Mole and Number of Particles Calculations (n = N/N_A)Chemistry Tutorial

⚛ 1 mole of any substance contains 6.022 × 10²³ particles.

⚛ 6.022 × 10²³ 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 × 10²³ particles mol^-1

N = n × (6.022 × 10²³)

⚛ To calculate the number of particles, N, in a substance:

N = n × N_A

N = n × (6.022 × 10²³)

⚛ To calculate the amount of substance in moles, n :

n = N ÷ N_A

n = N ÷ (6.022 × 10²³)

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1 mole of a pure substance contains N_A 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

Imagine we now add another mole of helium gas, He(g), also represented in the diagram on the right as an X

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:

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 × 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 × 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_aY_b:

⚛ 1 molecule of compound X_aY_b contains

a atoms of element X

b atoms of element Y

⚛ 1 mole of compound X_aY_b contains

a moles of atoms of element X

b moles of atoms of element Y

⚛ n moles of compound X_aY_b contains

(n × a) moles of atoms of element X

(n × b) moles of atoms of element Y

⚛ n moles of compound X_aY_b contains

(n × a) × N_A atoms of element X

(n × b) × N_A atoms of element Y

⚛ n moles of compound X_aY_b 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:

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:

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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 × 10²³ 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 or 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):

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 ÷ 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_aY_b:

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) ÷ N_A

moles of atoms of element Y = n(Y) = (N × b) ÷ N_A

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

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The solutions to these problems are given as a set of 6 general steps to help structure your approach to problem solving:

1. Step 1: What is the question asking you to do?
2. Step 2: What information (data) has been given in the question?
3. Step 3: What is the general relationship between the moles of particles and the number of particles?
4. Step 4: Write the equation for the specific relationship you need to solve the problem
5. Step 5: Substitute in the vales and solve for the unknown
6. Step 6: Write the answer

Calculating the number of particles (N = n × N_A)

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 × N_A  

where
N = number of particles
n = moles of particles
N_A = 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) × N_A

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 × N_A  

where
N = number of particles
n = moles of particles
N_A = 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) × N_A

where
N(H₂O molecules) = number of water molecules
n(H₂O molecules) = moles of water molecules = 1.5 mol
N_A = 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/N_A)

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 ÷ N_A
where N_A = 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) ÷ N_A

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 ÷ N_A
where N_A = 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) ÷ N_A
where N_A = 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):

Step 5: Substitute in the values and solve the equation:

(Note: 2 significant figures are justified)

Step 6: Write the answer

moles of oxygen atoms = 0.080 mol

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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:

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Which of the following pure substances contains the lowest number of oxygen atoms?

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