Chapter 2 Standard System of Units and Measurement
Chemistry is an empirical and natural science relying on observations of nature and follow-up
experiments; on “experience” of the surrounding world. Like other physical sciences, it depends
on data collection and its interpretation for theory and praxis. The data are only as good as the
observer and the tools used for measurements. All collected data are expressed in dimensions,
which consist of a quantity and a quality. The quantity implies how much of a certain unit is
expressed and the quality refers to the type of unit used to characterize the measured quantity.
Dimensions always contain a factor and a label or, in other words, a digit (numerical value) and a
unit. The number in front of the unit is also referred to as the coefficient.
For example: A measured distance turns out to be 3.0 meters or 3.0 m. The quantity is 3.0
(the coefficient, digit, integer, or factor); the quality is m (unit, label). The label can also be
a combination of individual units such as m/s meters per second for velocity
In order to compare dimensions worldwide, a standardized set of units was devised. This was
necessary, since so many, often arbitrary units evolved throughout the ages often confined to
specific cultures. Scientists of different countries came together in France and adopted the
“Système International” of units and measures by mutual consensus, now simply known as the
SI-system. This system is comprised of 7 standard units, representing 7 general qualities, from
which all others can be derived, and is based for the most part on the metric system or the
powers of 10. The units for time, however, are also based on the ancient Sumerian/Babylonian
(sexagesimal) number system based on multiples of 60 (seconds, minutes, etc.)
The prefixes for the various powers of ten are listed below:
Table 2.1
Prototypes for each quality of measurement (mass, length, etc.) are housed in a museum in
Sèvres, France, close to the capital, Paris, serving as actual physical references for anybody in
the world to use as a standard. Occasionally the prototypes are updated to more modern or more
sophisticated means of standardization.
The seven major qualities with their standard units are listed below:
Table 2.2
The standard units can be combined or given in multiples or quotients, differing from each other
in the powers of ten. All these are referred to as derived units and can have their own names and
symbols. If the units measure the same quantity only in different units of the same quality such
as length or distance, the dimensions are called equalities. For instance 1m is equivalent to
100cm. They both provide the same information. The equality can be expressed as an equation
1m=100cm or as a ratio 1m:100cm (1m/100cm; 100cm/1m) when used in dimensional analysis.
Another important relationship is the conversion ratio, which represents a ratio involving
different units, each relating to a different quality. Density, for instance, describes the
“compactness” of matter or how much volume is occupied by a certain amount of mass, usually
expressed in g/mL. The density for water at room temperature is about 1 g/ml. So the ratio
compares grams with milliliters, a ratio that can either be set up as g:mL or mL:g. A very
effective and streamlined way for calculating with conversion ratios and equalities of dimensions
to solve for unknowns is the method of dimensional analysis. Here a given or known dimension
is written on the left of a dimensional analysis equation, followed by a set of equality and/or
conversion ratios, all written in such a way, that undesired , ”old” units can be cross-cancelled
and that the desired “new” unit occurs to the right of the equal-sign.
Example Problem 2.1
How many cm are in 12 km?
12 km (
𝟏𝟎𝟎𝟎 𝒎
𝟏 𝒌𝒎
)(
𝟏𝟎𝟎 𝒄𝒎
𝟏𝒎
) = 1200000 cm or 1.2 x 106 cm
Example Problem 2.2
If the density of a certain substance is 2.3 g/mL, how may mg are in 5.0 mL?
𝟐.𝟑 𝒈 𝟏𝟎𝟎𝟎 𝒎𝒈
5.0 mL (𝟏 𝒎𝑳)(
𝟏𝒈
) = 11500 mg or 1.2 x 104 mg
The answers in the two examples above are shown in the full expression with all the zeros
involved as “place-holders” but also in the form of scientific notation and the proper number of
significant figures or digits.
Scientific notation was introduced to prevent the occurrence and “clutter” of so many zeros. The
definition requires the numerical value of any measurement to be expressed by the first digit
followed by a decimal and then by all the other numbers of measured significance times a power
of ten and the involved units. The coefficient times the power of ten when multiplied out should
give the original number, expressed in a row of zeros, back again.
