mark s. cracolice edward i. peters mark s. cracolice the university of montana chapter 3...

Mark S. CracoliceEdward I. Peters

http://academic.cengage.com/chemistry/cracolice

Mark S. Cracolice • The University of Montana

Chapter 3Measurement and

Chemical Calculations

Introduction to MeasurementMeasure

Comparison of the dimensions, quantity, or capacity of something with a standard.

Thus, to measure something, members of a society have to first agree on a standard for comparison.

For example, in the United States, people agree on thedistance represented by the unit called the inch.

The dimensions of objects can then be expressed in inches,and all members of society understand

the meaning of the measurement.

Introduction to MeasurementHow tall are you?

To answer this question, you use an agreed-upon standard to express the value of the measured quantity:

Feet and inches are typically used in the

United States to express a person’s height.

Centimeters are typically used in the

rest of the world to express a person’s height.

Introduction to MeasurementMeasurements everywhere in the world, with the

exception of the U.S., are made in the metric system.

U.S. scientists, as well as all scientists in every country, also make and express measurements in the metric system.

SI units are a subset of all metric units.

SI is an abbreviation for the French name for the International System of Units (the metric system was invented in France).

Introduction to MeasurementThe SI system is defined by seven base units.

Examples of base units include:

Quantity Base Unit

Mass (weight) Kilogram

Length Meter

Temperature Kelvin

Time Second

Other measurement units are derived from the base units; accordingly, they are called derived units.

Exponential NotationGoal 1

Write in exponential notation a number given in ordinary decimal form; write in ordinary decimal form a number given in exponential notation.

Goal 2

Using a calculator, add, subtract, multiply, and divide numbers expressed in exponential notation.

Exponential NotationExponentials

BP

B is the base

p is the power or exponent

104 = 10 10 10 10 = 10,000

10–4 = = = = 0.0001

1104

110

110

110

110

110,000

Exponential NotationExponential Notation

a.bcd 10e

Coefficient: a.bcd

Usually 1 ≤ coefficient < 10

Exponent: e

A whole number

Exponential NotationConversion Between Decimal Numbers and

Standard Exponential Notation

Example:Convert 724,000 to standard exponential notation

7.24 10e

7 2 4 0 0 0 .

Five places

7.24 105

Exponential NotationConversion Between Decimal Numbers and

Standard Exponential Notation

Example:Convert 0.000427 to standard exponential notation

4.27 10e

0 . 0 0 0 4 2 7

Four places

4.27 10–4

Dimensional AnalysisGoal 3

In a problem, identify given and wanted quantities that are related by a PER expression. Set up and solve the problem by dimensional analysis.

Dimensional AnalysisDimensional Analysis

A quantitative problem-solving method featuring algebraic cancellation of units and the use of PER expressions.

PER Expression

A mathematical statement expressing the relationship between two quantities that are directly proportional to one another.

Examples:

24 hours PER day

10 cents PER dime

Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis

Sample Problem:

How many days are in

23 weeks?

Step 1:

Identify and write down the

GIVEN quantity, including GIVEN: 23 weeks

units.


Sample Problem:


23 weeks?

Step 2:

Identify and write down the GIVEN: 23 weeks

units of the WANTED WANTED: days

quantity.


Sample Problem:


23 weeks?

Step 3: GIVEN: 23 weeks

Write down the WANTED: days

PER/PATH. PER: 7 days/week

PATH: wk days


Sample Problem:


23 weeks?


Write the calculation WANTED: days

setup. Include units. PER: 7 days/week

PATH: wk days

23 weeks =

7 daysweek


Sample Problem:


23 weeks?


Calculate the answer. WANTED: days PER: 7 days/week

PATH: wk days

23 weeks = 161 days

7 daysweek


Sample Problem:How many days are in23 weeks?

Step 6: GIVEN: 23 weeksCheck the answer to be sure WANTED: days both the

number and the PER: 7 days/weekunits make sense. PATH: wk days

23 weeks = 161 days

More days (smaller unit)than weeks (larger unit).OK.

7 daysweek

Metric UnitsGoal 4

Distinguish between mass and weight.

Goal 5

Identify the metric units of mass, length, and volume.

Metric UnitsMass and Weight

Mass is a measure of quantity of matter.

Weight is a measure of the force of gravitational attraction.

Mass and weight are directly proportional to each other.

Metric UnitsThe SI unit of mass is the

kilogram, kg.

It is defined as the mass of a platinum-iridium cylinder stored in a vault in France.

