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Deriving and Applying the Equation N = n × NA
1 mole of a pure substance contains NA particles, or 6.022 × 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:
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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:
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The number of helium atoms (N) in the box is equal to the moles of helium atoms (n) multiplied by the Avogadro number (NA):
N = n × NA
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 (NA) as shown in the table below:
n (amount of substance in moles) | × | NA (Avogadro's number) | = | N (number of particles) |
---|---|---|---|---|
1 mol | × | 6.022 × 1023 | = | 6.022 × 1023 particles |
2 mol | × | 6.022 × 1023 | = | 1.204 × 1024 particles |
10 mol | × | 6.022 × 1023 | = | 6.022 × 1024 particles |
0.5 mol | × | 6.022 × 1023 | = | 3.011 × 1023 particles |
The mathematical equation, N = n × NA, 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 × 1023) = 6.022 × 1024 helium atoms
- 10 moles of sodium ions = 10 × (6.022 × 1023) = 6.022 × 1024 sodium ions
- 10 moles of water molecules = 10 × (6.022 × 1023) = 6.022 × 1024 water molecules
The mathematical equation, N = n × NA, 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 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 × a) moles of atoms of element X
(n × b) moles of atoms of element Y
⚛ n moles of compound XaYb contains
(n × a) × NA atoms of element X
(n × b) × NA atoms of element Y
⚛ n moles of compound XaYb contains
(n × a) × 6.022 × 1023 atoms of element X
(n × b) × 6.022 × 1023 atoms of element Y
Consider n moles of each of these compounds with the general formula XY2.
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:
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 × 1 = n mol of C atoms | n × NA atoms of C | n × 2 = 2n mol of O atoms | 2n × NA atoms of O |
NO2 | n | n × 1 = n mol of N atoms | n × NA atoms of N | n × 2 = 2n mol of O atoms | 2n × NA atoms of O |
SCl2 | n | n × 1 = n mol of S atoms | n × NA atoms of S | n × 2 = 2n mol of Cl atoms | 2n × NA 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:
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 × 1 = 5 mol of C | 5 × NA C atoms | 5 × 2 = 10 mol of O | 10 × NA O atoms |
NO2 | 5 | 5 × 1 = 5 mol of N | 5 × NA N atoms | 5 × 2 = 10 mol of O | 10 × NA O atoms |
SCl2 | 5 | 5 × 1 = 5 mol of S | 5 × NA S atoms | 5 × 2 = 10 mol of Cl | 10 × NA Cl atoms |
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Deriving and Applying the Equation n = N ÷ NA (n=N/NA)
In the previous section we derived the mathematical equation:
N = n × NA
where:
N = the number of particles present in the substance
n = the amount of particles in the substance in moles (mol)
NA = Avogadro number = 6.022 × 1023 particles mol-1
If we divide both sides of this equation by NA as shown below:
N NA | = | n × |
We arrive at the equation shown below:
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 n=N/NA or n = N ÷ NA 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) | ÷ | NA (Avogadro's number) | = | n (moles of particles) |
---|---|---|---|---|
(3.011 × 1023) | ÷ | (6.022 × 1023) | = | 0.5 mol |
(1.204 × 1024) | ÷ | (6.022 × 1023) | = | 2 mol |
(6.022 × 1024) | ÷ | (6.022 × 1023) | = | 10 mol |
If you know a substance contains 3.011 × 1023 particles of the substance, then the moles of substance will be (3.011 × 1023) ÷ (6.022 × 1023) = 0.5 mol
3.011 × 1023 helium atoms = 0.5 mol of helium atoms
3.011 × 1023 sodium ions = 0.5 mol of sodium ions
3.011 × 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 × a
number of atoms of element Y = N(Y) = N × b
moles of atoms of element X = n(X) = (N × a) ÷ NA
moles of atoms of element Y = n(Y) = (N × b) ÷ NA
Consider the following examples in which 1.927 × 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 × 1024 | 2 × (1.927 × 1024) H atoms = 3.854 × 1024 H atoms | (3.854 × 1024) ÷ (6.022 × 1023) = 6.4 mol H atoms | 1 × (1.927 × 1024) S atoms = 1.927 × 1024 S atoms | (1.927 × 1024) ÷ (6.022 × 1023) = 3.2 mol S atoms |
H2O | 1.927 × 1024 | 2 × (1.927 × 1024) H atoms = 3.854 × 1024 H atoms | (3.854 × 1024) ÷ (6.022 × 1023) = 6.4 mol H atoms | 1 × (1.927 × 1024) O atoms = 1.927 × 1024 O atoms | (1.927 × 1024) ÷ (6.022 × 1023) = 3.2 mol O atoms |
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Worked Examples: Moles-Avogadro Number Calculations
The solutions to these problems are given as a set of 6 general steps to help structure your approach to problem solving:
- Step 1: What is the question asking you to do?
