Numbers in Science

Chapter 2

Measurement What is measurement?

Quantitative Observation Based on a comparison to an accepted scale.

A measurement has 2 Parts – the Number and the Unit Number Tells Comparison Unit Tells Scale

There are two common unit scales English Metric

The Unit

The measurement System units English (US)

Length – inches/feet Distance – mile Volume –

gallon/quart Mass- pound

Metric (rest of the world)

Length – meter Distance – kilometer Volume – liter Mass - gram

Related Units in the Metric System 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 unit

Fundamental Unit 100

Fundamental SI Units Established in 1960 by an international

agreement to standardize science units These units are in the metric system

Physical Quantity Name of Unit Abbreviation

Mass Kilogram kg

Length Meter m

Time Second s

Temperature Kelvin K

Energy Joules J

Pressure Pascal Pa

Volume Cubic meters m3

Length…..

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

1 meter = distance between marks on standard metal rod in a Paris vault or distance covered by a certain number of wavelengths of a special color of light

Commonly use centimeters (cm)

1 inch (English Units) = 2.54 cm (exactly)

Figure 2.1: Comparison of English and metric units for length on a ruler.

Volume

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) Commonly measure liquid or gas volume in milliliters (mL)

◦ 1 L is slightly larger than 1 quart◦ 1 mL = 1 cm3

Mass

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

Temperature Scales

Any idea what the three most common temperature scales are?

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 (SI unit)◦ Temperature unit same size as Celsius◦ Water’s freezing point = 273 K, boiling point = 373 K

Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

The number

Scientific Notation Technique Used to Express Very Large or

Very Small Numbers 135,000,000,000,000,000,000 meters 0.00000000000465 liters

Based on Powers of 10 What is power of 10 Big?

0,10, 100, 1000, 10,000 100, 101, 102, 103, 104

What is the power of 10 Small? 0.1, 0.01, 0.001, 0.0001 10-1, 10-2, 10-3, 10-4

Writing Numbers in Scientific Notation1. Locate the Decimal Point : 1,438.2. Move the decimal point to the

right of the non-zero digit in the largest place- The new number is now between 1 and 10

- 1.4383. Now, multiply this number by a

power of 10 (10n), where n is the number of places you moved the decimal point- In our case, we moved 3 spaces, so n = 3 (103)

The final step for the number……

4. Determine the sign on the exponent nIf 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

- We moved left, so 3 is positive- 1.438 x 103

Writing Numbers in Standard Form

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 point

3 Move the decimal point and rewrite the number

Try it for these numbers: 2.687 x 106 and 9.8 x 10-2

We reverse the process and go from a number in scientific notation to standard form…..

Let’s Practice….. Change these numbers to Scientific Notation:

1,340,000,000,000 697, 000 0.00000000000912

Change these numbers to Standard Form: 3.76 x 10-5

8.2 x 108

1.0 x 101

1.34 x 1012

6.97 x 105

9.12 x 10-

12

0.0000376820,000,00010

Are you sure about that number?

Uncertainty in Measured Numbers

A measurement always has some amount of uncertainty, you always seem to be guessing what the smallest division is…

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

cm

Rules, Rules, Rules…. We follow guidelines (i.e. rules) to determine

what numbers are significant Nonzero integers are always significant

2753 89.659 .281

Zeros Captive zeros are always significant (zero sandwich)

1001.4 55.0702 4780.012

Significant Figures – Tricky Zeros

Zeros Leading zeros never count as significant figures

0.00048 0.0037009 0.0000000802

Trailing zeros are significant if the number has a decimal point 22,000 63,850. 0.00630100 2.70900 100,000

Significant FiguresScientific 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

How many significant figures are in these numbers? 102,340 0.01796 92,017 1.0 x 107 1,200.00 0.1192 1,908,021.0 0.000002 8.01010 x 1014

Have a little fun remembering sig figs http://www.youtube.com/watch?v=ZuVPkBb-z

2I

Exact Numbers

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

Calculations with 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

For addition and subtraction, the last digit to the right is the uncertain digit. Use the least number of decimal places

For multiplication, count the number of sig figs in each number in the calculation, then go with the smallest number of sig figs Use the least number of significant figures

Rules for Rounding Off

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

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

Examples of Sig Figs in Math

1) 5.18 x 0.0208

2) 21 + 13.8 + 130.36

3) 116.8 – 0.33

Answers must be in the proper number of significant digits!!!

