Definitions of a Mole Chemistry Tutorial

Key Concepts

Mole is the SI based unit for measuring the amount of substance⁽¹⁾.

Mole is abbreviated to mol

Mole has the symbol n (in calculations)

1 mole of a substance is defined as the amount of substance which contains the same number of particles as there are in 0.012 kg of carbon-12 atoms.⁽²⁾
0.012 kg of carbon-12 atoms contains 6.022 × 10²³ carbon-12 atoms

since 1 kg = 1000 g

0.012 kg of carbon-12 atoms = 0.012 kg × 1000 g ~~kg^-1~~ = 12 g of carbon-12 atoms

therefore 12 g of carbon-12 atoms contains 6.022 × 10²³ carbon-12 atoms

1 mole of "particles" of any substance must contain 6.022 × 10²³ "particles".

6.022 × 10²³ is called the Avogadro number or Avogadro constant
N_A is the symbol used to represent the Avogadro Number⁽³⁾ .

The number of "particles" in 1 mole of "particles"
= Avogadro number of particles

= N_A particles

= 6.022 × 10²³ particles

The mass of 1 mole of carbon-12 atoms is 12 g
The atomic weight of an element is defined as its mass relative to carbon-12.

So the mass of 1 mole of atoms of an element is equal to its atomic weight expressed in grams.

The mass of 1 mole of atoms of any element

= relative atomic mass of the element in grams

= M_r(element) in grams

= molar mass of the element ⁽⁴⁾ (symbol M, units⁽⁵⁾ g mol^-1)

The mass of molecules in 1 mole of molecules must be equal to the relative molecular mass of the molecule expressed in grams
The mass of 1 mole of molecules of a compound

= relative molecular mass of the molecule in grams

= M_r(molecule) in grams

= molar mass of the molecule (symbol M, units g mol^-1)

Equal volumes of gas at the same temperature and pressure contain the same number of "particles" (Avogadro's Principle)

We can therefore define a new term, the molar volume of a gas (symbol V_m), as the volume of 1 mole of gas "particles" at a particular temperature and pressure.

1 mole of any ideal gas has a volume V_m

Molar Volume of Gas (V_m)	Conditions			Name given to these conditions
V_m = 22.71 L	at 0°C (273.15K)	and	100 kPa ⁽⁶⁾ (0.987atm)	Standard Temperature and Pressure (STP)
V_m = 24.79 L	at 25°C (298.15K)	and	100 kPa ⁽⁷⁾ (0.987atm)	Standard Laboratory Conditions (SLC) Standard Ambient Temperature and Pressure (SATP)

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Mole Concepts

Avogadro Number, N_A

On the right is a diagram of 1 dozen eggs, each egg represented as 0.

"Dozen" is the term used to refer to a particular number, 12 in fact.

1 dozen eggs = 12 eggs

	0		0		0		0		0		0
0		0		0		0		0		0

If you had 12 cartons of a dozen eggs you would have a gross of eggs.

"Gross" is the term used to refer to 12 dozen of something.

The term "Avogadro number" is just the name given to a particular number.

The Avogadro number is an extremely large number: 602 200 000 000 000 000 000 000

which is why we usually write it using scientific notation (exponential notation): 6.022 × 10²³

A Mole, n

On the right is a diagram of 1 dozen eggs, each egg represented as 0 and each egg is sitting in its own space in a carton.

1 carton of eggs contains 1 dozen eggs

1 carton of eggs = 1 dozen eggs

The word "carton" refers to the collection of 1 dozen eggs.

1 carton of eggs = 1 dozen eggs = 12 eggs

	0		0		0		0		0		0
0		0		0		0		0		0

When you have a collection of 6.022 × 10²³ things, the collection is referred to as a mole.

1 mole of anything contains the Avogadro number of things:

1 mole of eggs = N_A eggs = 6.022 × 10²³ eggs

1 mole of dust particles = N_A dust particles = 6.022 × 10²³ dust particles

1 mole of sand grains = N_A sand grains = 6.022 × 10²³ sand grains

1 mole of water droplets = N_A water droplets = 6.022 × 10²³ water droplets

1 mole of water molecules = N_A water molecules = 6.022 × 10²³ water molecules

1 mole of hydrogen atoms = N_A hydrogen atoms = 6.022 × 10²³ hydrogen atoms

Molar Mass, M

On the right is a diagram of 1 dozen extra large eggs, each extra large egg represented as 0 and each egg is sitting in its own space in a carton.

