Chapter 1: MeasurementsThe metric system

Becoming more common in our country.

The SI systemDeveloped in 1960, also called the international system of units.Set of seven standard quantities of which all others can be derived.

Common Metric UnitsLength

Metric and SI unit is the Meter (m)1 m = 39.4 inches1 inch = 2.54 centimeters (cm)

VolumeAmount of space occupied by a substanceMetric unit = Liter (L), SI unit = m3

1 L = 1.06 quarts946 milliliters = 1 quart

Common Metric UnitsMass

Metric unit is the gram (g), SI unit is the kilogram (kg)1 kg = 2.2 pounds454 g = 1 pound

TimeMetric and SI unit is the second (s)

Common Metric UnitsTemperature

Indicates how hot or cold an object isMetric unit is Celsius (C) and SI unit is Kelvin (K)These will be discussed in more detail in Chapter 2.

Scientific NotationUsed when numbers are really small or really large.General format:A x 10n

A is a number between 1 and 10.n is the exponent on 10 (called the

power) and is equal to the number of places that the decimal place is moved.

Scientific NotationExamples0.000058 = 5.8 x 10-5

• The decimal is moved five places to the right, hence the –5 power. Note that small numbers (0< x <1) will always have a negative exponent.

58,000 = 5.8 x 104

• The decimal is moved four places to the left, hence the power of 4 is used. Note that large numbers (>1) will always have a positive exponent.

Scientific NotationOn your calculator, use the EE or EXP key when entering these numbers.

Example:(1.8 x 105) x (3.2 x 10-3) =

576 or 5.76 x 102

Measured & Exact Numbers

Measured numbers are those which come from using any kind of a measuring device.

Examples are: ruler, graduated cylinder, thermometer, scale, etc.Include both certain digits and one uncertain digit.Certain digits are digits that all would agree on.Uncertain digit is the last digit that requires YOU to make a “best guess.”

Measured & Exact Numbers

“Best Guess” depends on the smallest increments on the measuring device.

A ruler may have markings for every 0.1cm. Your “best guess” might be to 0.05cm or even 0.02cm.

Measured & Exact Numbers

A graduated cylinder – depends on size.A 10-mL has markings every 0.2-mL.A 50-mL has markings every 1-mL.

Measured & Exact Numbers

A Thermometer has markings every degree.

Measured & Exact Numbers

If a measurement results where it is exactly lined up with a mark, then a zero at the end may be needed.Not everyone will agree on the “best guess,” which means that all measured numbers have some uncertainty.Digital devices – the last digit in display is uncertain.

Measured & Exact Numbers

An exact number is a either a counted number or an established relationship between two units.

Examples are:12 computers in the room15 apples in the bag1 foot = 12 inches1 pound = 16 ounces

Learning CheckAre the following numbers exact or measured?

There are 50 pages in a bookA coin weighs 2.87 grams There are 100 centimeters in one meterA graduated cylinder contains 40.0mL of water

Significant FiguresGoal: Report answers from calculations by rounding the answer to the correct number of significant figures.Accuracy refers to measurement(s) that are close to the true (accepted) measurement.Precision refers to the agreement of values obtained while repeating the same measurement.In the lab, we desire both of these.

Significant FiguresRules are:

1. A number is significant if it is:a. A non-zero digitb. A zero between two non-zero digitsc. A zero at the end of a decimal numberd. Any digit in the coefficient of a number written

in scientific notation

2. A number is not significant if it is:a. A zero at the beginning of a decimal numberb. A zero used as the placeholder in a large

number without a decimal point.

Learning CheckHow many significant digits do each of the following measured numbers have?

1.02 0.0045 91,0002.50 x 10-8

0.0150250.

Significant Figures in Calculations

A calculation (using measured numbers) is only as accurate as the number that had the least number of significance.Calculators almost NEVER provide answers with the proper significance.Thus, answers will need to be rounded (or possibly require additional zero’s).

