Section 7
MEASUREMENT: QUANTITIES, NUMBERS AND UNITS
Quantity: A property that is measured.
Example: mass, length, time, volume, pressure.
Unit: A standard quantity against which a quantity is measured.
Example: gram, metre, second, litre, pascal; which are units of the above quantities.
Chemists measure various quantities. If the mass of a substance was found to be 6.0 grams this can be expressed as an equation
m = 6.0 g
Here m is the symbol for the quantity mass, 6.0 is a number and g is the symbol for the gram, a standard quantity of mass. The equation is shorthand for
mass = 6.0 × (1 gram)
In general an equation of measurement takes the form
quantity = (number) × (unit)
In print, the symbols for quantities are shown in italics and for units in regular type as is done above.
The equation may be manipulated by the normal rules of algebra. Thus the heading of a table or the axis of a graph could be labelled m/g (i.e. mass in grams) and numbers only tabulated or shown on the axis.
It is important to realise that the magnitude of a quantity expressed as a number without units for the quantity is meaningless unless the quantity is in fact a ratio and is dimensionless (i.e. has no units since the units in the numerator and denominator cancel).
Many quantities are defined in terms of other quantities. Thus density, symbol r, is defined as mass per unit volume.
Thus if the mass and volume of a sample of a substance are known, the density is calculated by dividing the mass by the volume,
For many quantities there are different systems of units. It is essential that one can convert from one set of units to another. This is simply done by replacing the unit in the above equation by its value in terms of the desired unit.
International System of Units (SI units): The internationally adopted system which defines or expresses all quantities in terms of seven basic units, the six used by chemists being:
| length: | metre | m
| mass:
| kilogram
| kg
| time:
| second
| s
| temperature:
| kelvin
| K
| amount of substance:
| mole
| mol
| electric current:
| ampere
| A
| |
Other quantities commonly used in chemistry, and which have special names for the units derived from these basic units are:
| frequency: | hertz | Hz | s−1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| energy: | joule | J | kg m2 s¯2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| force: | newton | N | kg m s¯2 = J m¯1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| pressure: | pascal | Pa | kg m¯1 s¯2 = N m¯2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| power: | watt t | W | kg m2 s¯3 = J s¯1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| electric charge: | coulomb | C | A s | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| electric potential difference: | volt | V | kg m2 s¯3 A¯1 = J A¯1 s¯1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Further quantities used in chemistry but without special names for the derived units are:
area, m2, volume, m3, density, kg m¯3, absorbance, dimensionless (therefore no units).
Coherent SI units: The base units, and those derived from them. Thus the units shown above are coherent SI units. Note that the base unit of mass is the kilogram. This is the only base unit which has a multiple prefix (see below). If, in a calculation of a quantity involving several other quantities, only coherent SI units are used, the units of the required quantity will be coherent (i.e. the appropriate base unit).
The sizes of these units are often unsuitable for some measurements and the decimal multiples, shown below with the name and symbol of the prefix, are used:
| atto | a | 10−18
| femto
| f
| 10−15
| pico
| p
| 10−12
| nano
| n
| 10−9
| micro
| μ
| 10−6
| milli
| m
| 10−3
| centi
| c
| 10−2
| deci
| d
| 10−1
| deca
| da
| 101
| hecto
| h
| 102
| kilo
| k
| 103
| mega
| M
| 106
| tera
| T
| 109
| peta
| P
| 1015
| |
Some multiple units have their own name, the three relevant for chemists being:
| mass | tonne (t) | 1 t = 103 kg = 1 Mg
| volume
| litre (L)
| 1 L = 10¯3 m3 = 1 dm3 = 103 cm3
| length
| ångström (Å)
| 1 Å = 10¯8 cm = 10¯10 m = 100 pm
| | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Units may be written out in full or the symbol used. A mixture of full word and symbol is not permitted. Example: kgram is not permitted. The letter s is never added to the symbol to indicate a plural. A full stop is not written after symbols except at the end of a sentence. Those symbols named after a person have a capital first letter, but when the name of the unit is written out in full a lower case first letter is used. Example: J, joule.
When two or more symbols are combined to indicate a derived unit, a space is left between them. A space is also left between the number and the symbol for the unit, but no space is left between the prefix indicating powers of ten and the symbol to which it applies. When symbols are combined as a quotient, (for example metres per second), either power to the minus one or the solidus may be used (m s¯1 or m/s in the above case). But the solidus may only be used once in a derived unit to avoid ambiguity. Thus writing kg/m/s2 for pascal could be interpreted as kg m¯1 s¯2 or as kg m¯1 s2.
Quantity calculus: The manipulation of the mathematical equations relating quantities and their measured values using the rules of algebra. All calculations in this manual follow the rules of quantity calculus. When one quantity is multiplied by another, no space is left between their symbols. Example: m = ρV. In calculations prefixes for multiples can be replaced by their numerical values.
Some non-SI units are commonly used by chemists:
| temperature | oCelsius (the temperature intervals 1 oC and 1 K are identical)
0 oC = 273.15 K 0 K = −273.15 oC time
| minute (min) = 60 s
| hour (h) = 60 min = 3600 s day (d) = 24 h pressure
| atmosphere (atm) = 1.013 × 105 Pa = 760 torr
| bar (bar) = 105 Pa torr (mmHg) = 133.3 Pa energy
| calorie (cal) = 4.184 J or J = 0.2390 cal
| electron volt (eV) = 1.602 × 10¯19 J kilowatt hour (kWh) = 3.6 × 106 J |
The calorie, still widely used in USA, is the energy required to raise the temperature of 1 g of liquid water 1 oC.
Electron volts are useful at the atomic or molecular level, and kilowatt hours are the common unit for power consumption.
EXERCISES
Write the equations for the following statements using powers of 10 in place of prefixes.
1. Example: The density, ρ , is 3 microgram per cubic millimetre
Answer: ρ = 3 × μg mm−3 = 3 × 10−6 g mm−3 = 3 × 10−6 g (10−3 m)−3 = 3 × 103 g m−3
2. The velocity, v, is 50 millimetres per second
3. The pressure, p, is 64 meganewton per square centimetre
4. The concentration, c, is 2.8 nanogram per microlitre
Answers:- 2,3,4
Express the following numbers under the given heading in a table as a normal equation.
5. Example: 4.2 under the heading 105 m/g
Answer: 105 m/g = 4.2 m = 4.2 × 10¯5 g
6. 23 under the heading 10¯2 t/s
7. 7.8 under the heading 105 p/Pa
Convert the following quantities to SI units.
8. q = 230 cal
9. T = 27 oC
10. t = 32 h
11. p = 61 mmHg (mmHg = 133.3 Pa)
12. ρ = 34 lb ft−3 (lb = 454 g, ft = 12 in, in = 2.54 cm)