Each extra large egg has a mass of 55 grams.

mass of eggs in 1 carton of extra large eggs

= 1 dozen eggs × mass of each extra large egg

= 12 × 55 g = 660 g

	0		0		0		0		0		0
0		0		0		0		0		0

On the right is a diagram of 1 dozen small eggs, each small egg represented as 0 and each egg is sitting in its own space in a carton.

Each small egg has a mass of 25 grams.

mass of eggs in 1 carton of small eggs

= 1 dozen eggs × mass of each small egg

= 12 × 25 g = 300 g

	0		0		0		0		0		0
0		0		0		0		0		0

A carton always holds 1 dozen eggs.

1 dozen eggs is always equal to 12 eggs.

But, the mass of a carton of 1 dozen eggs depends on the "identity" of the eggs, that is, small eggs have less mass than extra large eggs.

The same is true of moles of atoms and molecules.

A mole of atoms always contains 6.022 × 10²³ atoms, but the mass of that mole depends on the mass of the atoms.

Atoms of different elements have a different mass, so a mole of atoms of one element will have a different mass to a mole of atoms of a different element.

Molar Mass of Atoms

Moles of atoms of an element are just like the carton of eggs analogy above:

1 mole of atoms always contains Avogadro's number of atoms

1 mole of atoms always contains N_A atoms

1 mole of atoms always contains 6.022 × 10²³ atoms

But, the mass of 1 mole of atoms will depend on the "identity" of the atoms.

The "identity" of an atom is determined by which element the atoms belong to.

Atoms of different elements have different relative atomic masses (atomic weights).

You can find the relative atomic mass (or atomic weight), M_r, for an atom of any element in the Periodic Table of the Elements.

The mass of 1 mole of atoms of an element is the relative atomic mass (atomic weight) of the element expressed in grams.

The mass of 1 mole of atoms of an element is known as the molar mass of the element and has the units grams per mole, g mol^-1

Below is a table of some of the elements you will encounter during your chemistry course:

Name of Element	Symbol of Element	Relative Atomic Mass (Atomic Weight)	Number of Atoms in 1 Mole (N_A)	Mass of 1 Mole of Atoms (g)	Molar Mass of Atoms (g mol^-1)
hydrogen	H	1.008	6.022 × 10²³ H atoms	1.008 g	1.008 g mol^-1
carbon	C	12.01	6.022 × 10²³ C atoms	12.01 g	12.01 g mol^-1
nitrogen	N	14.01	6.022 × 10²³ N atoms	14.01 g	14.01 g mol^-1
oxygen	O	16.00	6.022 × 10²³ O atoms	16.00 g	16.00 g mol^-1

Molar Mass of Molecules⁽⁸⁾

A molecule is made up of 2 or more atoms chemically joined (bonded) together.

1 mole of molecules always contains Avogadro's number of molecules

1 mole of molecules always contains N_A molecules

1 mole of molecules always contains 6.022 × 10²³ molecules

But the mass of 1 mole of molecules will depend on the "identity" of the molecules.

The "identity" of a molecule is determined by the number of atoms of each element making up the molecule.

For every molecule we can write a molecular formula.

For every molecular formula we can write, we can determine the relative molecular mass of the molecule.

And, just like we did for the examples of atoms of elements above, 1 mole of molecules will have a mass equal to its relative molecular mass expressed in grams.

Relative molecular mass expressed in grams is known as the molar mass of the molecule.

Below is a table of some of the molecules you will encounter during your chemistry course:

Name of Molecule	Molecular Formula	Relative Molecular Mass	Number of Molecules in 1 Mole (N_A)	Mass of 1 Mole of Molecules (g)	Molar Mass of Molecule (g mol^-1)
oxygen	O₂	32.00	6.022 × 10²³ O₂ molecules	32.00 g	32.00 g mol^-1
nitrogen	N₂	28.02	6.022 × 10²³ N₂ molecules	28.02 g	28.02 g mol^-1
water	H₂O	18.016	6.022 × 10²³ H₂O molecules	18.016 g	18.016 g mol^-1
ammonia	NH₃	17.034	6.022 × 10²³ NH₃ molecules	17.034 g	17.034 g mol^-1

Molar Volume of Gas, V_m

You've probably watched as a balloon is filled with helium from a gas cylinder.