A kilogram weighs 2.2 pounds.

Metric Units

For most laboratory work, the basic metric mass unit is used:

the gram, g.

Metric UnitsIn the metric system, units that are larger than the

basic unit are larger by multiples of 10.

For example, the kilounit is 1000 times

larger than the basic unit.

Units that are smaller than the basic unit are smaller

by fractions that are also multiples of 10.

For example, the milliunit is 1/1000 times

smaller than the basic unit.

Metric UnitsMetric Prefixes

Large Units Small Units

Metric Metric Metric Metric

Prefix Symbol Multiple Prefix Symbol Multiple

tera- T 1012 Unit 1

giga- G 109 deci- d 0.1

mega- M 106 centi- c 0.01

kilo- k 1000 milli- m 0.001

hecto- h 100 micro- µ 10–6

deca- da 10 nano- n 10–9

Unit 1 pico- p 10–12

Metric UnitsLength

The SI unit of length is the meter, m.

It is defined as the distance light travels in a vacuum in 1/299,792,458 second

The meter is 39.37 inches:

1 m 100 cm

m

1 in.2.54 cm

= 39.37 in.

Metric Units

One inch is defined as 2.54 centimeters.

Metric UnitsVolume

The SI unit of volume is the cubic meter, m3.

This is a derived unit. V = l w h.

A more practical unit for laboratory work is the

cubic centimeter, cm3.

Metric UnitsOne liter (L) is defined as exactly 1000 cubic centimeters.

1 mL = 0.001 L = 1 cm3

Metric UnitsGoal 6

State and write with appropriate metric prefixes the relationship between any metric unit and its corresponding kilounit, centiunit, and milliunit.

Goal 7

Using Table 3.1, state and write with appropriate metric prefixes the relationship between any metric unit and other larger and smaller metric units.

Goal 8

Given a mass, length, or volume expresed in metric units, kilounits, centiunits, or milliunits, express that quantity in the other three units.

Metric UnitsMetric Relationship Example

1000 units per kilounit 1000 meters per kilometer

1000 m/km

100 centiunits per unit 100 centigrams per gram

100 cg/g

1000 milliunits per unit 1000 milliliters per liter

1000 mL/L

Metric UnitsExample:

How many centigrams are in 0.87 gram?

Solution:

Use dimensional analysis.

GIVEN: 0.87 g

WANTED: cg

PER: 100 cg/g

PATH: g cg

0.87 g = 87 cg

More centigrams (smaller unit) than grams (larger unit). OK.

100 cgg


How many kilometers are in 2,335 meters?

Solution:


GIVEN: 2335 m

WANTED: km

PER: 1000 m/km

PATH: m km

2335 m = 2.335 km

More meters (smaller unit) than kilometers (larger unit). OK.

1 km1000 m


How many milliliters are in 0.00339 liter?

Solution:


GIVEN: 0.00339 L

WANTED: mL

PER: 1000 mL/L

PATH: L mL

0.00339 L = 3.39 mL

More milliliters (smaller unit) than liters (larger unit). OK.

1000 mLL

Significant FiguresGoal 9

State the number of significant figures in a given quantity.

Significant FiguresUncertainty in Measurement

No measurement is exact.

In scientific writing, the uncertainty associated with a measured quantity is always included.

By convention, a measured quantity is expressed by stating all digits known accurately plus one uncertain digit.

Significant Figures

Significant Figures

The bottom board is one meter long.

How long is the top board?

More than half as long as the meter stick,

but less than one meter—about 6/10 of a meter.

The uncertain digit is the last digit written: 0.6 m

Significant Figures

Now the meter stick has marks every 0.1 m,

numbered in centimeters. How long is the board?

Between 0.6 m and 0.7 m with certainty, and the uncertain digit must be estimated—the board is about 4/10 of the way

between 0.6 m and 0.7 m: 0.64 m.

Significant Figures

The measuring instrument now has centimeter marks.

How long is the board?

Between 0.64 m and 0.65 m with certainty.

It is about 3/10 of the way between the two marks,

so we record 0.643 m as the length of the board.

Significant Figures

The measuring instrument now has millimeter marks.

We could estimate between the millimeter marks, but the alignment of the board and the meter stick has an uncertainty

of a millimeter or so.

We have reached the limit of this measuring instrument: 0.643 m.

Significant FiguresSignificant Figures

Significant figures are applied to measurements

and quantities calculated from measurements.

They do not apply to exact numbers.

An exact number has no uncertainty.

