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Units of Length and Unit Conversions
You are probably already very familiar with some units for measuring distance.
We measure the distance between one town and another in kilometres.
We measure the height of a person in metres or centimetres.
My ruler is 30 centimetres long and each centimetre is divided into 10 millimetres.
But all the distances above are large enough to see.
Chemists are interested in atoms and the distances between them which are much too small to be seen using only your eyes.
For example, some of the lengths, or distances, chemists are interested in are:
- The size of an atom which is of the order of 10-10 metres
- The distance between atoms in a molecule which is of the order of 10-10 metres
- The wavelength of visible light (distance between adjacent wave peaks) which is of the order of 10-7 metres
- The wavelength of X-rays which are of the order of 10-10 metres
- The diameter of a nanoparticle, like a buckyball, which is generally between 10-9 and 10-7 metres
All of the lengths listed above are much, much, smaller than a metre.
Because the size of an atom and the distance between atoms in a molecule is of the order of 10-10 metres, a convenient unit of measurement is the Ångstrom (Å)(3) defined as 10-10 m
1 Å = 10-10 m
If an atom with has a radius of 1.54 Å we can convert this radius to metres as shown below:
| 1 Å |
= |
10-10 m |
| multiply both sides of the equation by 1.54 |
| 1.54 × 1 Å |
= |
1.54 × 10-10 m |
| 1.54 Å |
= |
1.54 × 10-10 m |
But the Ångstrom, although very convenient, is not an SI unit and has been largely replaced by other units such as the nanometre (nm) and the picometre (pm):
- the nanometre (nm)
1 nm = 10-9 m (1 nm is very small)
1 m = 109 nm (lots of nanometres make up 1 m)
- the picometre (pm)
1 pm = 10-12 m (1 pm is even smaller than 1 nm)
1 m = 1012 pm (more picometres than nanometres make up 1 m)
If an atom has a radius of 1.54 × 10-10 m we can convert this to nanometres (nm) as shown below:
| 1 m |
= |
109 nm |
| multiply both sides of the equation by 1.54 × 10-10 |
| (1.54 × 10-10) × 1 m |
= |
(1.54 × 10-10) × 109 nm |
| 1.54 × 10-10 m |
= |
0.154 nm |
| in scientific notation this is 1.54 × 10-1 nm |
We can also convert this radius of 1.54 × 10-10 m to picometres (pm) as shown below:
| 1 m |
= |
1012 pm |
| multiply both sides of the equation by 1.54 × 10-10 |
| (1.54 × 10-10) × 1 m |
= |
(1.54 × 10-10) × 1012 pm |
| 1.54 × 10-10 m |
= |
154 pm |
| in scientific notation this is 1.54 × 102 pm |
Therefore, the radius of our atom can be represented in a number of different, but equivalent, ways:
- radius = 1.54 Å
- radius = 0.154 nm
- radius = 1.54 × 10-1 nm
- radius = 154 pm
- radius = 1.54 × 102 pm
We can also convert lengths in nanometres to picometres in a similar way.
An atom of fluorine has a radius of 71 pm.
