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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 Xrays 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 = 10^{9} 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 = 10^{12} 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 
= 
10^{9} nm 
multiply both sides of the equation by 1.54 × 10^{10} 
(1.54 × 10^{10}) × 1 m 
= 
(1.54 × 10^{10}) × 10^{9} 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 
= 
10^{12} pm 
multiply both sides of the equation by 1.54 × 10^{10} 
(1.54 × 10^{10}) × 1 m 
= 
(1.54 × 10^{10}) × 10^{12} pm 
1.54 × 10^{10} m 
= 
154 pm 
in scientific notation this is 1.54 × 10^{2} 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 × 10^{2} 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 = 10^{9} nm 
multiply 71 × 10^{12} m by 10^{9} nm/m 
71 × 10^{12} m 
= 
(71 × 10^{12}) × 10^{9} 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 
10^{12} 
10^{9} 
10^{6} 
10^{3} 
10^{2} 
10^{1} 
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:
 10^{3} 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 ÷ 10^{3} = 10^{3} km/m
number of centimetres in a metre = 1 cm ÷ 10^{2} = 10^{2} cm/m
number of millimetres in a metre = 1 mm ÷ 10^{3} = 10^{3} mm/m
number of micrometres in a metre = 1 μm ÷ 10^{6} = 10^{6} μm/m
number of nanometres in a metre = 1 nm ÷ 10^{9} = 10^{9} nm/m
number of picometres in a metre = 1 pm ÷ 10^{12} = 10^{12} pm/m
For chemistry students, the most important conversions are probably between metres (m), nanometres (nm) and picometres (pm):
1 nm = 10^{9 } m 
10^{9} nm = 1 m 
1 pm = 10^{12 } m 
10^{12} pm = 1 m 
1 pm = 10^{3 } nm 
10^{3} 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 = 10^{9} nm
Multiply both sides of the equation by 1.4 × 10^{10}
1 m × (1.4 × 10^{10}) = 10^{9} 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 = 10^{12} pm
Multiply both sides of the equation by 1.02 × 10^{10}
1 m × (1.02 × 10^{10}) = 10^{12} pm × (1.02 × 10^{10})
1.02 × 10^{10} m = 102 pm = 1.02 × 10^{2} 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 = 10^{9} nm
Multiply both sides of the equation by 1.81 × 10^{10}
1 m × (1.81 × 10^{10}) = 10^{9} 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 = 10^{12} pm
Multiply both sides of the equation by 7.8 × 10^{8}
1 m × (7.8 × 10^{8}) = 10^{12} pm × (7.8 × 10^{8})
7.8 × 10^{8} m = 78000 pm = 7.8 × 10^{4} pm