1.1: Units of Measurement--Readings

Metric Units

Scientific measurements all over the world, except in the United States, are made using the metric system. Scientists prefer the metric system because it is internationally standardized. Scientists adopted the metric system for their standard of measurement because it is decimal-based. Units differ by powers of ten. There are 10 millimeters in a centimeter, 10 centimeter in a decimeter, and 10 decimeter in a meter. Compare that to the English system used in the US, where 12 inches per foot, 3 feet per yard, 1760 yards per mile, and so on, are used.

Modern scientists use the SI units, which are a subset of units in the metric system. SI is an abbreviation for the French name for the International System of Units. The SI system is based on seven fundamental units, or base units, each identified with a physical quantity (Table 1.1). Other quantities are made up of combinations of base units; these are called derived units. In this section, we will describe three of the base units-units of mass, length, and temperature. Two of the derived units, units for volume and density, will be described here as well.

Table 1.1 SI Base Units
Quantity	Unit	Symbol
Length	Meter	m
Mass	Kilogram	Kg
Time	Second	s
Temperature	Kelvin	K
Amount of Substance	Mole	mol
Electric Current	Ampere	A
Luminous Intensity	Candela	cd

For quantities much smaller or larger than the base unit, we use decimal prefixes and exponential (scientific) notation (Table 1.2). Because the prefixes are based on powers of 10, SI units are easier to use in calculations than English units.

Table 1.2 Common Decimal Prefixes Used with SI Units
Prefix^*	Prefix Symbol	Word	Conventional Notation	Exponential Notation
tera	T	trillion	1,000,000,000,000	1 x 10¹²
giga	G	billion	1,000,000,000	1 x 10⁹
mega	M	million	1,000,000	1 x 10⁶
kilo	k	thousand	1,000	1 x 10³
--	--	one	1	1 x 10¹
deci	d	tenth	0.1	1 x 10^-1
centi	c	hundredth	0.01	1 x 10^-2
milli	m	thousandth	0.001	1 x 10^-3
micro	µ	millionth	0.000001	1 x 10^-6
nano	n	billionth	0.000000001	1 x 10^-9
pico	p	trillionth	0.000000000001	1 x 10^-12
femto	f	quadrillionth	0.000000000000001	1 x 10^-15

*The prefixes most frequently used by chemists appear in bold type.

Important SI Units in Chemistry

In this section we will discuss units for length, mass, temperature, volume and density. Table 1.3 shows some SI quantities for length, volume, and mass, along with their English-system equivalents.

Length The SI unit of length is the meter (m). The definition is exact and invariant: 1 meter is the distance light travels in a vacuum in 1/299,792,458 of a second. A meter is a little longer than a yard (1 m = 1.094 yd). Biological cells are often measured in micrometers (1 µm = 10^-6 m). On the atomic scale, nanometers (10^-9 m) and picometers (10^-12 m) are used. Many proteins have diameters of about 2 nm; atomic diameters are about 200 pm (0.2 nm). An older unit still is use is the angstrom (1 Å = 10^-10 m = 0.1 nm = 100 pm).

Table 1.3 Common SI-English Equivalent Quantities
Quantity	SI to English Equivalent	English to SI Equivalent
Length	1 km = 0.6214 mile (mi) 1 m = 1.094 yards (yd) 1 m = 39.37 inches (in) 1 cm = 0.3937 inch	1 mile = 1.609 km 1 yd = 0.9144 m 1 foot (ft) = 0.3048 m 1 inch = 2.54 cm (exactly)
Volume	1 cubic meter (m³) = 35.31 ft³ 1 cubic decimeter (dm³) = 0.2642 gallon (gal) 1 dm3 = 1.057 quartz (qt) 1 cubic centimeter (cm³) = 0.03381 fluid ounce	1 ft3 = 0.02832 m³ 1 gal = 3.785 dm³ 1 qt = 0.9464 dm³ 1 qt = 946.4 cm³ 1 fluid ounce = 29.57 cm³
Mass	1 kg = 2.205 pounds (lb) 1 g = 0.03527 ounce (oz)	1 lb = 0.4536 kg 1 oz = 2835 g

Mass The quantity of matter an object contains is its mass. The SI unit of mass is the kilogram (kg). The terms mass and weight have distinct meanings:

* Mass is constant because an object's quantity of matter cannot change.

* Weight is variable because it depends on the local gravitational field.

Because the strength of gravitational field varies with altitude, you (and other objects) weigh slightly less on a high mountain that at sea level.

Temperature is a measure of how hot or cold one object is relative to another. In the laboratory, we measure temperature with a thermometer. We'll consider three temperature scales: the Celsius (°C); the Kelvin (K), and the Fahrenheit (°F) scales. The SI base unit of temperature is the kelvin (K, with no degree sign, °). In the Unites States, the Fahrenheit scale is the one generally used.