Significant figures are considered numbers of actual significance or “meaningfulness” as part of
the measurement. As pointed out earlier, a measurement is only as good as the tool and the
observer. How accurate or precise a measurement is, depends on how carefully a person acquires
the digits from scales of a well- standardized measuring tool. Any number (digit) derived from
an actual measurement is called a significant figure (digit).
The object being measured “knows” it’s true or actual value, but doesn’t reveal it. The observer
needs the “crutch” of the measuring tool to get close to the true value. The closer the
measurement gets to the actual value, the more accurate it is. The more consistent the tool is in
giving repeated accurate measurements, the more precise it is. This also means that the standard
deviation (spread) of data-points and degree of error is minimal.
A measuring tool should be precise (reliable and consistent) and accurate (true to the real
immanent value. Sometimes a measuring device can have a systemic error, like a bathroom that
consistently gives the same faulty weight, being precise but not accurate. By compensating for
the repeated error, one can still get the proper result.
Fig. 2.1
The measurement tools have a gradation of the units they measure. The smallest intervals
separated by lines are called increments. Large intervals that are indicated by consecutively
increasing whole numbers and are mostly separated by 10 smaller subdivisions which are not
marked by numbers. In this case each increment measures one tenth (1/10) of the larger intervals.
Occasionally five increments separate the larger intervals, which indicate that every line
increment measures one fifth (1/5) of the larger subdivisions. This is sometimes seen on
graduated cylinders. The observer must be careful when evaluating the gradation and increments
before measuring a quantity.
Fig. 2.2
The more subdivisions the more accurate and precise data can be collected. Precision also means
how many places after the decimal can still be registered with significance – in good faith. This
is called the degree of precision. The last digit recorded, however, has some degree of
uncertainty and is often estimated but still in good faith (“guesstimated”). It is still considered
significant. A measurement with all the recorded significant figures allows a bit of a latitude for
the last digit of +
1 for the same degree of precision or decimal space such as 2.33 +
0.01, which
−
−
means that the true value is somewhere between 2.32 and 2.34, with the range of uncertainty
being in the hundredths.
Fig. 2.3
When calculating with dimensions certain rules must be applied. These are the
addition/subtraction rule and the division/multiplication rule. When adding or subtracting,
the dimensions are lined up vertically as a column with the decimal points as a type border line.
Once the numbers are subtracted or added, the final result has to be tailored down to the least
precision, based on the number with least precision occurring in the set. For this operation the
degree of precision is the determining factor.
Example Problem 2.3
How much is 10.002m + 12 m + 11.5m?
10.002
12
11.5
m
m
m
33.502 m
precision)
= 33 m
(rounded according to lowest degree of
Example Problem 2.4
How much is 10.002m × 12m × 11.5m ?
(10.002m)(12m)(11.5m) = 1380.276 m3 = 1400 m3 or 1.4 x 103 m3 (rounded to the lowest
occurring number of sig figs and expressed in scientific notation)
When multiplying or dividing, the numbers are computed in the normal way but then modified or
tailored down to the lowest number of significant figures, based on the number of the set that
has the lowest number of significant figures.
In both cases the principle “the weakest link determines the strength of a chain” is the guiding
factor. In other words, the answer cannot be more accurate and/or precise then the lowest quality
measurement of a set.
When addition/ subtraction and division/multiplication are part of the same calculation, the
order of operations should be applied, meaning addition/subtraction goes before
division/multiplication. If the number of significant figures changes during addition or
subtraction, then this new value needs to be considered for the follow-up division or
multiplication operations and thus for the final answer.
Example Problem 2.5
Calculate
(𝟐𝟐.𝟓𝒎+𝟐𝟑𝒎)
(𝟐𝟐.𝟓𝒎+𝟐𝟑𝒎)
𝟏𝟐𝟓𝒔
𝟏𝟐𝟓𝒔
𝟒𝟔𝒎
applying the order of operations in regards to significant figures.
= 𝟏𝟐𝟓𝒔 =0.368
𝒎
𝒔
𝒎
= 0.39 𝒔 = 3.9 x 10-1
𝒎
𝒔