As more helium enters the balloon, the volume of the balloon increases.

This is because the atoms of helium making up the helium gas occupy as large a volume as possible.

And that volume is determined by the number of helium atoms there are in the balloon as well as by the air temperature and pressure.

If you were to simultaneously fill one balloon with 1 mole of helium atoms, and another balloon with 1 mole of argon atoms, you would find that both balloons would fill up to occupy the same volume.

The volume of 1 mole of gas depends on the surrounding temperature and pressure, and not really on the "identity" of the atoms or molecules making up the gas!⁽⁹⁾

The volume of 1 mole of gas is called its molar volume and is given the symbol V_m.

Molar volume of gas is a useful term ONLY if you known the prevailing conditions of temperature and pressure.

Molar Volume of Gas, V_m, at Standard Temperature and Pressure

Standard Temperature is defined as 0°C (≈273.15 K)

Standard Pressure is defined as 100 kPa (or 0.987 atm)

Therefore the conditions of standard temperature and pressure are 0°C (273.15 K) and 100 kPa (0.987 atm).

Standard temperature and pressure are usually abbreviated as STP.

The volume of 1 mole of any ideal gas at a temperature of 0°C and a pressure of 100 kPa is 22.71 L.

The volume of 1 mole of any ideal gas at standard temperature and pressure is 22.71 L.

The volume of 1 mole of any ideal gas at STP is 22.71 L.

For an ideal gas at 0°C and 100 kPa, the molar volume of gas , V_m, is 22.71 L.

For an ideal gas at standard temperature and pressure the molar gas volume V_m, is 22.71 L.

For an ideal gas at STP the molar volume of gas, V_m, is 22.71 L.

Below is a table of the molar volume of some gases you will encounter during your chemistry course:

Name of Gas	Molecular Formula	Relative Molecular Mass	Molar Mass of Gas (g mol^-1)	Number of Gas Molecules in 1 Mole (N_A)	Molar Volume of Gas at STP (L)
helium	He	4.003	4.003 g mol^-1	6.022 × 10²³ He atoms	22.71 L
nitrogen	N₂	28.02	28.02 g mol^-1	6.022 × 10²³ N₂ molecules	22.71 L
carbon monoxide	CO	28.01	28.01 g mol^-1	6.022 × 10²³ CO molecules	22.71 L
carbon dioxide	CO₂	44.01	44.01 g mol^-1	6.022 × 10²³ CO₂ molecules	22.71 L

Molar Volume of Gas, V_m, at 25°C and 100 kPa

While 0°C is a very useful standard for temperature, it isn't really a temperature you'd like to keep the laboratory at while you work.

So, Chemists will define as standard a useful temperature and pressure based on the kind of work that they do.

In some circumstances, 25°C (298.15 K) and 100 kPa (0.987 atm) is used to define the standard, and is then referred to as Standard Laboratory Conditions (abbreviated as SLC) or as Standard Ambient Temperature and Pressure (abbreviated as SATP).

1 mole of any ideal gas at 25°C and 100 kPa has a volume of 24.79 L.

The molar volume, V_m, of any ideal gas at 25°C and 100 kPa is 24.79 L.

Below is a table of the molar volume of some gases you will encounter during your chemistry course:

Name of Gas	Molecular Formula	Relative Molecular Mass	Molar Mass of Gas (g mol^-1)	Number of Gas Molecules in 1 Mole (N_A)	Molar Volume of Gas at SLC (L)
helium	He	4.003	4.003 g mol^-1	6.022 × 10²³ He atoms	24.79 L
nitrogen	N₂	28.02	28.02 g mol^-1	6.022 × 10²³ N₂ molecules	24.79 L
carbon monoxide	CO	28.01	28.01 g mol^-1	6.022 × 10²³ CO molecules	24.79 L
carbon dioxide	CO₂	44.01	44.01 g mol^-1	6.022 × 10²³ CO₂ molecules	24.79 L