Rounding NumbersRules:

1. If the first digit to be dropped is a 4 or less, it and all other following digits can be dropped.

2. If the first digit to be dropped is a 5+, then the last retained digit is increased by one.

3. If the digit to be dropped is exactly 5 (nothing after it), then round up if it makes the digit even or down if that makes the digit even.

3.46, round to two s.f.’s.54.48, round to two s.f.’s.135.51, round to three s.f.’s.8.74528, round to three s.f.’s.

Learning Check

Multiplication & DivisionThe final answer is rounded to have the same number of significant digits as the measurement with the fewest s.f.’s.

Example: 24.65 x 0.67 = 16.5155 (calc.)

Example: (2.85 x 67.4) / 4.39 = 43.756264 (calc.)

Example: (8.00) / (0.250) = 32 (calc.)

Addition & SubtractionThe final answer is rounded to have the same number of decimal places as the measured number with the fewest decimal places.

Example: 2.045 + 34.1 = 36.145 (calc.)

Example: 255 – 175.65 = 79.35 (calc.)

Example: 89.15 – 82.95 = 6.2 (calc.)

Mixed CalculationsApply the rules for each type of calculation.Don’t round intermediate answers – round only at the very end.

Example: (23.8 + 4.25) / 67.85 = 0.413411938

Example: (17.92 – 16.82) x 0.01957 = 0.021527

SI & Metric Prefixes To increase or decrease metric units, prefixes are used as a multiplier.Prefixes that increase the size: G = giga,

M = mega, 106 or 1,000,000k = kilo, 103 or 1,000

Prefixes that decrease the size:d = deci, 10-1 or 0.1c = centi, 10-2 or 0.01m = milli, 10-3 or 0.001m = micro, 10-6 or 0.000001

Some equalities 1 m = 100 cm 1 kg = 1000 g 1 L = 1,000,000 mL 1000 m = 1 km 10 dL = 1 L Note: Volume – a milliliter is equivalent

to a cube that measures 1 cm x 1cm x 1cm or 1 cm3 (also referred to as a cubic centimeter – cc).

Conversion FactorsTo change from one unit to another, you will multiply by the appropriate conversion factor.Conversion factors are written in the form of a fraction.

Example: 100 cm = 1 m can be written as:

cm 100

m 1or

m 1

cm 100

Using Conversion Factors / Dimensional Analysis

Fence-post method uses number + unit.

When conversion factor is multiplied correctly, then all of the units – except those desired in the answer – will cancel out.

Dimensional Analysis Ex) Convert 35 inches to

centimeters

Ex) Convert 160 pounds to kilograms

Dimensional AnalysisFor problems with more than one step, they can be done either as a series of steps or as one continuous problem. Note that you will only round answers at the very END!

Dimensional AnalysisConvert 25,000 feet to kmAs one continuous conversion…

Dimensional AnalysisProblems with units in both the numerator and denominator can create problems.Will see some problems involving “clinical” calculations.

Dimensional AnalysisEx) Convert 35 miles per hour to meters per second

Dimensional AnalysisEx) A certain medicine requires that 250mg per kilogram of body mass is to be given. What dose should be given to a child that weighs 48lbs?

Dimensional AnalysisArea and volume conversions can also be easily missed. Ex) A concrete footing measuring 3.0 feet by 2.0 feet by 1.5 feet is poured. What volume of concrete is needed, in cm3?Note: 1 ft3 12 in3, Rather: 1 ft3 = (12)3 in3 = 1728 in3 !

Dimensional AnalysisVolume of concrete = 3.0ft x 2.0ft x 1.5ft = 9.0ft3

Alternatively, EACH dimension could be converted to cm first, then multiplied out to yield the volume.

DensityThe density of an object is equal to its mass divided by its volume.

You will be asked to solve for any one of the D, m, or V’s.

V

m D

Density ProblemsEx) An object has a mass of 35.0g and occupies 5.2mL. What is its density?

Density ProblemsEx) The mass of an iron bar is 1500g. What volume does it occupy? The density of iron is 7.9g/mL.Can rearrange formula or use Density as conversion factor.