How to calculate number of molecules

Molecules and moles are important to understand when studying chemistry and physical science. Here's an explanation of what these terms mean, how they relate to Avogadro's number, and how to use them to find molecular and formula weight.

Table of Contents Show

Key Concepts
Deriving and Applying the Equation N = n × NA
Deriving and Applying the Equation n = N ÷ NA (n=N/NA)
Worked Examples: Moles-Avogadro Number Calculations
Calculating the number of particles (N = n × NA)
Calculating the moles of substance (n=N/NA)
Problem Solving: Moles, Number of Particles, and Avogadro Number
Sample Question: Moles, Number of Particles and Avogadro's Number

A molecule is a combination of two or more atoms that are held together by chemical bonds, such as covalent bonds and ionic bonds. A molecule is the smallest unit of a compound that still displays the properties associated with that compound. Molecules may contain two atoms of the same element, such as O2 and H2, or they may consist of two or more different atoms, such as CCl4 and H2O. A chemical species consisting of a single atom or ion is not a molecule. So, for example, an H atom is not a molecule, while H2 and HCl are molecules. In the study of chemistry, molecules are usually discussed in terms of their molecular weights and moles.

A related term is a compound. In chemistry, a compound is a molecule consisting of at least two different types of atoms. All compounds are molecules, but not all molecules are compounds! Ionic compounds, such as NaCl and KBr, do not form traditional discrete molecules like those formed by covalent bonds. In their solid state, these substances form a three-dimensional array of charged particles. In such a case, molecular weight has no meaning, so the term formula weight is used instead.

The molecular weight of a molecule is calculated by adding the atomic weights (in atomic mass units or amu) of the atoms in the molecule. The formula weight of an ionic compound is calculated by adding its atomic weights according to its empirical formula.

A mole is defined as the quantity of a substance that has the same number of particles as are found in 12.000 grams of carbon-12. This number, Avogadro's number, is 6.022x1023. Avogadro's number may be applied to atoms, ions, molecules, compounds, elephants, desks, or any object. It's just a convenient number to define a mole, which makes it easier for chemists to work with very large numbers of items.

The mass in grams of one mole of a compound is equal to the molecular weight of the compound in atomic mass units. One mole of a compound contains 6.022x1023 molecules of the compound. The mass of one mole of a compound is called its molar weight or molar mass. The units for molar weight or molar mass are grams per mole. Here is the formula for determining the number of moles of a sample:

mol = weight of sample (g) / molar weight (g/mol)

Converting between molecules and moles is done by either multiplying by or dividing by Avogadro's number:

To go from moles to molecules, multiply the number of moles by 6.02 x 1023.
To go from molecules to moles, divide the numbers of molecules by 6.02 x 1023.

For example, if you know there are 3.35 x 1022 water molecules in a gram of water and want to find how many moles of water this is:

moles of water = molecules of water / Avogadro's number

moles of water = 3.35 x 1022 / 6.02 x 1023

moles of water = 0.556 x 10-1 or 0.056 moles in 1 gram of water

Key Concepts

⚛ 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 NA(1)

⚛ N = n × NA

· N = number of particles in the substance

· n = amount of substance in moles (mol)

· NA = 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 × NA

N = n × (6.022 × 1023)

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

n = N ÷ NA

n = N ÷ (6.022 × 1023)

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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:

1 mole of He atoms
= Avogadro number of He atoms
= NA He atoms
= 6.022 × 1023 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
= NA + NA = 2 × NA He atoms
= (6.022 × 1023) + (6.022 × 1023) = 2 × (6.022 × 1023) 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 (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:

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/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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Problem Solving: 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 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 = ?

PAUSE!

Pause to Plan.

What chemical principle will you need to apply?

Apply stoichoimetry (N = n × 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 )

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-) × NA
Use a data sheet to find the value of Avogadro 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-) × NA
Use a data sheet to find the value of Avogadro Number, NA:
NA = 6.022 × 1023

N(Cl- in solution) = 0.50 × (6.022 × 1023)

N(Cl- in solution) = 3.011 × 1023 chloride ions

Step 2: Write an equation for the relationship between number of chloride ions, sodium ions and potassium ions

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 × 1023 = 2 × N(Na+)

3.011 × 1023 2	=	2 × N(Na+) 2
1.5 × 1023	=	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.

Have you used the correct number of significant figures?

When multiplying or dividing, the number of significant figures in the result is the same as the least number of significant figures in any of the terms: number of significant figures in moles of chloride ions (0.50) = 2

number of significant figures in NA (6.022 ×1023) = 4

We are justified in using 2 significant figures in our answer.

1.5 × 1023 has 2 significant figures.

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 × 1023
N(Cl- from NaCl) = N(Na+) = 1.5 × 1023
N(K+) = N(Na+) = 1.5 × 1023
N(Cl- from KCl) = 1.5 × 1023
total number of Cl- in solution = N(Cl- from NaCl) + N(Cl- from KCl) = (1.5 × 1023) + (1.5 × 1023) = 3.0 × 1023
n(Cl-) = N(Cl- in solution) ÷ NA = (3.0 × 1023) ÷ (6.022 × 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. What is the number of sodium ions in the solution?

1.5 × 1023 sodium ions in the solution

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