N(Cl^{} from NaCl) = N(Na^{+}) = 1.5 × 10^{23}
N(K^{+}) = N(Na^{+}) = 1.5 × 10^{23}
N(Cl^{} from KCl) = 1.5 × 10^{23}
total number of Cl^{} in solution = N(Cl^{} from NaCl) + N(Cl^{} from KCl) = (1.5 × 10^{23}) + (1.5 × 10^{23}) = 3.0 × 10^{23}
n(Cl^{}) = N(Cl^{} in solution) ÷ N_{A} = (3.0 × 10^{23}) ÷ (6.022 × 10^{23}) = 0.50 mol
Since the moles of Cl^{} in solution we have calculated is equal to the moles of Cl^{} given in the question, we are confident our answer for the number of sodium ions is correct.

STOP!
 State the solution. 
What is the number of sodium ions in the solution?
1.5 × 10^{23} sodium ions in the solution

Footnotes:
(1) The Avogadro number is sometimes referred to as Loschmidt's number and is given the symbol L.