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Summary of How A Mole Can Be Defined

1 mole of molecules:

contains N_A molecules
1 mole of molecules = 6.022 × 10²³ molecules

has mass equal to its relative molecular mass in grams
mass of 1 mole of molecules = M_r g
has molar mass equal to relative molecular mass in grams per mole
molar mass in g mol^-1 = M_r in grams = M g mol^-1

1 mole of gas molecules also has a volume of

22.71 L at 0°C (273.15 K) and 100 kPa (0.987 atm) (STP)

24.79 L at 25°C (298.15 K) and 100 kPa (0.987 atm) (SLC or SATP)

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Worked Examples Using the Definition of a Mole

Worked Example of Mole Definition: Avogadro Number (N_A)

Question: How many chlorine molecules, Cl₂, are present in 1 mole of chlorine gas, Cl₂(g)?

Solution:

What is the question asking you to do?
Calculate the number of chlorine molecules in 1 mole of chlorine gas.

What information (data) has been given in the question?
molecular formula for chlorine molecules: Cl₂

amount of chlorine gas = 1 mole

What relationship exits between moles and the number of molecules present?
1 mole molecules = N_A molecules = 6.022 × 10²³ molecules

Substitute the word "molecules" above with the particular molecule given in the question, "Cl₂ molecules":
1 mole Cl₂ molecules = N_A Cl₂ molecules = 6.022 × 10²³ Cl₂ molecules

State your answer:
1 mole of chlorine gas contains 6.022 × 10²³ chlorine molecules.

Worked Example of Mole Definition: Molar Mass (M)

Question: What is the mass of 1 mole of methane, CH₄, molecules?

Solution:

What is the question asking you to do?
Calculate the mass of 1 mole of methane molecules.

What information (data) has been given in the question?
molecular formula for methane molecules: CH₄

amount of methane = 1 mole

What relationship exits between moles and the mass of molecules?
mass of 1 mole molecules = relative molecular mass expressed in grams

Calculate the relative molecular mass of methane, M_r(CH₄)
Use the Periodic Table to find the relative atomic mass (atomic weight) of carbon, M_r(C), and hydrogen, M_r(H):

M_r(C) = 12.01

M_r(H) = 1.008

CH₄ contains 1 carbon atom and 4 hydrogen atoms, so

M_r(CH₄) = 1 × M_r(C) + 4 × M_r(H) = 1 x 12.01 + 4 × 1.008 = 16.042

Calculate the mass of 1 mole of CH₄:
mass of 1 mole CH₄ = M_r(CH₄) expressed in grams

mass of 1 mole CH₄ = 16.042 g

State your answer:
The mass of 1 mole of methane, CH₄, is 16.042 g

Worked Example of Mole Definition: Molar Volume of Gas (V_m)

Question: What is the volume occupied by 32.00 g of oxygen gas, O₂(g), at a temperature of 0°C and a pressure of 100 kPa?

Solution:

What is the question asking you to do?
Calculate the volume of oxygen gas.

What information (data) has been given in the question?
molecular formula for oxygen gas: O₂(g)

mass of oxygen gas = m(O_2(g)) = 32.00 g

conditions for the experiment: temperature = 0°C, pressure = 100 kPa

Recall: 0°C and 100 kPa = Standard Temperature and Pressure (STP)

What relationship exists between the mass of a gas and its volume at STP?
At STP 1 mole of gas has a volume of 22.71 L

1 mole of gas has a mass equal to its relative molecular mass expressed in grams.

Calculate the relative molecular mass of oxygen gass, M_r(O_2(g)):
Use the Periodic Table to find the relative atomic mass (atomic weight) for oxygen atoms, M_r(O):

M_r(O) = 16.00

Each oxygen molecule, O₂, contains 2 oxygen atoms (O), so

M_r(O₂) = 2 × M_r(O) = 2 × 16.00 = 32.00

Calculate the mass of 1 mole of oxygen molecules, m(O_2(g))
mass of 1 mole of O₂ molecules = m(O₂) = M_r(O₂) in grams = 32.00 g

Note that the mass of O₂(g) given in the question is 32.00 g, so the question is asking for the volume of 1 mole of O₂(g).

