Mole-Number of Particles Calculations

Key Concepts

• 1 mole of any substance contains 6.022 x 1023 particles.

• 6.022 x 1023 is known as the Avogadro Number or Avogadro Constant and is given the symbol NA1

• N = n x NA
where N = number of particles in the substance
and n = amount of substance in moles (mol)
and NA = Avogardro Number = 6.022 x 1023 particles mol-1

• To find the number of particles, N, in a substance: N = n x NA

• To find the moles, n, of substance, n = N ÷ NA

Deriving and Applying the Equation N = n x NA

1 mole of a pure substance contains NA particles, or 6.022 x 1023 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
• = NA He atoms
• = 6.022 x 1023 He atoms
 X

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
• = NA + NA = 2 x NA He atoms
• = 6.022 x 1023 + 6.022 x 1023 = 2 x 6.022 x 1023 He atoms
 X X

The number of helium atoms (N) in the box is equal to the moles of helium atoms (n) multiplied by Avogadro's number (NA):

N = n x NA

We can use this equation (formula or 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 (NA):

n
(amount of substance in moles)
x NA
= N
(number of particles)
1 mol x 6.022 x 1023 = 6.022 x 1023 particles
2 mol x 6.022 x 1023 = 1.204 x 1024 particles
10 mol x 6.022 x 1023 = 6.022 x 1024 particles
0.5 mol x 6.022 x 1023 = 3.011 x 1023 particles

The equation, N = n x NA, can be used to find the number of atoms, ions or molecules in any amount (moles) of atoms, ions or molecules:

• 10 moles of helium atoms = 10 x 6.022 x 1023 = 6.022 x 1024 helium atoms
• 10 moles of sodium ions = 10 x 6.022 x 1023 = 6.022 x 1024 sodium ions
• 10 moles of water molecules = 10 x 6.022 x 1023 = 6.022 x 1024 water molecules

The equation, N = n x NA, can also be used to find the number of atoms of each element in a known amount (moles) of a compound.
For a compound with the molecular formula XaYb:

• 1 molecule of compound XaYb contains
a atoms of element X
b atoms of element Y

• 1 mole of compound XaYb contains
a moles of atoms of element X
b moles of atoms of element Y

• n moles of compound XaYb contains
(n x a) moles of atoms of element X
(n x b) moles of atoms of element Y

• n moles of compound XaYb contains
(n x a) x NA atoms of element X
(n x b) x NA atoms of element Y

• n moles of compound XaYb contains
(n x a) x 6.022 x 1023 atoms of element X
(n x b) x 6.022 x 1023 atoms of element Y

Consider n moles of each of these compounds with the general formula XY2:

XY2
formula
n(XY2)
moles of XY2
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
CO2 n n x 1
= n mol of C atoms
n x NA atoms of C n x 2
= 2n mol of O atoms
2n x NA atoms of O
NO2 n n x 1
= n mol of N atoms
n x NA atoms of N n x 2
= 2n mol of O atoms
2n x NA atoms of O
SCl2 n n x 1
= n mol of S atoms
n x NA atoms of S n x 2
= 2n mol of Cl atoms
2n x NA atoms of Cl

If we have 5 moles of each the compounds above then we can calculate the moles of each element, and the number of atoms of each element:

XY2
formula
n(XY2)
moles of XY2
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
CO2 5 5 x 1
= 5 mol of C
5 x NA C atoms 5 x 2
= 10 mol of O
10 x NA O atoms
NO2 5 5 x 1
= 5 mol of N
5 x NA N atoms 5 x 2
= 10 mol of O
10 x NA O atoms
SCl2 5 5 x 1
= 5 mol of S
5 x NA S atoms 5 x 2
= 10 mol of Cl
10 x NA Cl atoms

Deriving and Applying the Equation n = N ÷ NA

In the previous section we derived the equation:

N = n x NA
where:
N = the number of particles present in the substance
n = the amount of particles in the substance in moles (mol)
NA = Avogadro's number = 6.022 x 1023 particles

If we divide both sides of this equation by NA:

