Measurements and

Calculations

Chapter 2

Quantitative Observation Comparison Based on an Accepted Scale

◦ e.g. Meter Stick Has 2 Parts – the Number and the Unit

◦ Number Tells Comparison◦ Unit Tells Scale

Technique Used to Express Very Large or Very Small Numbers

Based on Powers of 10

1. Move the decimal point so there is only one non-zero number to the left of it. The new number is now between 1 and 9

2. Multiply the new number by 10n

◦ where n is the number of places you moved the decimal point

3. Determine the sign on the exponent n◦ If the decimal point was moved left, n is +◦ If the decimal point was moved right, n is –◦ If the decimal point was not moved, n is 0

1 Determine the sign of n of 10n

◦ If n is + the decimal point will move to the right◦ If n is – the decimal point will move to the left

2 Determine the value of the exponent of 10◦ Tells the number of places to move the decimal

point3 Move the decimal point and rewrite the

number

75,000,000

0.0005710

0.0000000011

8,031,000,000

2.75 x 10-7

5.22 x 104

7.10 x 10-5

9.38 x 1012

Change to scientific notation

41080.642

1.8732

Change to standard notation

0.00065 x 106

391 x 10-2

All units in the metric system are related to the fundamental unit by a power of 10

The power of 10 is indicated by a prefix

The prefixes are always the same, regardless of the fundamental or basic unit

SI unit = meter (m)◦ About 3½ inches longer than a yard

1 meter = one ten-millionth the distance from the North Pole to the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light

Commonly use centimeters (cm)

◦ 1 m = 100 cm◦ 1 cm = 0.01 m = 10 mm◦ 1 inch = 2.54 cm (exactly)

Measure of the amount of three-dimensional space occupied by a substance

SI unit = cubic meter (m3) Commonly measure solid volume in cubic

centimeters (cm3 (cm x cm x cm))◦ 1 m3 = 106 cm3 ◦ 1 cm3 = 10-6 m3 = 0.000001 m3

Commonly measure liquid or gas volume in milliliters (mL)◦ 1 L is slightly larger than 1 quart◦ 1 L = 1 dL3 = 1000 mL = 103 mL ◦ 1 mL = 0.001 L = 10-3 L◦ 1 mL = 1 cm3

Measure of the amount of matter present in an object

SI unit = kilogram (kg) Commonly measure mass in grams (g) or

milligrams (mg)◦ 1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g◦ 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg◦ 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3

g

250 mL to Liters

1.75 kg to grams

88 daL to mL

475 cg to g

328 hm to Mm

0.00075 nL to cL

A measurement always has some amount of uncertainty

Uncertainty comes from limitations of the techniques used for comparison

To understand how reliable a measurement is, we need to understand the limitations of the measurement

To indicate the uncertainty of a single measurement scientists use a system called significant figures

The last digit written in a measurement is the number that is considered to be uncertain

Unless stated otherwise, the uncertainty in the last digit is ±1

Nonzero integers are always significant

◦ How many significant figures are in the following examples: 2753

89.659

0.281

Zeros ◦ Captive zeros are always significant◦ How many significant figures are in the

following examples: 1001.4

55.0702

0.4900008

Zeros ◦ Leading zeros never count as significant figures◦ How many significant figures are in the

following examples: 0.00048

0.0037009

0.0000000802

Zeros ◦ Trailing zeros are significant if the number has a

decimal point◦ How many significant figures are in the

following examples: 22,000 63,850. 0.00630100 2.70900 100,000

Scientific Notation◦ All numbers before the “x” are significant. Don’t

worry about any other rules.◦ 7.0 x 10-4 g has 2 significant figures◦ 2.010 x 108 m has 4 significant figures

If the digit to be removed• is less than 5, the preceding digit stays the

same Round 87.482 to 4 sig figs.

• is equal to or greater than 5, the preceding digit is increased by 1 Round 0.00649710 to 3 sig figs.