- Step 2: What information (data) has been given in the question?
- Step 3: What is the general relationship between the moles of particles and the number of particles?
- Step 4: Write the equation for the specific relationship you need to solve the problem
- Step 5: Substitute in the vales and solve for the unknown
- Step 6: Write the answer
Calculating the number of particles (N = n × NA)
Question 1: Calculate the number of ammonia, NH3, molecules in 3.5 moles of ammonia.
Solution:
Step 1: What is the question asking you to do?
Calculate the number of ammonia molecules.
N(ammonia) = number of ammonia molecules = ?
Step 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
Step 3: What is the relationship between the moles of particles and the number of particles?
N = n × NA
where
N = number of particles
n = moles of particles
NA = Avogadro number = 6.022 × 1023
Step 4: Write the equation for the relationship between between moles of ammonia molecules and number of ammonia molecules:
N(NH3) = n(NH3) × NA
N(NH3) = n(NH3) × (6.022 x 1023)
Step 5: Substitute in the vales and solve for N:
N(NH3) = 3.5 × (6.022 × 1023)
N(NH3) = 2.1 × 1024 ammonia (NH3) molecules
(Note: 2 significant figures are justified)
Step 6: Write the answer
number of ammonia molecules = 2.1 × 1024 molecules
Question 2. Determine the number of hydrogen atoms in 1.5 moles of water, H2O, molecules.
Solution:
Step 1: What is the question asking you to do?
Calculate the number of hydrogen atoms.
N(H atoms) = number of hydrogen atoms = ?
Step 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
Step 3: What is the relationship between moles of particles and number of particles?
N = n × NA
where
N = number of particles
n = moles of particles
NA = Avogadro number = 6.022 × 1023
Step 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) × NA
where
N(H2O molecules) = number of water molecules
n(H2O molecules) = moles of water molecules = 1.5 mol
NA = Avogadro number = 6.022 × 1023
N(H2O molecules) = 1.5 × (6.022 × 1023) = 9.033 × 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 × N(H2O molecules)
Step 5: Substitute in the values and solve the equation:
N(H atoms) = 2 × N(H2O molecules)
N(H atoms) = 2 × (9.033 × 1023)
N(H atoms) = 1.8 × 1024 hydrogen atoms
(Note: 2 significant figures are justified)
Step 6: Write the answer
number of hydrogen atoms = 1.8 × 1024 hydrogen atoms
Calculating the moles of substance (n=N/NA)
Question 1. A sample of gas contains 4.4 × 1024 carbon dioxide molecules.
How many moles of carbon dioxide molecules are present in the sample?
Solution:
Step 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 = ? mol
Step 2: What information (data) has been given in the question?
N(carbon dioxide molecules) = number of carbon dioxide molecules
N(carbon dioxide molecules) = 4.4 × 1024 carbon dioxide molecules
Step 3: What is the relationship between moles (n) of particles and number (N) of particles?
n = N ÷ NA
where NA = Avogadro number = 6.022 × 1023
Step 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) = N(carbon dioxide molecules) ÷ (6.022 × 1023)
Step 5: Substitute the values into the equation and solve:
n(carbon dioxide molecules) = N(carbon dioxide molecules) ÷ (6.022 × 1023)
n(carbon dioxide molecules) = (4.4 × 1024) ÷ (6.022 × 1023)
n(carbon dioxide molecules) = 7.3 moles of carbon dioxide molecules
(Note: 2 significant figures are justified)
Step 6: Write the answer
moles of carbon dioxide molecules = 7.3 mol
Question 2. A sample contains 2.4 × 1022 molecules of oxygen gas (O2).
How many moles of oxygen atoms are present in the sample?
Solution:
Step 1: What is the question asking you to do?
Calculate the moles of oxygen atoms.
n(O atoms) = moles of oxygen atoms = ? mol
Step 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 × 1022
Step 3: What is the relationship between moles (n) of particles and number (N) of particles?
n = N ÷ NA
where NA = Avogadro number = 6.022 × 1023
Step 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 number = 6.022 × 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 × number of oxygen molecules
N(O atoms) = 2 × 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 × N(O2 molecules) NA | |
= | 2 × N(O2 molecules) 6.022 × 1023 |
Step 5: Substitute in the values and solve the equation:
n(O atoms) | = | 2 × N(O2 molecules) 6.022 × 1023 |
= | 2 × (2.4 × 1022) 6.022 × 1023 | |
= | 0.080 moles of oxygen atoms |
(Note: 2 significant figures are justified)
Step 6: Write the answer
moles of oxygen atoms = 0.080 mol
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