The three scales differ in the size of the unit and/or the temperature of the zero point. Figure 1.1 shows the freezing and boiling points of water in the three scales. The Celsius scale set water's freezing point at 0°C and its boiling point (at normal atmospheric pressure) at 100°C. The Kelvin (absolute) scale uses the same size degree as the Celsius scale, but it has a different zero point; that is 0 K, or absolute zero, equals -273.15°C. Thus, in the Kelvin scale, all temperatures are positive.

Figure 1.1 Freezing and boiling points of water in the Fahrenheit, Celsius, and Kelvin (absolute) scales. Because the difference between the freezing point of water and the boiling point of water is 100° on both the Celsius and Kelvin scales, the size of a degree Celsius (°C) and a kelvin (K) are precisely the same. In contrast, both a degree Celsius and a kelvin are 9/5 the size of a degree Fahrenheit (°F). https://saylordotorg.github.io/text_...-skills-4.html

We convert between Celsius and Kelvin scales by using the following equations:

T (in K) = T (in °C) + 273.15

And therefore,

T (in °C) = T (in K) - 273.15

The Fahrenheit scale differs from the the other scales in its zero point and in the size of its degree. Water freezes at 32°F and boils at 212°F. Therefore, 180 Fahrenheit degrees (212°F - 32°F) represents the same temperature change as 100 Celsius degrees or (100 kelvins).

To convert a temperature from °C to °F, we use the following equation:

T (in °F) = 9/5 T (in °C) + 32

To convert a temperature from °F to °C , we use the following equation:

T (in °C) = [ T (in °F) - 32] x 5/9

Volume The amount of space that any sample of matter occupies is defined as the volume. The SI unit of volume is the cubic meter (m³). In chemistry, we often use the non-SI units liter (L) and milliliter (mL). Medical practitioners measure body fluids in cubic decimeters (dm³), which are equivalent to liters:

1 L = 1 dm³ = 10^-3 m³ and,

1 mL = 1 cm³ = 10^-3 dm³ = 10^-3 L = 10^-6 m³

A liter is slightly larger that a quart (1 L = 1.057 qt; 1 qt = 946.4 mL); 1 fluid ounce equals 29.57 mL (2957 cm³).

Density The density (d) of an object is its mass divided by its volume:

Density = mass/volume

We can also rearrange this equation and solve for mass or volume:

Mass = volume x density

or Volume = mass/density

Because volume can change with temperature, so can density. But, at a given temperature and pressure, the density of a substance is a characteristic physical property and, thus, has a specific value.

The SI unit of density is kilograms per cubic meter (kg/m³), but in chemistry, density has units of g/L (g/dm³) or g/mL (g/cm³). Table 1.4 show densities of some common substances. Note that the densities of gases are much lower than those of liquids or solids.

Table 1.4 Densities of Some Common Substances^*
Substance	Physical State	Density (g/cm³)
Hydrogen	gas	0.0000899
Oxygen	gas	0.00133
Grain alcohol	liquid	0.789
Water	liquid	0.998
Table salt	solid	2.16
Aluminum	solid	2.70
Lead	solid	11.3
Gold	solid	19.3

* At room temperature (20°C) and normal atmospheric pressure (1 atm).

Significant Figures in Measurements

Why use significant figures?

Significant figures are the digits in a number that carry meaningful information about its precision. They include all the certain digits plus the first uncertain (or estimated) digit in a measurement. Significant figures indicate the precision of a measurement or calculation. Using the correct number of significant figures helps ensure that reported results are neither over- nor under-stated in terms of accuracy. They help communicate the reliability of measurements and results. For example, if you measure something as 12.3 cm, it implies precision to the nearest tenth of a centimeter, not to the nearest millimeter.

Rules for Identifying Significant Figures:

1. All non-zero digits ARE significant.

Example: 123.45 has 5 significant figures.

2. Zeros between two non-zero digits ARE significant.

Example: 1002 has 4 significant figures; 2025 has 4 significant numbers; 309 has 3 significant numbers)

3. Leading zeros (zeros before the first non-zero digit) are NOT significant.
Example: 0.0045 has 2 significant figures; 0.54 has only TWO significant figures.

4. Trailing zeros to the right of the decimal ARE significant.
Example: 45.00 has 4 significant figures; 0.0350 (3 significant numbers)

45.00 is different from 45: a scientist who measures 45.00 grams knows their value to the nearest 1/100th gram; meanwhile their colleague who measured 92 grams only knows their value to the nearest 1 gram. It's important to understand that "zero" does not mean "nothing." Zero denotes actual information, just like any other number.

5. Trailing zeros in a whole number without a decimal point are NOT significant.
Example: Writing 540 indicates that the zero is NOT significant, and there are only TWO significant figures in this value.

Scientific notation can be used to change the number of significant numbers in a value. For example, let's try writing 1500 with THREE significant figures. 1500 has only 2 significant figures since the trailing zeros are not significant, If we add a decimal at the end, 1500., this means we now have FOUR significant figures. But by using scientific notation: 1.50 x 10³, we create a THREE significant figure value.