Types of exact numbers:

Counting numbers

Numbers fixed by definition


The number of significant figures in a quantity is the number of digits that are known accurately plus the one that is uncertain

—the uncertain digit.

The uncertain digit is the last digit written

when expressing a scientific measurement.


The measurement process, not the unit in which

the result is expressed, determines the

number of significant figures in a quantity.

The length of the board in the previous illustrations

was 0.643 m. Expressed in centimeters, it is 64.3 cm.

They are the same measurement with the same uncertainty.

Both must have the same number of significant figures.


The location of the decimal point has nothing

to do with significant figures.

The same 0.643 m board is 0.000643 km.

The three zeros before the decimal point are not significant.

Begin counting significant figures at the first nonzero digit,

not at the decimal point.


The uncertain digit is the last digit written.

If the uncertain digit is a zero to the right of the decimal point,

that zero must be written.

If the mass of a sample on a triple beam balance is 15.10 g, and the balance is accurate to ±0.01 g, the last digit recorded must

be zero to indicate the correct uncertainty.


Exponential notation must be used for very large numbers to show if final zeros are significant.

If the length of the 0.643 m board is expressed

in micrometers, its length is 643,000 µm.

The uncertainty is ±1,000 µm.

The ordinary decimal number makes this ambiguous.

Writing 6.43 105 µm shows clearly the correct

location of the uncertain digit.


Round off given numbers to a specified number of significant figures.

Significant FiguresRounding a Calculated Number

If the first digit to be dropped is less than 5,

leave the digit before it unchanged.

Examples:

Round to three significant figures. Answers

1.743 m 1.74 m

0.041239 kg 0.0412 kg

Significant FiguresRounding a Calculated Number

If the first digit to be dropped is 5 or more,

increase the digit before it by 1.

Examples:

Round to three significant figures. Answers

32.88 mL 32.9 mL

0.0097761 km 0.00978 km


Add or subtract given quantities and express the result in the proper number of significant figures.

Significant FiguresSignificant Figure Rule

for Addition and Subtraction

Round off the answer to the first column

that has an uncertain digit.

Significant FiguresExample:

The following is a list of masses of items to be shipped. What is the total mass of the package?

Carton: 226 g; Item 1: 33.5 g; Item 2: 589 g; Packaging: 11.88 g

Answer:

2 2 6 g

3 3 . 5 g

5 8 9 g

1 1 . 8 8 g

8 6 0 . 3 8 g = 860 g = 8.60 102 g


Multiply or divide given measurements and express the result in the proper number of significant figures.

Significant FiguresSignificant Figure Rule

For Multiplication and Division

Round off the answer to the same number of significant figures as the smallest number of significant figures in any factor.

Significant FiguresExample:

What is the volume of a cube that is 34.49 cm long, 23.0 cm wide and 15 cm high?

Solution:

Use algebra because the GIVENS and WANTED are related by a formula, volume = length width height.

V = l w h = 34.49 cm 23.0 cm 15 cm

4 sf 3 sf 2 sf

= 11,899.05 cm3 (unrounded)

The answer is rounded to 2 sf, 1.2 104 cm3

Metric–USCS ConversionsGoal 13

Given a metric–USCS conversion factor and a quantity expressed in any unit in Table 3.2, express that quantity in corresponding units in the other system.

Metric–USCS ConversionsConversions between the United States Customary System

(USCS) and the metric system are made by applying dimensional analysis.

Length

1 in. 2.54 cm (definition of an inch)

Mass

1 lb 453.59237 g (definition of a pound)

Volume

1 gal 3.785411784 L (exactly)

Metric–USCS ConversionsCitizens of the U.S. should know USCS–USCS conversions

Length1 ft 12 in.1 yd 3 ft

1 mi 5280 ft

Mass (Weight)1 lb = 16 oz

Volume1 qt = 32 fl oz1 gal = 4 qt

Metric–USCS ConversionsExample:

How many milliliters are in 1.0 quart?

Solution:

PER: 1 gal/4 qt 3.785 L/gal 1000 mL/L

PATH: qt gal L mL

1.0 qt = 9.4 102 mL

CHECK: More milliliters (smaller unit) than quarts (larger unit). OK.

1 gal4 qt

3.785 Lgal

1000 mLL

TemperatureGoal 14

Given a temperature in either Celsius or Fahrenheit degrees, convert it to the other scale.

Goal 15

Given a temperature in Celsius degrees or kelvins, convert it to the other scale.

Temperature

TemperatureFahrenheit Temperature Scale

Water freezes at 32°F and boils at 212°F.

