CHAPTER TWO
MEASUREMENT I
To measure, we need to know or define the quantity to be measured and the units for measuring it. In 1971, a system known as the International System of Units (Système International) and seven basic units were agreed upon as follows. Other quantities can be obtained from these basic quantities and are referred to as derived quantities.
| Basic quantity | SI units | Symbols |
|---|---|---|
| Length | Metre | m |
| Mass | Kilogram | kg |
| Time | Second | s |
| Electric current | Ampere | A |
| Thermodynamic temperature | Kelvin | K |
| Luminous intensity | Candela | Cd |
| Amount of substance | Mole | mol |
Length
This is the measure of distance between two points in space. The SI unit for length is the metre (m). Therefore:
- 1 km = 1000 m
- 1 Hm = 100 m
- 1 Dm = 10 m
- 1 mm = 0.001 m
Length is measured using a metre rule (100 cm) or a tape measure (100 m, 300 m, 500 m).
Area
This is the measure of the extent of a surface. It is a derived quantity of length. Its SI unit is square metres (m2). Other units include cm2, km2, etc. Formulas are used to determine areas of regular bodies, while for irregular bodies an approximation of area is used.
Volume
This is the amount of space occupied by matter. The SI unit for volume is cubic metre (m3).
Other submultiples are cm3, mm3, and l. Hence, 1 m3 = 1,000,000 cm3 and 1 l = 1,000 cm3. Volume can be measured using a measuring cylinder, eureka can, pipette, burette, volumetric flask, beaker, etc.
Mass
This is the quantity of matter contained in a substance. Matter is anything that occupies space and has weight. The SI unit for mass is the kilogram (kg). Other submultiples used are grams (g), milligrams (mg), and tonnes (t). 1 kg = 1,000 g = 1,000,000 mg = 0.001 tonnes. A beam balance is used to measure mass.
Density
This is mass per unit volume of a substance. It is symbolized by rho (ρ) and its SI unit is kg/m3. Density = mass / volume.
Examples
A block of glass of mass 187.5 g is 5.0 cm long, 2.0 cm thick, and 7.5 cm high. Calculate the density of the glass in kg/m3.
Solution:
Density = mass / volume = (187.5 / 1000) / (2.0 × 7.5 × 5.0 / 1,000,000) = 2,500 kg/m3.
The density of concentrated sulphuric acid is 1.8 g/cm3. Calculate the volume of 3.1 kg of the acid.
Solution:
Volume = mass / density = 3,100 / 1.8 = 1,722 cm3 or 0.001722 m3.
The following is a list of densities of some common substances:
| Substance | Density (g/cm3) | Density (kg/m3) |
|---|---|---|
| Platinum | 21.4 | 21,400 |
| Gold | 19.3 | 19,300 |
| Lead | 11.3 | 11,300 |
| Silver | 10.5 | 10,500 |
| Copper | 8.93 | 8,930 |
| Iron | 7.86 | 7,860 |
| Aluminium | 2.7 | 2,700 |
| Glass | 2.5 | 2,500 |
| Ice | 0.92 | 920 |
| Mercury | 13.6 | 13,600 |
| Sea water | 1.03 | 1,030 |
| Water | 1.0 | 1,000 |
| Kerosene | 0.80 | 800 |
| Alcohol | 0.79 | 790 |
| Carbon (IV) oxide | 0.00197 | 1.97 |
| Air | 0.00131 | 1.31 |
| Hydrogen | 0.000089 | 0.089 |
Example
The mass of an empty density bottle is 20 g. Its mass when filled with water is 40.0 g and 50.0 g when filled with liquid X. Calculate the density of liquid X if the density of water is 1,000 kg/m3.
Solution:
Mass of water = 40 – 20 = 20 g = 0.02 kg.
Volume of water = 0.02 / 1,000 = 0.00002 m3. Volume of liquid = volume of bottle.
Mass of liquid = 50 – 20 = 30 g = 0.03 kg.
Therefore, density of liquid = 0.03 / 0.00002 = 1,500 kg/m3.
Relative density
This is the density of a substance compared to the density of water. It is symbolized by d and has no units since it is a ratio.
Relative density (d) = density of substance / density of water. It is measured using a relative density bottle.
Example
The relative density of some type of wood is 0.8. Find the density of the wood in kg/m3.
Solution:
Density of substance = d × density of water
Density of substance = 0.8 × 1,000 = 800 kg/m3.
Densities of mixtures
We use the following formula to calculate densities of mixtures:
Density of the mixture = mass of the mixture / volume of the mixture
Example
100 cm3 of fresh water of density 1,000 kg/m3 is mixed with 100 cm3 of sea water of density 1,030 kg/m3. Calculate the density of the mixture.
Solution:
Mass = density × volume
Mass of fresh water = 1,000 × 0.0001 = 0.1 kg.
Mass of sea water = 1,030 × 0.0001 = 0.103 kg.
Mass of mixture = 0.1 + 0.103 = 0.203 kg.
Volume of mixture = 100 + 100 = 200 cm3 = 0.0002 m3.
Therefore, density = mass / volume = 0.203 / 0.0002 = 1,015 kg/m3.
Time
This is a measure of the duration of an event. The SI unit for time is the second (s). Submultiples of the second include milliseconds, microseconds, minutes, hours, days, weeks, and years. Time is measured using clocks, stopwatches, wristwatches, and digital watches.
Accuracy and errors
Accuracy is the closeness of a measurement to the correct value of the quantity being measured. It is expressed as an error. An error is therefore the deviation of a measurement from the correct value being measured. The smaller the error, the more accurate the measurement.
% error = (sensitivity / size measured) × 100.
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