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Definitions of a mole

Key Concepts

  • Mole is the generally accepted SI unit for measuring the amount of substance1.

  • Mole is abbreviated to mol

  • Mole has the symbol n

  • 1 mole contains the same number of particles as there are in 12 g of carbon-12 atoms by definition.
        12 g of carbon-12 atoms = 6.022 x 1023 carbon atoms
        6.022 x 1023 is called the Avogadro number or Avogadro constant
        NA is the symbol used to represent the Avogadro Number2 .

  • 1 mole of any sort of particle has the number of particles
      = Avogadro's number of particles

    = NA particles

    = 6.022 x 1023 particles

  • 1 mole of any pure substance has a mass
      = relative molecular mass of substance in grams

    = Mr(substance) in grams

    = molar mass3 (symbol M, units4 g mol-1)

  • 1 mole of any ideal gas has a volume Vm
      Vm = 22.71 L at 0oC
    (273.15K)
    and 100 kPa 5
    (0.987atm)
    Standard Temperature and Pressure (STP)
      Vm = 24.79 L at 25oC
    (298.15K)
    and 100 kPa 6
    (0.987atm)
    Standard Laboratory Conditions (SLC)
    Standard Ambient Temperature and Pressure (SATP)

    Vm is the symbol used to represent the molar volume of the gas at a specified temperature and pressure.

Mole Concepts

Avogadro's Number, NA

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 boxes 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 (exponential) notation: 6.022 x 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 x 1023 things, the collection is referred to as a mole.
1 mole of anything contains Avogadro's number of things:
    1 mole of eggs = NA eggs = 6.022 x 1023 eggs
    1 mole of dust particles = NA dust particles = 6.022 x 1023 dust particles
    1 mole of sand grains = NA sand grains = 6.022 x 1023 sand grains
    1 mole of water droplets = NA water droplets = 6.022 x 1023 water droplets
    1 mole of water molecules = NA water molecules = 6.022 x 1023 water molecules
    1 mole of hydrogen atoms = NA hydrogen atoms = 6.022 x 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 x mass of each extra large egg
                = 12 x 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 x mass of each small egg
                = 12 x 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.

Molar Mass of Atoms

Moles of atoms 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 NA atoms
        1 mole of atoms always contains 6.022 x 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), Mr, 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 (NA) Mass of 1 Mole of Atoms (g) Molar Mass of Atoms (g mol-1)
hydrogen H 1.008 6.022 x 1023 H atoms 1.008 g 1.008 g mol-1
carbon C 12.01 6.022 x 1023 C atoms 12.01 g 12.01 g mol-1
nitrogen N 14.01 6.022 x 1023 N atoms 14.01 g 14.01 g mol-1
oxygen O 16.00 6.022 x 1023 O atoms 16.00 g 16.00 g mol-1

Molar Mass of Molecules7

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 NA molecules
    1 mole of molecules always contains 6.022 x 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 (NA) Mass of
1 Mole of
Molecules (g)
Molar Mass of
Molecule (g mol-1)
oxygen O2 32.00 6.022 x 1023 O2 molecules 32.00 g 32.00 g mol-1
nitrogen N2 28.02 6.022 x 1023 N2 molecules 28.02 g 28.02 g mol-1
water H2O 18.016 6.022 x 1023 H2O molecules 18.016 g 18.016 g mol-1
ammonia NH3 17.034 6.022 x 1023 NH3 molecules 17.034 g 17.034 g mol-1

Molar Volume of Gas, Vm

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!8
The volume of 1 mole of gas is called its molar volume and is given the symbol Vm.
Molar gas volume is a useful term ONLY if you known the prevailing conditions of temperature and pressure.

Molar Gas Volume, Vm, at Standard Temperature and Pressure

Standard Temperature is defined as 0oC (or 273.15 K)
Standard Pressure is defined as 100 kPa (or 0.987 atm)
Therefore the conditions of standard temperature and pressure are 0oC (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 0oC 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 0oC and 100 kPa, the molar gas volume, Vm, is 22.71 L.
For an ideal gas at standard temperature and pressure the molar gas volume Vm, is 22.71 L.
For an ideal gas at STP the molar gas volume Vm, is 22.71 L.

Below is a table 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 (NA) Molar Volume of Gas at STP (L)
helium He 4.003 4.003 g mol-1 6.022 x 1023 He atoms 22.71 L
nitrogen N2 28.02 28.02 g mol-1 6.022 x 1023 N2 molecules 22.71 L
carbon monoxide CO 28.01 28.01 g mol-1 6.022 x 1023 CO molecules 22.71 L
carbon dioxide CO2 44.01 44.01 g mol-1 6.022 x 1023 CO2 molecules 22.71 L

Molar Gas Volume, Vm, at 25oC and 100 kPa

While 0oC 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, 25oC (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 25oC and 100 kPa has a volume of 24.79 L.
The molar volume, Vm, of any ideal gas at 25oC and 100 kPa is 24.79 L.