Calculate the volume of 1 mole of O₂(g):
At STP, 1 mole of gas has a volume of 22.71 L

At STP, 1 mole of O₂(g) has a volume of 22.71 L

State your answer:
32.00 g of oxygen gas has a volume of 22.71 L at 0°C and 100 kPa.

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Problem Solving Using Definitions of a Mole

The Problem: Bo the Biologist has been collecting the gas emitted overnight by pea plants in a sealed plastic bag. Bo suspects the gas is either oxygen, carbon monoxide or carbon dioxide, but would like to know for sure. Bo took the gas sample in the sealed plastic bag to Chris the Chemist for analysis. Chris the Chemist cooled the gas down to 0°C at 100 kPa and weighed it. The mass of gas was found to be 44.01 g. Next Chris plunged the sealed plastic bag full of gas into a filled container of ice cold water at 0°C and measured the volume of water displaced by the plastic bag full of gas. The volume of the gas was then found to be 22.71 L.

What was the chemical formula for the gas in the plastic bag?

Solving the Problem

Using the StoPGoPS model for problem solving:

STOP!	State the question.	What is the question asking you to do? Determine the molecular formula of the gas.
PAUSE!	Plan.	What chemical principle will you need to apply? Apply stoichoimetry (definitions of a mole) What information (data) have you been given? substance is a gas gas could be oxygen, carbon monoxide or carbon dioxide m(gas) = mass of gas = 44.01 g V(gas) = volume of gas = 22.71 L conditions of experiment: 0°C and 100 kPa (standard temperature and pressure, STP) What is the relationship between what you know and what you need to find out? Step 1: Determine the amount, in moles, of gas present Assume gas is behaving like an ideal gas At STP, 1 mole of ideal gas has a volume of 22.71 L At STP, volume of unknown gas, V_(gas) = 22.71 L moles of unknown gas = ? mol Step 2: Write the molecular formula for each of the three possible gasses: molecular formula of oxygen gas: molecular formula of carbon monoxide gas: molecular formula of carbon dioxide gas: Step 3: Determine the molar mass, M, for each of the three possible gases Use the Periodic Table to find the relative atomic mass (atomic weight) of oxygen and carbon: M_r(O) = atomic weight of oxygen M_r(C) = atomic weight of carbon Use the molecular formula above to determine the relative molecular mass of each of the three possible gasses: relative molecular mass of oxygen gas: relative molecular mass of carbon monoxide gas: relative molecular mass of carbon dioxide gas: Convert each relative molecular mass, M_r, to a molar mass, M: molar mass = relative molecular mass expressed as grams per mole molar mass of oxygen gas: molar mass of carbon monoxide gas: molar mass of carbon dioxide gas: Step 4: Determine which of the three gases is present Assume the gas in the plastic bag is a pure substance, that is, only one type of molecule is present Compare the molar mass of each gas to that of the amount of gas in the plastic bag.
GO!	Go with the Plan.	Step 1: Determine the amount, in moles, of gas present Assume gas is behaving like an ideal gas At STP, 1 mole of ideal gas has a volume of 22.71 L At STP, volume of unknown gas, V_(gas) = 22.71 L moles of unknown gas = 1 mol because its volume at STP is 22.71 L Step 2: Write the molecular formula for each of the three possible gasses: molecular formula of oxygen gas: O₂ molecular formula of carbon monoxide gas: CO molecular formula of carbon dioxide gas: CO₂ Step 3: Determine the molar mass, M, for each of the three possible gases Use the Periodic Table to find the relative atomic mass (atomic weight) of oxygen and carbon: M_r(O) = atomic weight of oxygen = 16.00 M_r(C) = atomic weight of carbon = 12.01 Use the molecular formula above to determine the relative molecular mass of each of the three possible gasses: relative molecular mass of oxygen gas = M_r(O₂) = 2 × 16.00 = 32.00 relative molecular mass of carbon monoxide gas = M_r(CO) = 12.01 + 16.00 = 28.01 relative molecular mass of carbon dioxide gas = M_r(CO₂) = 12.01 + 2 × 16.00 = 12.01 + 32.00 = 44.01 Convert each relative molecular mass, M_r, to a molar mass, M: molar mass = relative molecular mass expressed as grams per mole molar mass of oxygen gas = M(O₂) = 32.00 g mol^-1 molar mass of carbon monoxide gas = M(CO) = 28.01 g mol^-1 molar mass of carbon dioxide gas = M(CO₂) = 44.01 g mol^-1 Step 4: Determine which of the three gases is present Assume the gas in the plastic bag is a pure substance, that is, only one type of molecule is present Compare the molar mass of each gas to that of the amount of gas in the plastic bag. molar mass of unknown gas = mass of 1 mole of unknown gas = 44.01 g mol^-1 M(CO₂) = 44.01 g mol^-1 Unknown gas most likely to be CO₂
PAUSE!	Ponder Plausability.	Have you answered the question that was asked? Yes, we have determined the molecular formula for the gas most likely to be in the plastic bag. Is your solution to the question reasonable? Let's work backwards to see if the formula for the gas we have chosen will give us the correct mass and volume at STP. M_r(CO₂) = 12.01 + 2 × 16.00 = 12.01 + 32.00 = 44.01 g 1 mole CO₂(g) has volume = 22.71 L at STP (0°C, 100 kpa) Experiment has conducted at STP, and unknown gas had volume of 22.71 L = 1 mole, and this 1 mole had a mass of 44.01 g, CO₂ does look like a reasonable answer to the question.
STOP!	State the solution.	What is the chemical formula of the gas in the bag? CO₂(g)

