Chapter 2

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Chapter 2

Measurements and Calculations

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Lab Demo

Page 69: Zinc and HCl

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Objectives:

Describe the difference between a qualitative and a quantitative measurement.

Describe the difference between accuracy and precision. Write a number in scientific notation. State the appropriate units for measuring length, volume,

mass, density, temperature and time in the metric system. Determine the number of significant figures in a

measurement or calculation. Calculate the percent error in a measurement. Calculate density given the mass and volume, the mass

given the density and volume, and the volume given the density and mass.

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Chapter 2Section 1

Scientific Method

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Scientific Method is a logical approach to solving problems by observing and collecting data, formulating hypotheses, testing hypotheses and formulating theories that are supported by data.

Observations Hypothesis Experimentation

Theory

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Observations

• Collecting data

• Measuring

• Communicating with other scientists

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MeasurementsMeasurements are divided into two sets:

Qualitative – a descriptive measurement.Color, hardness, shininess, physical state.(non-numerical)

Quantitative – a numerical measurement.Mass in grams, volume in milliliters, length in meters.

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Hypothesis

A tentative explanation that is consistent with the observations (educated guess).

An experiment is then designed to test the hypothesis.

Predict the outcome from the experiments.

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Theory

Attempts to explain why something happens.

Has experimental evidence to support the theory.

Observations, data and facts.

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ClassworkWhat is the scientific theory?

What is the difference between qualitative and quantitative measurements?

Which of the following are quantitative?a. The liquid floats on water?b. The metal is malleable?c. A liquid has a temperature of 55.6 oC?

How do hypothesis and theories differ?

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Section 2

Units of Measurement

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Measurements represent quantities.

A quantity is something that has magnitude, size or amount.

All measurements are a number plus a unit (grams, teaspoon, liters).

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Number vs. Quantity Quantity = number + unit

UNITS MATTER!!

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Use SI units — based on the metric Use SI units — based on the metric systemsystem

Length Length

MassMass

VolumeVolume

TimeTime

TemperatureTemperature

UNITS OF MEASUREMENTUNITS OF MEASUREMENT

Meter, mMeter, m

kilogram, kgkilogram, kg

Seconds, sSeconds, s

Celsius degrees, ˚CCelsius degrees, ˚Ckelvins, Kkelvins, K

Liter, LLiter, L

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UNITS OF MEASUREMENTUNITS OF MEASUREMENT

Use SI units — based on the metric Use SI units — based on the metric systemsystem

Amount Amount

Electric currentElectric current

Luminous IntensityLuminous Intensity

mole, molmole, mol

ampere, Aampere, A

candela, cdcandela, cd

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SI Prefix Conversions1 T(base) = 1 000 000 000 000(base) = 1012 (base)

1 G(base) = 1 000 000 000 (base) = 109 (base)

1 M(base) = 1 000 000 (base) = 106 (base)

1 k(base) = 1 000 (base) = 103 (base)

1 h(base) = 100 (base) = 102 (base)

1 da(base) = 101 (base)

1 (base) = 1 (base) meter, gram, liter

1 d(base) = 10-1 (base)

1 c(base) = 10-2 (base)

1 m (base) = 10-3(base)

1 µ(base) = 1 000 000 (base) = 10-6(base)

1 n(base) = 1 000 000 000 (base) = 10-9(base)

1 p(base) = 1 000 000 000 000(base) = 10-12(base)

Tera-

Giga-

Mega-

Kilo-

Hecto-

Deka-

Base

Deci-

Centi-

Milli-

Micro-

Nano-

Pico-

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mega- M 106

deci- d 10-1

centi- c 10-2

milli- m 10-3

Prefix Symbol Factor

micro- 10-6

nano- n 10-9

kilo- k 103

BASE UNIT --- 100

giga- G 109

deka- da 101

hecto- h 102

tera- T 1012

mo

ve le

ft

mo

ve r

igh

tSI Prefix Conversions

pico- p 10-12

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1. 1000 m = 1 ___ a) mm b) km c) dm

2. 0.001 g = 1 ___ a) mg b) kg c) dg

3. 0.1 L = 1 ___ a) mL b) cL c) dL

4. 0.01 m = 1 ___ a) mm b) cm c) dm

Learning Check

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SI Prefix Conversions

1) 20 cm = ______________ m

2) 0.032 L = ______________ mL

3) 45 m = ______________ m

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Derived SI Units

Many SI units are combinations of the quantities shown earlier.

Combinations of SI units form derived units.

Derived units are produced by multiplying or dividing standard units.

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VolumeVolume (m3) is the amount of space occupied by an object.

length x width x height

Also expressed as cubic centimeter (cm3).

When measuring volumes in the laboratory a chemist typically uses milliliters (mL).

1 mL = 1 cm3

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Density

Density – the ratio of mass to volume, or mass divided by volume.

Density = D =

Density is often expressed in grams/milliliter or g/mL

massvolume

m v

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Density is a characteristic physical property of a substance.

It does not depend on the size of the sample.

As the sample’s mass increases, its volume increases proportionally.

The ratio of mass to volume is constant.

Density

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DensityCalculating density is pretty straightforward.

