# 1.3 Units and Conversions

Learning Objectives

- Learn the units that go with various quantities.
- Express units using their abbreviations.
- Make new units by combining numerical prefixes with units.
- Convert from one unit to another unit of the same type.

A number indicates “how much,” but the unit indicates “of what.” The “of what” is important when communicating a quantity. For example, if you were to ask a friend how close you are to Lake Erie and your friend says “six,” then your friend isn’t giving you complete information. Six *what*? Six miles? Six inches? Six city blocks? The actual distance to the lake depends on what units you use.

Chemistry, like most sciences, uses the International System of Units, or SI for short. The letters *SI* stand for the French “le Système International d’unités”. SI specifies certain units for various types of quantities based on seven fundamental units for various quantities (see Table 1.3.1).

Table 1.3.1 SI base units

Parameter |
SI unit |
Symbol |

Time | seconds | s |

Length | meter | m |

Mass | kilograms | kg |

Electric current | Ampere | A |

Temperature | Kelvin | K |

Amount | mole | mol |

Luminous intensity | candela | cd |

To express a quantity, you need to combine a number with a unit. If you have a length that is [latex]2.4 m[/latex], then you express that length as simply [latex]2.4 m[/latex]. A time of [latex]15,000 s[/latex] can be expressed as [latex]{1.5×10^{4}}[/latex] [latex]{s}[/latex] in scientific notation.

Sometimes, a given unit is not an appropriate size to easily express a quantity. For example, the width of a human hair is very small, and it doesn’t make much sense to express it in metres. SI also defines a series of numerical prefixes that refer to multiples or fractions of a fundamental unit to make a unit more conveniently sized for a specific quantity. Table 1.3.2 “Multiplicative Prefixes for SI Units” lists the prefixes, their abbreviations, and their multiplicative factors. Some of the prefixes, such as kilo-, mega-, and giga-, represent more than one of the fundamental units, while other prefixes, such as centi-, milli-, and micro-, represent fractions of the original unit. Note, too, that once again, we are using powers of 10. Each prefix is a multiple of or fraction of a power of 10.

Table 1.3.2 Multiplicative prefixes for SI units.

Multiply |
Prefix |
Symbol |

10 | deca | d |

10^{2} |
hecto | h |

10^{3} |
kilo | k |

10^{6} |
Mega | M |

10^{9} |
Giga | G |

10^{12} |
Tera | T |

10^{-1} |
deci | d |

10^{-2} |
centi | c |

10^{-3} |
milli | m |

10^{-6} |
micro | μ |

10^{-9} |
nano | n |

10^{-12} |
pico | p |

To use fractions to generate new units, simply combine the prefix with the unit itself; the abbreviation for the new unit is the combination of the abbreviation for the prefix and the abbreviation of the unit. For example:

- The kilometre (km) is [latex]1,000 × {metre}[/latex], or [latex]1,000 m[/latex]. Thus,[latex]5[/latex] kilometres ([latex]5 km[/latex]) is equal to [latex]5,000 m[/latex].
- Similarly, a millisecond (ms) is [latex]{1/1000}×{second}[/latex] one-thousandth of a second. Thus, [latex]25 ms[/latex] is [latex]25[/latex] thousandths of a second. You will need to become proficient in combining prefixes and units.
- You may recognise that one of our fundamental units, the kilogram, automatically has a prefix-unit combination, the kilogram. The word
*kilogram*means [latex]1,000 g[/latex].

In addition to the fundamental units, SI also allows for derived units based on a fundamental unit or units. There are many derived units used in science. For example, the derived unit for area comes from the idea that area is defined as width times height. Because both width and height are lengths, they both have the fundamental unit of metre, so the unit of area is [latex]{metre} × {metre}[/latex] or, [latex]{metre^{2}}[/latex] [latex]{(m^{2})}[/latex]. This is sometimes referred to as “square metres.” A unit with a prefix can also be used to derive a unit for area, so we can also have [latex]cm^{2}[/latex], [latex]mm^{2}[/latex], or [latex]km^{2}[/latex] as acceptable units for area.

Volume is defined as length times width times height, so it has units of metre × metre × metre or metre^{3} ([latex]m^{3}[/latex]), sometimes referred to as “cubic metres.” The cubic metre is a rather large unit. So, another unit is defined that is somewhat more manageable: the litre ([latex]{L}[/latex]). A litre is one-thousandth of a cubic metre. Prefixes can also be used with the litre unit so that we can speak of millilitres (one-thousandth of a litre;[latex]{mL}[/latex]) and kilolitres ([latex]1,000 {L}[/latex]; [latex]{kL}[/latex]).