Solutions:

1) 0.107744 round to proper # sig fig1) 5.18 has 3 sig figs, 0.0208 has 3 sig figs so

answer is 0.108

2) 165.471) Limiting number of sig figs in addition is the

smallest number of decimal places = 12 (no decimals) answer is 165

3) 116.471) Same rule as above so answer is 116.5

Moving unit to unit: Conversion

Exact Numbers

Exact Numbers are numbers known with certainty

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

The Metric System

Fundamental Unit 100

Movement in the Metric system

In the metric system, it is easy it is to convert numbers to different units. Let’s convert 113 cm to meters

Figure out what you have to begin with and where you need to go.. How many cm in 1 meter?

100 cm in 1 meter Set up the math sentence, and check that the

units cancel properly. 113 cm [1 m/100 cm] = 1.13 m

Let’s Practice converting metric units

250 mL to Liters0.250 mL

1.75 kg to grams1,750 grams

88 µL to mL0.088 mL

475 cg to kg47,500,000 or4.75 x 107

328 mm to dm 3.28 dm

0.00075 nL to µL0.75 µL

Converting Between Metric and non-Metric

(English) units

Converting non-Metric Units

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

Conversion factors are relationships between two units

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

in1

cm54.2

cm54.2

in1

Converting non-Metric Units Arrange conversion factor so starting unit is on

the bottom of the conversion factor Convert kilometers to miles

You may string conversion factors together for problems that involve more than one conversion factor. Convert kilometers to inches

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.

Practice Convert 1.89 km to miles

Find equivalence statement 1mile = 1.609 km 1.89 km (1 mile/1.609 km) 1.17 miles

Convert 5.6 lbs to grams Find equivalence statement 454 grams = 1 lb 5.6 lbs(454 grams/1 lb) 2500 grams

Convert 2.3 L to pints Find equivalence statements: 1L = 1.06 qts, 1 qt

= 2 pints 2.3 L(1.06 qts/1L)(2 pints/1 qt) 4.9 pints

Temperature Conversions To find Celsius from Fahrenheit

oC = (oF -32)/1.8 To find Fahrenheit from Celsius

oF = 1.8(oC) +32 Celsius to Kelvin

K = oC + 273 Kelvin to Celsius

oC = K – 273

Temperature Conversion Examples1) 180°C to Kelvin

1) To convert Celsius to Kelvin add 2732) 180+ 273 = 453 K

2) 23°C to Fahrenheit1) Use the conversion factor: F = (1.80)C + 322) F = (1.80)23 + 323) F=73.4 or 73°F

3) 87°F to Celsius1) Use the conversion factor C=5/9(F-32)2) C = 5/9(87-32)3) C = 30.5555555… or 31°C

4) 694 K to Celsius1) To convert K to C, subtract 2732) 694-273= 421°C

Measurements and

Calculations

Density Density is a physical 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

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

Using Density in Calculations

Volume

MassDensity

Density

Mass Volume

Volume Density Mass

Density Example Problems What is the density of a metal with a

mass of 11.76 g whose volume occupies 6.30 cm3?

What volume 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?

How could you find your density?

Volume by displacement 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?

A student masses a piece of unusually shaped metal and determines the mass to be 187.7 grams. After placing the metal in a graduated cylinder, the water level rose from 50.0 mL to 60.2 mL. What is the density of the metal?

A piece of lead (density = 11.34 g/cm3) has a mass of 162.4 g. If a student places the piece of lead in a graduated cylinder, what is the final volume of the graduated cylinder if the initial volume is 10.0 mL?

Percent Error 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.

Percent Error Example 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?

Joe measured the boiling point of hexane to be 66.9 °C. If the actual boiling point of hexane is 69 °C , what is the percent error?

A student calculated the volume of a cube to be 68.98 cm3. If the true volume is 71.08 cm3, what is the student’s percent error?

Tom used the density of copper and the volume of water displaced to measure the mass of a copper pipe to be 145.67 g. When he actually weighed the sample, he found a mass of 146.82 g. What was his percent error?