Below is a table 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 (NA) Molar Volume of Gas at SLC (L)
helium He 4.003 4.003 g mol-1 6.022 x 1023 He atoms 24.79 L
nitrogen N2 28.02 28.02 g mol-1 6.022 x 1023 N2 molecules 24.79 L
carbon monoxide CO 28.01 28.01 g mol-1 6.022 x 1023 CO molecules 24.79 L
carbon dioxide CO2 44.01 44.01 g mol-1 6.022 x 1023 CO2 molecules 24.79 L

Summary

1 mole of molecules:

  • contains NA molecules
    1 mole of molecules = 6.022 x 1023 molecules

  • has mass equal to its relative molecular mass in grams
    mass of 1 mole of molecules = Mr g

  • has molar mass equal to relative molecular mass in grams per mole
    molar mass in g mol-1 = Mr in grams = M g mol-1

1 mole of gas molecules also has a volume of

  • 22.71 L at 0oC (273.15 K) and 100 kPa (0.987 atm) (STP)

  • 24.79 L at 25oC (298.15 K) and 100 kPa (0.987 atm) (SLC)

Examples

Avogadro Number (NA)

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

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

  2. What information (data) has been given in the question?
    molecular formula for chlorine molecules: Cl2
    amount of chlorine gas = 1 mole

  3. What relationship exits between moles and the number of molecules present?
    1 mole molecules = NA molecules = 6.022 x 1023 molecules

  4. Substitute the word "molecules" above with the particular molecule given in the question, "Cl2 molecules":
    1 mole Cl2 molecules = NA Cl2 molecules = 6.022 x 1023 Cl2 molecules

  5. 1 mole of chlorine gas contains 6.022 x 1023 chlorine molecules.

Molar Mass (M)

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

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

  2. What information (data) has been given in the question?
    molecular formula for methane molecules: CH4
    amount of methane = 1 mole

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

  4. Calculate the relative molecular mass of methane, Mr(CH4),
    Use the Periodic Table to find the relative atomic mass (atomic weight) of carbon, Mr(C), and hydrogen, Mr(H):
        Mr(C) = 12.01
        Mr(H) = 1.008
    CH4 contains 1 carbon atom and 4 hydrogen atoms, so
    Mr(CH4) = 1 x Mr(C) + 4 x Mr(H) = 1 x 12.01 + 4 x 1.008 = 16.042

  5. Calculate the mass of 1 mole of CH4:
    mass of 1 mole CH4 = Mr(CH4) expressed in grams
    mass of 1 mole CH4 = 16.042 g

  6. The mass of 1 mole of methane, CH4, is 16.042 g

Molar Volume of Gases

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

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

  2. What information (data) has been given in the question?
    molecular formula for oxygen gas: O2(g)
    mass of oxygen gas = m(O2(g)) = 32.00 g
    conditions for the experiment: temperature = 0oC, pressure = 100 kPa
        recognise that 0oC and 100 kPa = Standard Temperature and Pressure (STP)

  3. 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.

  4. Calculate the relative molecular mass of oxygen gass, Mr(O2(g)):
    Use the Periodic Table to find the relative atomic mass (atomic weight) for oxygen atoms, Mr(O):
        Mr(O) = 16.00
    Each oxygen molecule, O2, contains 2 oxygen atoms (O), so
        Mr(O2) = 2 x Mr(O) = 2 x 16.00 = 32.00

  5. Calculate the mass of 1 mole of oxygen molecules, m(O2(g))
    mass of 1 mole of O2 molecules = m(O2) = Mr(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).

  6. 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

  7. 32.00 g of oxygen gas has a volume of 22.71 L at 0oC and 100 kPa.

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 0oC 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 0oC 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.

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: 0oC and 100 kPa (standard temperature and pressure, STP)
PAUSE!
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 = ? 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:
    Mr(O) = atomic weight of oxygen
    Mr(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, Mr, 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:
    Mr(O) = atomic weight of oxygen = 16.00
    Mr(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 = Mr(O2) = 2 x 16.00 = 32.00
    relative molecular mass of carbon monoxide gas = Mr(CO) = 12.01 + 16.00 = 28.01
    relative molecular mass of carbon dioxide gas = Mr(CO2) = 12.01 + 2 x 16.00 = 12.01 + 32.00 = 44.01
Convert each relative molecular mass, Mr, 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.
Mr(CO2) = 12.01 + 2 x 16.00 = 12.01 + 32.00 = 44.01 g
1 mole CO2(g) has volume = 22.71 L at STP (0oC, 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. The molecular formula for the gas in the plastic bag is CO2(g).


What would you like to do now?


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

2Avogadro's number is sometimes referred to as Loschmidt's number and is given the symbol L.

3 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.

4 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.

5 Prior to 1982, standard temperature and pressure were defined as 0oC (273.15 K) and 1 atm (101.3 kPa), so 1 mole of gas would occupy 22.41 L

6 Prior to 1982, standard laboratory conditions were defined as 25oC (298.15 K) and 1 atm (101.3 kPa), so 1 mole of gas would occupy 24.47 L.

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

8 This is true as long as the gas is behaving as an ideal gas.

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