There are 180 Fahrenheit degrees between freezing and boiling.

Celsius Temperature Scale

Water freezes at 0°C and boils at 100°C.

There are 100 Celsius degrees between freezing and boiling.

T°F – 32 = T°C

T°F – 32 = 1.8 T°C

180100

TemperatureKelvin (Absolute) Temperature Scale

The degree is the same size as a Celsius degree,

but 0 on the Kelvin scale is set at the lowest

temperature possible, which is –273°C.

TK = T°C + 273

TemperatureExample:

Convert 65°F to its equivalent in degrees Celsius and kelvins.

Solution:

GIVEN: 65°F WANTED: °C and K

EQUATIONS: T°F – 32 = 1.8 T°C and TK = T°C + 273

T°C = = = 18°C

TK = T°C + 273 = 18 + 273 = 291 K

TF

Š 321.8

65 Š 321.8

Proportionality and DensityGoal 16

Write a mathematical expression indicating that one quantity is directly proportional to another quantity.

Goal 17

Use a proportionality constant to convert a proportionality to an equation.

Goal 18

Given the values of two quantities that are directly proportional to each other, calculate the proportionality constant, including its units.

Proportionality and DensityGoal 19

Write the defining equation for a proportionality constant and identify units in which it might be expressed.

Goal 20

Given two of the following for a sample of a pure substance, calculate the third: mass, volume, and density.

Proportionality and DensityA direct proportionality exists between two quantities when

they increase or decrease at the same rate.

If a graph of two related measurements is a straight line that passes through the origin, the measured quantities are directly

proportional to each other.

a b

a is proportional to b

Proportionality and DensityWe can describe direct proportionalities between measured

quantities with PER expressions.

Direct proportionalities between measured quantities yield two conversion factors between the quantities.

Given either quantity in a direct proportionality and the conversion factor between the quantities, we can calculate the

other quantity with dimensional analysis.

Proportionality and DensityThe mass and volume of any pure substance at a given

temperature are directly proportional:

mass is proportional to volume

mass volume

m V

A proportionality is changed into an equation by inserting a multiplier called a proportionality constant.

Let D be the proportionality constant:

m = D V

Proportionality and DensitySolving for the proportionality constant yields the defining equation for a physical property of a pure substance called

density:

D

In words, density is the mass per unit volume of a substance:

Density

mV

massvolume

Proportionality and DensityThe definition of density establishes its common units:

Density

The common laboratory unit for mass is grams.

The common laboratory unit for volume ismilliliters or cubic centimeters.

Density is therefore typically expressed in g/mL or g/cm3.

Since volume varies with temperature,density is temperature dependent.

massvolume

Proportionality and DensityDensities of Some Common Substances

(g/cm3 at 20°C and 1 atm)

Substance Density Substance Density

Helium 0.00017 Aluminum 2.7

Air 0.0012 Iron 7.8

Pine lumber 0.5 Copper 9.0

Maple lumber 0.6 Silver 10.5

Oak lumber 0.8 Lead 11.4

Water 1.0 Mercury 13.6

Glass 2.5 Gold 19.3

Proportionality and Density

Proportionality and DensityWater is unusual in that its solid

phase, ice, will float on its liquid phase.

Solid ethanol sinks to the bottom of the liquid. The solid form of almost all substances is more dense than the liquid phase.

Proportionality and DensityExample:

What is the volume of 15 g of silver?

Solution:

GIVEN: 15 g silver

WANTED: volume (assume cm3)

PER: 10.5 g/cm3 (from table)

PATH: g cm3

15 g = 1.4 cm3

1 cm3

10.5 g

Strategy for Solving ProblemsThe only way to learn how to solve problems is to

solve them for yourself.

However, we can provide you with some general

guidelines for solving chemistry problems.

Strategy for Solving ProblemsA problem can be solved by dimensional analysis if the

GIVEN and WANTED can be linked by one or more

PER expressions and you know or can find the

conversion factor for each expression.

A problem can be solved by algebra if the GIVEN and

WANTED appear in an algebraic equation

in which the WANTED is the only unknown.

Strategy for Solving Problems

Reflective PracticeTo become a better problem solver, you must practice solving

problems. When you solve a problem and check the answer section of the textbook, reflect on both the chemistry concept

and the strategy you used in solving the problem.

The reason for solving problems is not to get the same answers as the textbook authors, but to learn how to solve the problem.

mark s. cracolice edward i. peters mark s. cracolice the university of montana chapter 3...

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