 N   NA = n x NANA

We arrive at the equation:
 N   NA = n

which we can use to find the moles of substance if we know how many particles of the substance are present.
The equation 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)

N
(number of particles)
÷ NA
= n
(moles of particles)
3.011 x 1023 ÷ 6.022 x 1023 = 0.5 mol
1.204 x 1024 ÷ 6.022 x 1023 = 2 mol
6.022 x 1024 ÷ 6.022 x 1023 = 10 mol

3.011 x 1023 helium atoms = 0.5 mol of helium atoms
3.011 x 1023 sodium ions = 0.5 mol of sodium ions
3.011 x 1023 water molecules = 0.5 mol of water molecules

The equation n = N ÷ NA 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 XaYb:

• number of atoms of element X = N(X) = N x a
number of atoms of element Y = N(Y) = N x b

• moles of atoms of element X = n(X) = (N x a) ÷ NA
moles of atoms of element Y = n(Y) = (N x b) ÷ NA

Consider the following examples in which 1.927 x 1024 molecules of a compound with the general formula X2Y are present

X2Y
formula
N(X2Y)
(number of X2Y 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)
H2S 1.927 x 1024 2 x 1.927 x 1024 H atoms
= 3.854 x 1024 H atoms
3.854 x 1024 ÷ 6.022 x 1023
= 6.4 mol H atoms
1 x 1.927 x 1024 S atoms
= 1.927 x 1024 S atoms
1.927 x 1024 ÷ 6.022 x 1023
= 3.2 mol S atoms
H2O 1.927 x 1024 2 x 1.927 x 1024 H atoms
= 3.854 x 1024 H atoms
3.854 x 1024 ÷ 6.022 x 1023
= 6.4 mol H atoms
1 x 1.927 x 1024 O atoms
= 1.927 x 1024 O atoms
1.927 x 1024 ÷ 6.022 x 1023
= 3.2 mol O atoms

Examples

Find the number of particles

1. Calculate the number of ammonia, NH3, molecules in 3.5 moles of ammonia.
1. What is the question asking you to do?
Calculate the number of ammonia molecules.
N(ammonia) = number of ammonia molecules = ?

2. What information (data) has been given in the question?
molecular formula for ammonia: NH3
n = amount of ammonia molecules in moles = 3.5 mol

3. What is the relationship between the moles of particles and the number of particles?

 N = n x NA where N = number of particles n = moles of particles NA = Avogadro's number = 6.022 x 1023

4. Write the equation for the relationship between between moles of ammonia molecules and number of ammonia molecules:
N(NH3) = n(NH3) x NA
= n(NH3) x 6.022 x 1023

5. Substitute in the vales and solve:
N(NH3) = 3.5 x 6.022 x 1023
= 2.1 x 1024 ammonia (NH3) molecules

2. Determine the number of hydrogen atoms in 1.5 moles of water, H2O, molecules.

1. What is the question asking you to do?
Calculate the number of hydrogen atoms.
N(H atoms) = number of hydrogen atoms = ?

2. What information (data) has been given in the question?
molecular formula for water: H2O
n(H2O molecules) = moles of water molecules = 1.5 mol

3. What is the relationship between moles of particles and number of particles?

 N = n x NA where N = number of particles n = moles of particles NA = Avogadro's number = 6.022 x 1023

4. 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(H2O molecules) = n(H2O molecules) x NA

 where N(H2O molecules) = number of water molecules n(H2O molecules) = moles of water molecules = 1.5 mol NA = Avogadro's number = 6.022 x 1023

 N(H2O molecules) = 1.5 x 6.022 x 1023 = 9.033 x 1023

(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 x N(H2O molecules)

5. Substitute in the values and solve the equation:
N(H atoms) = 2 x N(H2O molecules)
= 2 x 9.033 x 1023 = 1.8 x 1024 hydrogen atoms

Find the moles of substance

1. A sample of gas contains 4.4 x 1024 carbon dioxide molecules.
How many moles of carbon dioxide molecules are present in the sample?
1. What is the question asking you to do?
Calculate the moles of carbon dioxide molecules.
n(carbon dioxide molecules) = moles of carbon dioxide molecules = ?