You measure the mass of an object by using a balance and then determine the volume.

For a liquid the volume is easily measured using for example a graduated cylinder.

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DensityFor a solid the volume can be a little more difficult.

If the object is a regular solid, like a cube, you can measure its three dimensions and calculate the volume.

Volume = length x width x height

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Density

If the object is an irregular solid, like a rock, determining the volume is more difficult.

Archimedes’ Principle – states that the volume of a solid is equal to the volume of water it displaces.

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Density

Put some water in a graduated cylinder and read the volume. Next, put the object in the graduated cylinder and read the volume again.

The difference in volume of the graduated cylinder is the volume of the object.

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Volume Displacement

A solid displaces a matching volume of water when the solid is placed in water.

33 mL25 mL

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Learning Check

What is the density (g/cm3) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL?

1) 0.2 g/cm3 2) 6 g/cm3 3) 252 g/cm3

33 mL

25 mL

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PROBLEM: Mercury (Hg) has a density PROBLEM: Mercury (Hg) has a density of 13.6 g/cmof 13.6 g/cm33. What is the mass of 95 mL . What is the mass of 95 mL of Hg in grams? of Hg in grams?

PROBLEM: Mercury (Hg) has a density PROBLEM: Mercury (Hg) has a density of 13.6 g/cmof 13.6 g/cm33. What is the mass of 95 mL . What is the mass of 95 mL of Hg in grams? of Hg in grams?

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StrategyStrategy

Use density to calc. mass (g) from Use density to calc. mass (g) from volume.volume.

PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?

First, note thatFirst, note that 1 cm1 cm33 = 1 mL = 1 mL

Density mass (g)volume (ml )

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PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?

Density mass (g)volume ( ml )

13.6 g/mL mass (g)

95 ( ml)

Mass = 1,292 grams

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Learning Check

Osmium is a very dense metal. What is its

density in g/cm3 if 50.00 g of the metal occupies a volume of 2.22cm3?

1) 2.25 g/cm3

2) 22.5 g/cm3

3) 111 g/cm3

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Solution

Placing the mass and volume of the osmium metal into the density setup, we obtain

D = mass = 50.00 g = volume2.22 cm3

= 22.522522 g/cm3 = 22.5 g/cm3

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Learning Check

The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?

1) 0.614 kg

2) 614 kg

3) 1.25 kg

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Learning Check

The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?

1) 0.614 kg

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Homework

Density Worksheet

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Conversion FactorsConversion factor – a ratio derived from the equality between two different units that can be used to convert from one unit to the other.

Example: the conversion between quarters and dollars:

4 quarters 1 dollar 1 dollar or 4 quarters

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Conversion Factors

When you want to use a conversion factor to change a unit in a problem, set up the problem as follows:

quantity sought = quantity given x conversion factor

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Conversion Factors

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Learning Check

Write conversion factors that relate each of the following pairs of units:

1. Liters and mL

2. Hours and minutes

3. Meters and kilometers

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How many minutes are in 2.5 hours?

Conversion factor

2.5 hr x 60 min = 150 min

1 hr

cancel

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Sample Problem

• You have $7.25 in your pocket in quarters. How many quarters do you have?

7.25 dollars 4 quarters 1 dollar

X = 29 quarters

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Learning Check

A rattlesnake is 2.44 m long. How long is the snake in cm?

a) 2440 cm

b) 244 cm

c) 24.4 cm

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Solution

A rattlesnake is 2.44 m long. How long is the snake in cm?

b) 244 cm

2.44 m x 100 cm = 244 cm

1 m

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Classwork

Textbook page 45

Question 19 (a-g)

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Homework

Problem Set: 4

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Section 3

Using Scientific Measurements

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Accuracy and Precision

Accuracy – refers to how well the measurements agree with the accepted or true value.

Precision – refers to how well a set of measurements agree with each other.

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ACCURATE = CORRECT

PRECISE = CONSISTENT

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Three targets with arrows.

Accuracy and Precision

Both accurate and precise

Precise but not accurate

Neither accurate nor precise

How do How do they they compare?compare?

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Accuracy and Precision Student 1 Student 2 Student 3 Student 4

Trial 1 27.77 cm 27.30 cm 27.55 cm 27.30 cm

Trial 2 27.30 cm 27.60 cm 27.55 cm 27.29 cm

Trial 3 27.56 cm 27.97 cm 27.53 cm 27.31 cm

Average 27.54 cm 27.62 cm 27.54 cm 27.30 cm

The accepted length of the object is 27.55 cm.

Based on the average values of the measurements which students had the best accuracy?

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Accuracy and Precision Student 1 Student 2 Student 3 Student 4

Trial 1 27.77 cm 27.30 cm 27.55 cm 27.30 cm

Trial 2 27.30 cm 27.60 cm 27.55 cm 27.29 cm

Trial 3 27.56 cm 27.97 cm 27.53 cm 27.31 cm

Average 27.54 cm 27.62 cm 27.54 cm 27.30 cm

The accepted length of the object is 27.55 cm.

Based on the individual trials of the measurements which students had the best precision?