(1) Some Chemists argue that the mole is a derived quantity and therefore should not be a SI unit.

(2) The SI unit of mass is the kilogram, so the definition is based on the mass of carbon-12 atoms in kg.
The mole is defined in the International Union of Pure and Applied Chemistry (IUPAC) Green Book, Quantities, Units and Symbols in Physical Chemistry, Third Edition, 2007.

(3) The Avogadro number (Avogadro constant) is sometimes referred to as Loschmidt's number and is then given the symbol L.
N_A = L = 6.022 141 79 × 10²³ mol^-1

(4) Before the term "mole" came into widespread use, the term gram formula weight, gfw, (or gram formula mass, gfm,) would be used to describe the relative molecular mass of a compound expressed in grams. Some Chemists still use gfw instead of molar mass.

(5) The SI units for molar mass are actually kg mol^-1 but Chemists in the laboratory work with much smaller amounts so usually use g mol^-1.

(6) V_m (p = 100 kPa, t = 0°C) = 22.710 981 dm³ mol^-1 = 22.710 981 L mol^-1
Prior to 1982, standard temperature and pressure were defined as 0°C (273.15 K) and 1 atm (101.325 kPa), so 1 mole of gas would occupy 22.413 996 dm³ mol^-1 = 22.413 996 L mol^-1

(7) Prior to 1982, standard laboratory conditions were defined as 25°C (298.15 K) and 1 atm (101.325 kPa), so 1 mole of gas would occupy 24.47 L.

(8) The term "molecule" is being used in a generic sense to cover both covalent and ionic substances.

(9) This is true as long as the gas is behaving as an ideal gas.

Definitions of a Mole Chemistry Tutorial

Key Concepts

Mole Concepts

Avogadro Number, NA

A Mole, n

Molar Mass, M

Molar Mass of Atoms

Molar Mass of Molecules(8)

Molar Volume of Gas, Vm

Molar Volume of Gas, Vm, at Standard Temperature and Pressure

Molar Volume of Gas, Vm, at 25°C and 100 kPa

Summary of How A Mole Can Be Defined

Worked Examples Using the Definition of a Mole

Worked Example of Mole Definition: Avogadro Number (NA)

Worked Example of Mole Definition: Molar Mass (M)

Worked Example of Mole Definition: Molar Volume of Gas (Vm)

Problem Solving Using Definitions of a Mole

Avogadro Number, N_A

Molar Mass of Molecules⁽⁸⁾

Molar Volume of Gas, V_m

Molar Volume of Gas, V_m, at Standard Temperature and Pressure

Molar Volume of Gas, V_m, at 25°C and 100 kPa

Worked Example of Mole Definition: Avogadro Number (N_A)

Worked Example of Mole Definition: Molar Volume of Gas (V_m)