2. What information (data) has been given in the question?
 N(carbon dioxide molecules) = number of carbon dioxide molecules = 4.4 x 1024 carbon dioxide molecules

3. What is the relationship between moles (n) of particles and number (N) of particles?
 n = N   NA where NA = Avogadro's number = 6.022 x 1023

4. 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) NA = N(carbon dioxide molecules) 6.022 x 1023

5. Substitute the values into the equation and solve:
 n(carbon dioxide molecules) = N(carbon dioxide molecules) 6.022 x 1023 = 4.4 x 1024   6.022 x 1023 = 7.3 carbon dioxide molecules

2. A sample contains 2.4 x 1022 molecules of oxygen gas (O2).
How many moles of oxygen atoms are present in the sample?

1. What is the question asking you to do?
Calculate the moles of oxygen atoms.
n(O atoms) = moles of oxygen atoms = ?

2. What information (data) has been given in the question?
molecular formula for oxygen gas: O2
N(O2 molecules) = number of oxygen molecules (O2) = 2.4 x 1022

3. What is the relationship between moles (n) of particles and number (N) of particles?
 n = N   NA where NA = Avogadro's number = 6.022 x 1023

4. What is the relationship between moles of oxygen atoms, n(O atoms), and number of oxygen molecules, N(O2 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)   NA where NA = Avogadro's number = 6.022 x 1023

(ii) relationship between number of oxygen molecules, N(O2 molecules), and number of oxygen atoms, N(O atoms):
One O2 molecule is made up of 2 oxygen atoms
number of oxygen atoms = 2 x number of oxygen molecules
N(O atoms) = 2 x N(O2 molecules)

(iii) relationship between moles of oxygen atoms, n(O atoms), and number of oxygen molecules, N(O2 molecules):

 n(O atoms) = N(O atoms)   NA = 2 x N(O2 molecules)   NA = 2 x N(O2 molecules)   6.022 x 1023

5. Substitute in the values and solve the equation:
 n(O atoms) = 2 x N(O2 molecules)   6.022 x 1023 = 2 x 2.4 x 1022   6.022 x 1023 = 0.080 moles of oxygen atoms

Problem Solving Using Moles, Number of Particles, and, Avogadro's 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 de-ionised 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 = ?

What chemical principle will you need to apply?
Apply stoichoimetry (N = n x NA)

What information (data) have you been given?

• volume of solution = 1 L

• solvent is de-ionised water

• solute is a mixture of sodium chloride and potassium chloride

• formula of sodium chloride: NaCl

• formula of potassium chloride: KCl

• n(Cl-) = moles of chloride ions = 0.50 mol

• N(Na+) = N(K+)
number of sodium ions = number of potassium ions
PAUSE!
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-) x NA
Use a data sheet to find the value of Avogadro's Number, NA:
NA =

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-) x NA
Use a data sheet to find the value of Avogadro's Number, NA:
NA = 6.022 x 1023
N(Cl- in solution) = 0.50 x 6.022 x 1023
= 3.011 x 1023 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 x N(Na+)

Step 3: Solve the equation to determine the number of sodium ions
N(Cl- in solution) = 2 x N(Na+)
3.011 x 1023 = 2 x N(Na+)

 3.011 x 10232 = 2 x N(Na+)2 1.5 x 1023 = N(Na+)
PAUSE!
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 x 1023
N(Cl- from NaCl) = N(Na+) = 1.5 x 1023
N(K+) = N(Na+) = 1.5 x 1023
N(Cl- from KCl) = 1.5 x 1023
total number of Cl- in solution = N(Cl- from NaCl) + N(Cl- from KCl) = 1.5 x 1023 + 1.5 x 1023 = 3.0 x 1023
n(Cl-) = N(Cl- in solution) ÷ NA = 3.0 x 1023 ÷ 6.022 x 1023 = 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. There are 1.5 x 1023 sodium ions in the solution.

What would you like to do now?

1Avogadro's number is sometimes referred to as Loschmidt's number and is given the symbol L.

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