For example, a length measurement of 94,500 m can be expressed with 3, 4 or 5 significant figures based on how the measurement was obtained:

3 significant figures: 9.45 x 10⁴ m

4 significant figures: 9.450 x 10⁴ m

5 significant figures: 9.4500 x 10⁴ m

6. Exact numbers have an INFINITE number of significant numbers. This rule applies to numbers that are definitions. For example, 1 meter = 100 cm; 1 hour = 60 minutes; 1 in = 2.54 cm.

Rules for rounding off:

1. If the digit to be removed is 5 or above, the preceding number increased by 1 (so round up).

2. If the digit to be removed is less than 5, the preceding number remains the same (so drop without changing).

3. When doing a multistep calculation, always carry an extra one or 2 digits and round off at the end.

Significant figures in Calculations

a) Multiplication and Division

· The final answer should have the same number of significant figures as the measurement with the fewest number of significant figures.

Example: 10.5 cm x 2.33 cm x 0.70 cm = 17.1255 cm³ = 17 cm³

The measurement 0.70 has only 2 significant figures, so the final answer must only contain 2 significant figures.

Example: 9.235 g/1.55 cm³ = 5.958 g/cm³= 5.96 g/cm³

The measurement 1.55 has 3 significant figures, so the final answer must contain 3 significant figures, before we drop the next digit, we round up.

b) Addition and Subtraction

· The final answer should have the same number of digits after the decimal point as the measurement with the fewest number of digits after the decimal point.

Example: 7.55 g + 1.2 g + 0.875 g = 9.625 g = 9.6 g

The measurement 1.2 contains only 1 digit after the decimal point, so the final answer must only contain 1 digit after the decimal point.

Example: 16.125 mL - 5.95 mL = 10.175 mL = 10.18 mL

The measurement 5.95 contains 2 digits after the decimal point, so the final answer must only contain 2 digits after the decimail point.

c) Mixed Operations

· Do each step in order, keeping track of sig figs at each stage.

· Apply rounding at the final step only, not in intermediate steps (to avoid rounding error). It is recommended to carry an extra digit or 2 until the end.

Example:

Calculate the density of a cube that has a side length of 1.55 cm, and has a mass of 955 g.

Answer

Rounding after each step:

First we must calculate the volume = 1.55 cm x 1.55 cm x 1.55 cm = 3.72 cm³

Density = mass/volume = 955 g / 3.72 cm³ = 257 g/cm³

Rounding only at the final step:

First we must calculate the volume = 1.55 cm x 1.55 cm x 1.55 cm = 3.724 cm³(we will carry an extra digit until the end)

Density = mass/volume = 955 g / 3.724 cm³ = 256 g/cm³

*Always remember to carry an extra digit or 2 and round at the final step only!

Unit Conversions and Dimensional Analysis

n chemistry and science, measurements are only meaningful when they include units. Often we need to convert a measurement from one unit to another—for example meters to centimeters, grams to kilograms, or seconds to hours.

Unit conversions ensure that calculations are consistent and that quantities are expressed in useful forms.

A conversion factor is a ratio of two equivalent quantities. Because the numerator and denominator represent the same quantity, the ratio equals 1, so multiplying by it does not change the value of the measurement.

Example: 1 m=100 cm

Two conversion factors can be written:

We choose the one that cancels the unwanted unit.

Dimensional Analysis

Dimensional analysis is a method of solving conversion problems by treating units as algebraic quantities. We use conversion units to convert between different units. Units can multiply, divide, and cancel just like numbers.

Steps in Dimensional Analysis:

Write the given quantity
Multiply by the appropriate conversion factor
Arrange units so the unwanted unit cancels
Perform the calculation

Examples:

1. Convert 2.50 m to cm:

The conversion factor we need to use: 1 m = 100 cm

2. Convert 65 km/h to m/s:

This is an example where more than one conversion factor is needed, it is an example of a multi-step calculations. The conversion factors we need to use:

1 km = 1000 m and 1 h = 3600 s (you can also use 1 h = 60 min and 1 min = 60 s)

3. Convert 13.6 g/cm³ to lb/in³:

The conversion factors we need to use: 1 lb = 454 g and 1 in = 2.54 cm

= 0.491 lb/in³

Accuracy and Precision

Understanding the difference between accuracy and precision is essential in scientific measurements because it helps us evaluate the quality and reliability of experimental data.

Accuracy

Accuracy refers to how close a measured value is to the true or accepted value of a quantity.

A measurement is accurate if it is very close to the correct value.
Accuracy is affected mainly by systematic errors, such as faulty equipment or incorrect calibration.

Example

Accepted value of density of water at 25°C is 0.997 g/mL

Measurement	Accuracy
0.996 g/mL	Very accurate
0.998 g/mL	Very accurate
1.30 g/mL	Not accurate

If your measurement is close to the true value, it is accurate.

Precision

Precision refers to how close repeated measurements are to each other.

High precision means consistent or reproducible results.
Precision is mainly affected by random errors in measurement.