2 - Notre dame Chemistry



2.1 Units of Measurement

You probably know your height in feet and inches. Most people

outside the United States, however, measure height in meters and

centimeters. The system of standard units that includes the meter is

called the metric system. Scientists today use a revised form of the

metric system called the Système Internationale d’Unités, or SI.

Base units There are seven base units in SI. A base unit is a unit

of measure that is based on an object or event in the physical world.

Table 2-1 lists the seven SI base units, their abbreviations, and the

quantities they are used to measure.

SI is based on a decimal system. So are the prefixes in Table 2-2,

which are used to extend the range of SI units.

▲

SI Base Units

Quantity Base unit

Time second (s)

Length meter (m)

Mass kilogram (kg)

Temperature kelvin (K)

Amount of a mole (mol)

substance

Electric current ampere (A)

Luminous intensity candela (cd)

SI is based on a decimal system. So are the prefixes in Table 2-2,

which are used to extend the range of SI units.

Prefix Symbol Factor notation Example

giga G 1 000 000 000 109 gigameter (Gm)

mega M 1 000 000 106 megagram (Mg)

kilo k 1000 103 kilometer (km)

deci d 1/10 10_1 deciliter (dL)

centi c 1/100 10_2 centimeter (cm)

milli m 1/1000 10_3 milligram (mg)

micro _ 1/1 000 000 10_6 microgram (ug)

nano n 1/1 000 000 000 10_9 nanometer (nm)

pico p 1/1 000 000 000 000 10_12 picometer (pm)

Example Problem 2-1

Using Prefixes with SI Units

How many picograms are in a gram?

The prefix pico- means 10_12, or 1/1 000 000 000 000. Thus, there

are 1012, or 1 000 000 000 000, picograms in one gram.

Practice Problems

1. How many centigrams are in a gram?

2. How many liters are in a kiloliter?

3. How many nanoseconds are in a second?

4. How many meters are in a kilometer?

Derived units Not all quantities can be measured using SI base

units. For example, volume and density are measured using units

that are a combination of base units. An SI unit that is defined by a

combination of base units is called a derived unit. The SI unit for

volume is the liter. A liter is a cubic meter, that is, a cube whose

sides are all one meter in length. Density is a ratio that compares the

mass of an object to its volume. The SI units for density are often

grams per cubic centimeter (g/cm3) or grams per milliliter (g/mL).

One centimeter cubed is equivalent to one milliliter.

Example Problem 2-2

Calculating Density

A 1.1-g ice cube raises the level of water in a 10-mL graduated

cylinder 1.2 mL. What is the density of the ice cube?

To find the ice cube’s density, divide its mass by the volume of

water it displaced and solve.

density = mass/volume

density = 1.1 g / 1.2 mL = 0.92 g/mL

Example Problem 2-3

Using Density and Volume to Find Mass

Suppose you drop a solid gold cube into a 10-mL graduated

cylinder containing 8.50 mL of water. The level of the water rises

to 10.70 mL. You know that gold has a density of 19.3 g/cm3, or

19.3 g/mL. What is the mass of the gold cube?

To find the mass of the gold cube, rearrange the equation for density

to solve for mass.

density = mass/volume

mass =volume x density

Substitute the values for volume and density into the equation and

solve for mass.

mass = 2.20 mL x 19.3 g/mL = 42.5 g

Practice Problems

5. Calculate the density of a piece of bone with a mass of 3.8 g

and a volume of 2.0 cm3.

6. A spoonful of sugar with a mass of 8.8 grams is poured into a

10-mL graduated cylinder. The volume reading is 5.5 mL. What

is the density of the sugar?

7. A 10.0-gram pat of butter raises the water level in a 50-mL

graduated cylinder by 11.6 mL. What is the density of the

butter?

8. A sample of metal has a mass of 34.65 g. When placed in a

graduated cylinder containing water, the water level rises

3.3 mL. Which of the following metals is the sample made

from: silver, which has a density of 10.5 g/cm3; tin, which has

a density of 7.28 g/cm3; or titanium, which has a density of

4.5 g/cm3?

9. Rock salt has a density of 2.18 g/cm3. What would the volume

be of a 4.8-g sample of rock salt?

10. A piece of lead displaces 1.5 mL of water in a graduated

cylinder. Lead has a density of 11.34 g/cm3. What is the mass

of the piece of lead?

