Units and conversions
Check out this helpful resource to learn more about units and conversions.
Units and conversions
Units are used to describe physical quantities. Being able to express units in the most appropriate way is crucial for scientific research, engineering, everyday tasks, and effective communication across measurement systems.
If you expressed a quantity without a unit, it wouldn't mean much. Let's say that you tell a friend that you have gained \(5.68\textrm{ kg}\) over the past three months. If you only say \(5.68\) without kilograms, it won't make much sense to your friend. The number \(5.68\) can be in any units we use to measure mass, such as grams, milligrams, kilograms, pounds, etc. Your friend might even think you gained \(5.68\) centimetres in height!
Adding a unit next to the quantity is essential to communicate the correct information. We usually write the symbol for the unit, such as kg for kilograms.
5.68 kg, by RMIT, licensed under CC BY-NC 4.0
SI units
A physical quantity can be described with several different units. For instance, weight can be expressed in kilograms, grams, pounds, etc. To avoid confusion, scientists have introduced standard units for each physical quantity. This system is known as the international system of units (SI units). Some SI units and their symbols are shown.
| Quantity | Unit | Symbol |
|---|---|---|
| mass | kilogram | \(kg\) |
| length | metre | \(m\) |
| volume | cubic metre | \(m^{3}\) |
| temperature | Kelvin | \(K\) |
| time | second | \(s\) |
| electric current | ampere | \(A\) |
| amount of substance | mole | \(mol\) |
| luminous intensity | candela | \(cd\) |
SI prefixes
A common issue associated with any measuring system is that depending on the quantity, the unit we use might be inconvenient to express. For example, expressing the diameter of an atom \(\left(0.0000000002\textrm{ m}\right)\) in metres is challenging. Therefore, prefixes have been introduced to express such measurements.
| Prefix | Symbol | Multiplied by |
|---|---|---|
| giga | \(G\) | \(10^{9}\) |
| mega | \(M\) | \(10^{6}\) |
| kilo | \(k\) | \(10^{3}\) |
| deci | \(d\) | \(10^{-1}\) |
| centi | \(c\) | \(10^{-2}\) |
| milli | \(m\) | \(10^{-3}\) |
| micro | \(\mu\) | \(10^{-6}\) |
| nano | \(n\) | \(10^{-9}\) |
| pico | \(p\) | \(10^{-12}\) |
Prefixes allow us to express the diameter of an atom in a more sensible way: \(2 \times 10^{–10}\textrm{ m}\). This representation of a physical quantity is called scientific notation.
Example – converting between units
Convert the following quantities into the required units:
- \(150.0\textrm{ mg}\) to grams
- \(0.275\textrm{ g}\) to milligrams
Step 1: Write a conversion factor. In this case, we know from the prefixes that:
\[ 1000\textrm{ mg} = 1\textrm{ g}\] Both \(1000\textrm{ mg}\) and \(1\textrm{ g}\) refer to the same amount, but use different units. Based on this relationship, we can create pair of conversion factors:
\[ \frac{1000\textrm{ mg}}{1\textrm{ g}}\:\:\textrm{and}\:\:\frac{1\textrm{ g}}{1000\textrm{ mg}} \]
Step 2: Multiply the original quantity by the correct conversion factor. This is the conversion factor with the units we want in the numerator (top of the fraction) and the units we have in the denominator (bottom of the fraction).
For part a, we want to convert from mg to g. So, we need to use the conversion factor with g in the numerator: \(\dfrac{1\textrm { g}}{1000\textrm{ mg}}\). This gives us:
\[ 150.0\textrm{ mg}\times\frac{1\textrm{ g}}{1000\textrm{ mg}}=0.1500\textrm{ g}\]
For part b, we want to convert from g to mg. So, we need to use the conversion factor with mg in the numerator: \(\dfrac{1000\textrm{ mg}}{1\textrm{ g}}\). This gives us:
\[ 0.275\textrm{ g}\times\frac{1000\textrm{ mg}}{1\textrm{ g}}=275\textrm{ mg}\]
When you look at these two calculations, you can see that the conversion factor used from the pair depends on the required units.
Your turn – converting between units
Write the conversion factors for the following:
- centimetres and metres
\(1\textrm{ m} = 100\textrm{ cm}\)
\(\dfrac{1\textrm{ m}}{100\textrm{ cm}}\: \textrm{and} \dfrac{100\textrm{ cm}}{1\textrm{ m}} \)
- microlitres and millilitres
\(1000\textrm{ μL} = 1\textrm{ mL}\)
\(\dfrac{1000\textrm{ μL}}{1\textrm{ mL}}\: \textrm{and} \dfrac{1\textrm{ mL}}{1000\textrm{ μL}}\)
- seconds and hours
\(1\textrm{ h} = 60 \times 60\textrm{ s}=360\textrm{ s}\)
\(\dfrac{1\textrm{ h}}{360\textrm{ s}}\: \textrm{and} \dfrac{360\textrm{ s}}{1\textrm{ h}}\)
Convert the following quantities into the required units:
- \(250.0\textrm{ mg}\) to grams
\(250.0\textrm{ mg}\times\dfrac{1\textrm{ g}}{1000\textrm{ mg}}=0.2500\textrm{ g}\)
- \(0.567\textrm{ mL}\) to microlitres
\(0.56\textrm{ mL}\times\dfrac{1000\textrm{ μL}}{1\textrm{ mL}}=567\textrm{ μL}\)
- \(14.8\textrm{ mL}\) to litres
\(14.8\textrm{ mL}\times\dfrac{1\textrm{ L}}{1000\textrm{ mL}}=0.0148\textrm{ L}\)
- 0ºC to Kelvin (\(K=ºC+273\))
\(0ºC+273=273K\)