In a series of calculations, carry the extra digits to the final result and then round off◦ Ex: Convert 80,150,000 seconds to years

Don’t forget to add place-holding zeros if necessary to keep value the same!!◦ Round 80,150,000 to 3 sig figs.

Count the number of significant figures in each measurement

Round the result so it has the same number of significant figures as the measurement with the smallest number of significant figures

14.593 cm x 0.200 cm =

3.7 x 103 x 0.00340 =

Calculators/computers do not know about significant figures!!!

Exact numbers do not affect the number of significant figures in an answer

Answers to calculations must be rounded to the proper number of significant figures◦ round at the end of the calculation

Exact Numbers are numbers known with certainty

Unlimited number of significant figures They are either

◦ counting numbers number of sides on a square

◦ or defined 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 mL = 1 L; 1 gal = 4 qts. 1 minute = 60 seconds

Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another

Conversion factors are relationships between two units

Conversion factors generated from equivalence statements◦ e.g. 1 inch = 2.54 cm can give or

Arrange conversion factor so starting unit is on the bottom of the conversion factor

You may string conversion factors together for problems that involve more than one conversion factor.

Find the relationship(s) between the starting and final units.

Write an equivalence statement and a conversion factor for each relationship.

Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

Check that the units cancel properly

Multiply all the numbers across the top and divide by each number on the bottom to give the answer with the proper unit.

Round your answer to the correct number of significant figures.

Check that your answer makes sense!

28.5 inches to feet

4.0 gallons to quarts

48.39 minutes to hours

155.0 pounds to grams

2.00 x 108 seconds to hours

682 mg to pounds

3.5 x 10-4 L to cm3

0.091 ft2 to inches2

47.1 mm3 to kL

25 miles per hour to feet per second

4.70 gallons per minute to mL per year

5.6 x 10-6 centiliters per square meter (cL/m2) to cubic meters per square foot (m3/ft2)

Fahrenheit Scale, °F◦ Water’s freezing point = 32°F, boiling point = 212°F

Celsius Scale, °C◦ Temperature unit larger than the Fahrenheit◦ Water’s freezing point = 0°C, boiling point = 100°C

Kelvin Scale, K◦ Temperature unit same size as Celsius◦ Water’s freezing point = 273 K, boiling point = 373

K

Fahrenheit to Celsius oC = 5/9(oF -32)

Celsius to Fahrenheit oF = 1.8(oC) +32

Celsius to Kelvin K = oC + 273

Kelvin to Celsius oC = K – 273

86oF to oC

-5.0oC to oF

352 K to oC

12oC to K

248 K to oF

98.6oF to K

Density is a property of matter representing the mass per unit volume

For equal volumes, denser object has larger mass For equal masses, denser object has small

volume Solids = g/cm3

◦ 1 cm3 = 1 mL Liquids = g/mL Gases = g/L Volume of a solid can be determined by water

displacement Density : solids > liquids >>> gases In a heterogeneous mixture, denser object sinks

Volume

MassDensity

Volume

MassDensity

Density

Mass Volume

Volume Density Mass

What is the density of a metal with a mass of 11.76 g whose volume occupies 6.30 cm3?

What volume, in mL, of ethanol (density = 0.785 g/mL) has a mass of 2.04 lbs?

What is the mass (in mg) of a gas that has a density of 0.0125 g/L in a 500. mL container?

To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume.

A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from 50.0 mL to 58.5 mL. What is the density of the copper in g/mL?

Percent error – absolute value of the error divided by the accepted value, multiplied by 100%.

% error = measured value – accepted value x 100%

accepted value Accepted value – correct value based on

reliable sources. Experimental (measured) value – value

physically measured in the lab.

In the lab, you determined the density of ethanol to be 1.04 g/mL. The accepted density of ethanol is 0.785 g/mL. What is the percent error?

The accepted value for the density of lead is 11.34 g/cm3. When you experimentally determined the density of a sample of lead, you found that a 85.2 gram sample of lead displaced 7.35 mL of water. What is the percent error in this experiment?