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Percent ErrorThe accuracy of an individual value can be compared with the correct or accepted value by calculating the percent error.

Percent error is calculated by subtracting the accepted value from the experimental value, dividing the difference by the accepted value, and then multiplying by 100.

Percent error = x 100Value (experimental) – Value (accepted) Value (accepted)

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Percent Error

Indicates accuracy of a measurement

100accepted

acceptedalexperimenterror %

your valuegiven value

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Percent Error

A student determines the density of a substance to be 1.40 g/mL. Find the % error if the accepted value of the density is 1.36 g/mL.

100g/mL 1.36

g/mL 1.36g/mL 1.40error %

% error = 2.9%

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Classwork

Textbook page 49

Questions 32 and 33

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Review Homework

Problem Set: 4

Density Worksheet

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Rounding FiguresRule 1: If the first number to be dropped is 5 or greater, drop it and increase the last retained number by 1.

Rule 2: If the first number to be dropped is Less than 5, drop it and leave the last retained number unchanged.

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Homework

Problem Set: 5

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Scientific Notation

Scientific notation is a way of expressing really big numbers or really small numbers.

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Scientific NotationScientific notation – numbers are written in the form M x 10n, where M is a number greater than or equal to 1 but less than ten and n is a whole number.

Example: 65,000 = 6.5 x 104

When numbers are written in scientific notation, only the significant figures are shown.

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Small numbers are handled in a similar way; the decimal point is moved to the right:

0.00012 = 1.2 x 10-4

The decimal place is moves 4 places to the right.

There should be only one digit to the left of the decimal place.

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Examples

• Given: 289,800,000

• Use: 2.898 (moved 8 places)

• Answer: 2.898 x 108

• Given: 0.000567

• Use: 5.67 (moved 4 places)

• Answer: 5.67 x 10-4

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To change scientific notation to standard form…

• Simply move the decimal point to the right for positive exponent 10.

• Move the decimal point to the left for negative exponent 10.

(Use zeros to fill in places.)

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Example

• Given: 5.093 x 106

• Answer: 5,093,000 (moved 6 places to the right)

• Given: 1.976 x 10-4

• Answer: 0.0001976 (moved 4 places to the left)

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Learning Check

• Express these numbers in Scientific Notation:

1) 405789

2) 0.003872

3) 3000000000

4) 2

5) 0.478260

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Homework

Page 62 – 63

Questions: 76, 77, 80, 87, 93, 94

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Significant Figures

Significant figures – a measurement consists of all the digits known with certainty plus one final digit, which is uncertain or estimated.

Significant figures are the number of digits that you report in your final answer of a mathematical problem.

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Significant FiguresIndicates precision of a measurement.

Recording Sig Figs

Sig figs in a measurement include the known digits plus a final estimated digit

2.31 cm

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Determining Significant Figures

RULE 1. All non-zero digits (1-9) in a measured number are significant. (72.3)

RULE 2. Leading zeros in decimal numbers

are NOT significant. (0.0253)

RULE 3. Zeros between non-zero numbers

are significant. (60.5)

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RULE 4. Trailing zeros in numbers without

decimals are NOT significant. (4320)

RULE 5. Trailing zeros in numbers with decimals are significant. (6.20)

Counting Significant Figures

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Counting Significant Figures

RULE 1. All non-zero digits (1-9) in a measured RULE 1. All non-zero digits (1-9) in a measured number are significant. Only a zero could number are significant. Only a zero could indicate that rounding occurred.indicate that rounding occurred.

Number of Significant Figures

38.15 cm38.15 cm 44

5.6 ft5.6 ft 22

65.6 lb65.6 lb ______

122.55 m122.55 m ___

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Leading Zeros

RULE 2. Leading zeros in decimal numbers

are NOT significant.Number of Significant Figures

0.008 mm 1

0.0156 oz 3

0.0042 lb ____

0.000262 mL ____

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Sandwiched Zeros

RULE 3. Zeros between nonzero numbers are

significant.


50.8 mm 3

2001 min 4

0.702 lb ____

0.00405 m ____

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Trailing Zeros

RULE 4. Trailing zeros in numbers without decimals are

NOT significant. They are only serving as place holders.


25,000 in. 2

200 yr 1

48,600 gal ____

25,005,000 g ____

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Trailing Zeros

RULE 5. Trailing zeros in numbers with

decimals are significant.


3030.0 5

0.000230340 6

50.0 3

25,005,000.0 9

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4. 0.080

3. 5,280

2. 402

1. 23.50

Significant Figures

Counting Sig Fig Examples

1. 23.50

2. 402

3. 5,280

4. 0.080

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Learning Check

A. Which answers contain 3 significant figures?

1) 0.4760 2) 0.00476 3) 4760

B. All the zeros are significant in

1) 0.00307 2) 25.300 3) 2.050 x 103

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Learning Check

In which set(s) do both numbers contain the same number of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000015 and 150,000

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Learning Check

State the number of significant figures in each of the following:

A. 0.030 m 1 2 3

B. 4.050 L 2 3 4

C. 0.0008 g 1 2 4

D. 3.00 m 1 2 3

E. 2,080,000 bees 3 5 7

Chapter 2

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