Units not only are multiplied together but also can be divided. For example, if you are travelling at one metre for every second of time elapsed, your velocity is 1 metre per second, or [latex]1 m/s[/latex]. The word *per* implies division, so velocity is determined by dividing a distance quantity by a time quantity. Other units for velocity include kilometres per hour ([latex]km/h[/latex]) or even micrometres per nanosecond ([latex]μm/ns[/latex]). Later, we will see other derived units that can be expressed as fractions.

Examples 1.3.1

# Problems

- A human hair has a diameter of about [latex]6.0 × 10^{-5} m[/latex]. Suggest an appropriate unit for this measurement and write the diameter of a human hair in terms of that unit.
- What is the velocity of a car if it goes [latex]25 m[/latex] in [latex]5.0 s[/latex]?

## Solutions

- The scientific notation [latex]10^{-5}[/latex] is close to [latex]10^-6[/latex], which defines the micro- prefix. Let us use micrometres as the unit for hair diameter. The number [latex]6.0 × 10^{-5}[/latex] can be written as [latex]60 × 10^{-6}[/latex], and a micrometre is [latex]10^{-6}[/latex] m, so the diameter of a human hair is about [latex]60 μm[/latex].
- If velocity is defined as a distance quantity divided by a time quantity, then velocity is 25 metres/5.0 seconds. Dividing the numbers gives us [latex]25 ÷ 5.0 = 5.0[/latex], and dividing the units gives us metres/second or m/s. The velocity is [latex]5.0 m/s[/latex].

# Test Yourself

- Express the volume of an Olympic-sized swimming pool, [latex]2,500,000 L[/latex], in more appropriate units.
- A common garden snail moves about 6.1 m in 30 min. What is its velocity in metres per minute ([latex]m/min[/latex])?

## Answers

- [latex]2.5 ML[/latex]
- [latex]0.203 m/min[/latex]

# Unit Conversions

In this section, we learn how to replace initial units with other units of the same type to get a numerical value that is easier to comprehend. We can use conversion factors to convert one unit to another.

- For instance, the conversion of milligrams to grams. From prefixes, we know [latex]1000mg = 1g[/latex]

Both [latex]1000mg[/latex] and [latex]1g[/latex] refer to the same amount but in different units. Based on this relationship, we can create a pair of conversion factors:

[latex]\frac{1000mg}{1g}\:\:\textrm{and}\:\:\frac{1g}{1000mg}[/latex]

- Express [latex]150.0mg[/latex] in grams?

We can calculate [latex]150mg[/latex] in grams using the conversion factor [latex]\frac{1g}{1000mg}[/latex] as follows:

[latex]\frac{1g}{1000mg}\times150.0mg=0.1500g[/latex]

- Express 0.275g in milligrams?

Here we can use the other conversion factor [latex]\frac{1000mg}{1g}[/latex] as follows:

[latex]\frac{1000mg}{1g}\times0.275g=275mg[/latex]

When you observe the last two calculations, you can identify that the conversion factor used from the pair depends on the unit required to be converted.

Examples 1.3.2

# Problems

1. Write conversion factors for the following units

a) Litres and millilitres

b) Ounces and grams

2. Convert 3.500lb pounds to grams [latex](1lb=454g[/latex]).

## Solutions

1. a) The relationship between two units can be expressed as [latex]1L=1000mL[/latex]. Therefore, the conversion factors will be:

[latex]\frac{1L}{1000mL}[/latex] and [latex]\frac{1000mL}{1L}[/latex].

b) The relationship between two units can be expressed as [latex]1oz=28.3g[/latex]. Therefore, the conversion factors will be:

[latex]\frac{1oz}{28.3g}[/latex] and [latex]\frac{28.3g}{1oz}[/latex].

2. Relationship between two units – [latex]1lb=454g[/latex]. Conversion factors – [latex]\frac{1lb}{454g} and \frac{454g}{1lb}[/latex]. Only one of these conversion factors is required to convert pounds to grams. The conversion factor, [latex]\frac{454g}{1lb}[/latex], is required as it allows for the cancellation of the pound units, leaving the gram as the new unit.

[latex]\frac{454g}{1lb}\times3.500lb=1589g[/latex]

# Test Yourself

- Write the conversion factors for centimetres to metres
- Convert [latex]250.0mg[/latex] to grams

## Answers

- [latex]1m = 100cm[/latex]. Therefore, [latex]\frac{1m}{100cm}[/latex] and [latex]\frac{100cm}{1m}[/latex].
- [latex]\frac{1g}{1000mg}\times250.0mg=0.2500g[/latex]

Key Takeaways

- Numbers tell “how much,” and units tell “of what.”
- Chemistry uses a set of fundamental units and derived units from SI units.
- Chemistry uses a set of prefixes that represent multiples or fractions of units.
- Units can be multiplied and divided to generate new units for quantities.
- Units can be converted to other units using the proper conversion factors.
- Conversion factors are constructed from equalities that relate two different units.

Exercises