We can convert this to a radius in nanometres as shown below:
| 1 pm |
= |
10-12 m |
| multiply both sides of the equation by 71 |
| 71 × 1 pm |
= |
71 × 10-12 m |
| radius of atom is 71 × 10-12 m |
| recall that 1 m = 109 nm |
| multiply 71 × 10-12 m by 109 nm/m |
| 71 × 10-12 m |
= |
(71 × 10-12) × 109 nm |
| 71 × 10-12 m |
= |
71 × 10-3 nm |
| in scientific notation this is 7.1 × 10-2 nm |
The table below lists the common prefixes attached to the metre (the SI unit of length):
| |
larger than 1 metre |
smaller than 1 metre |
|---|
| factors |
1012 |
109 |
106 |
103 |
102 |
101 |
10-1 |
10-2 |
10-3 |
10-6 |
10-9 |
10-12 |
10-15 |
10-18 |
|---|
| prefix |
tera |
giga |
mega |
kilo |
hecto |
deca |
deci |
centi |
milli |
micro |
nano |
pico |
femto |
atto |
|---|
| symbol |
T |
G |
M |
k |
h |
da |
d |
c |
m |
µ |
n |
p |
f |
a |
|---|
This table tells us that:
- 103 m is called 1 kilometre (km)
- 10-2 m is called 1 centimetre (cm)
- 10-3 m is called 1 millimetre (mm)
- 10-6 m is called 1 micrometre (μm, previously known as a micron)
- 10-9 m is called 1 nanometre (nm)
- 10-12 m is called 1 picometre (pm)
We can use this determine how many of each unit are in a metre:
number of kilometres in a metre = 1 km ÷ 103 = 10-3 km/m
number of centimetres in a metre = 1 cm ÷ 10-2 = 102 cm/m
number of millimetres in a metre = 1 mm ÷ 10-3 = 103 mm/m
number of micrometres in a metre = 1 μm ÷ 10-6 = 106 μm/m
number of nanometres in a metre = 1 nm ÷ 10-9 = 109 nm/m
number of picometres in a metre = 1 pm ÷ 10-12 = 1012 pm/m
For chemistry students, the most important conversions are probably between metres (m), nanometres (nm) and picometres (pm):
| 1 nm = 10-9 m |
109 nm = 1 m |
| 1 pm = 10-12 m |
1012 pm = 1 m |
| 1 pm = 10-3 nm |
103 pm = 1 nm |
Worked Examples of Length Conversions Using SI and Metric Units
Question 1: Convert a bond length of 1.4 × 10-10 m to nanometres.
Solution:
From the table above we see that 1 m = 109 nm
Multiply both sides of the equation by 1.4 × 10-10
1 m × (1.4 × 10-10) = 109 nm × (1.4 × 10-10)
1.4 × 10-10 m = 0.14 nm = 1.4 × 10-1 nm
Question 2: A beam of yellow light has a wavelength of 500 nm. Convert this to a wavelength in metres.
Solution:
From the table above we see that 1 nm = 10-9 m
Multiply both sides of the equation by 500
1 nm × 500 = 10-9 m × 500
500 nm = 5.00 × 10-7 m
Question 3: An atom of iodine has a radius of 220 pm. Convert this to a radius in metres.
Solution:
From the table above we see that 1 pm = 10-12 m
Multiply both sides of the equation by 220
1 pm × 220 = 10-12 m × 220
220 pm = 2.20 × 10-10 m
Question 4: A sodium ion has a radius of 1.02 × 10-10 m. Convert this to a radius in picometres.
Solution:
From the table above we see that 1 m = 1012 pm
Multiply both sides of the equation by 1.02 × 10-10
1 m × (1.02 × 10-10) = 1012 pm × (1.02 × 10-10)
1.02 × 10-10 m = 102 pm = 1.02 × 102 pm
Question 5: A chloride ion has a radius of 181 pm. Convert this to a radius in nanometres.
Solution:
From the table above we see that 1 pm = 10-12 m
Multiply both sides of the equation by 181
1 pm × 181 = 10-12 m × 181
181 pm = 1.81 × 10-10 m
From the table above, 1 m = 109 nm
Multiply both sides of the equation by 1.81 × 10-10
1 m × (1.81 × 10-10) = 109 nm × (1.81 × 10-10)
1.81 × 10-10 m = 0.181 nm = 1.81 × 10-1 nm
Question 6: A nanoparticle is known to have a diameter of 78 nm. Convert this to a diameter in picometres.
Solution:
From the table above we see that 1 nm = 10-9 m
Multiply both sides of the equation by 78
1 nm × 78 = 10-9 m × 78
78 nm = 7.8 × 10-8 m
From the table above we see that 1 m = 1012 pm
Multiply both sides of the equation by 7.8 × 10-8
1 m × (7.8 × 10-8) = 1012 pm × (7.8 × 10-8)
7.8 × 10-8 m = 78000 pm = 7.8 × 104 pm