Temperature The temperature of an object describes how hot

or cold the object is relative to other objects. Scientists use two temperature

scales—the Celsius scale and the Kelvin scale—to measure

temperature. You will be using the Celsius scale in most of your

experiments. On the Celsius scale, the freezing point of water is

defined as 0 degrees and the boiling point of water is defined as

100 degrees.

A kelvin is the SI base unit of temperature. On the Kelvin scale,

water freezes at about 273 K and boils at about 373 K. One kelvin is

equal in size to one degree on the Celsius scale. To convert from

degrees Celsius to kelvins, add 273 to the Celsius measurement.

To convert from kelvins to degrees Celsius, subtract 273 from the

measurement in kelvins.

Practice Problems

11. Convert each temperature reported in degrees Celsius to

kelvins.

a. 54°C

b. -54°C

c. 15°C

12. Convert each temperature reported in kelvins to degrees

Celsius.

a. 32 K

b. 0 K

c. 281 K

2.2 Scientific Notation and Dimensional Analysis

Extremely small and extremely large numbers can be compared

more easily when they are converted into a form called scientific

notation. Scientific notation expresses numbers as a multiple of two

factors: a number between 1 and 10; and ten raised to a power, or

exponent. The exponent tells you how many times the first factor

must be multiplied by ten. When numbers larger than 1 are

expressed in scientific notation, the power of ten is positive. When

numbers smaller than 1 are expressed in scientific notation, the

power of ten is negative. For example, 2000 is written as 2 _ 103 in

scientific notation, and 0.002 is written as 2 _ 10_3.

Example Problem 2-4

Expressing Quantities in Scientific Notation

The surface area of the Pacific Ocean is 166 000 000 000 000 m2.

Write this quantity in scientific notation.

To write the quantity in scientific notation, move the decimal point

to after the first digit to produce a factor that is between 1 and 10.

Then count the number of places you moved the decimal point; this

number is the exponent (n). Delete the extra zeros at the end of the

first factor, and multiply the result by 10n. When the decimal point

moves to the left, n is positive. When the decimal point moves to the

right, n is negative. In this problem, the decimal point moves

14 places to the left; thus, the quantity is written as 1.66 _ 1014 in

scientific notation.

Practice Problems

13. Express the following quantities in scientific notation.

a. 50 000 m/s2

b. 0.000 000 000 62 kg

c. 0.000 023 s

d. 21 300 000 mL

e. 990 900 000 m/s

f. 0.000 000 004 L

Dimensional analysis Dimensional analysis is a method of

problem solving that focuses on the units that are used to describe

matter. Dimensional analysis often uses conversion factors. A

conversion factor is a ratio of equivalent values used to express the

same quantity in different units. A conversion factor is always equal

to 1. Multiplying a quantity by a conversion factor does not change

its value—because it is the same as multiplying by 1—but the units

of the quantity can change.

Example Problem 2-7

Converting From One Unit to Another Unit

How many centigrams are in 5 kilograms?

Two conversion factors are needed to solve this problem. Remember

that there are 1000 grams in a kilogram and 100 centigrams in a

gram. To determine the number of centigrams in 1 kilogram, set up

the first conversion factor so that kilograms cancel out. Set up the

second conversion factor so that grams cancel out.

Practice Problems

16. Mount Everest is 8847 m high. How many centimeters high is

the mountain?

17. Your friend is 1.56 m tall. How many millimeters tall is your

friend?

18. A family consumes 2.5 gallons of milk per week. How many

liters of milk do they need to buy for one week?

(Hint: 1 L _ 0.908 quart; 1 gallon _ 4 quarts.)

19. How many hours are there in one week? How many minutes are

there in one week?

2.3 How reliable are measurements?

When scientists look at measurements, they want to know how accurate

as well as how precise the measurements are. Accuracy refers

to how close a measured value is to an accepted value. Precision

refers to how close a series of measurements are to one another.

Precise measurements might not be accurate, and accurate measurements

might not be precise. When you make measurements, you

want to aim for both precision and accuracy.

Percent error Quantities measured during an experiment are

called experimental values. The difference between an accepted

value and an experimental value is called an error. The ratio of an

error to an accepted value is called percent error. The equation for

percent error is as follows.

Percent error = error/accepted value x 100

When you calculate percent error, ignore any plus or minus signs

because only the size of the error counts.

Example Problem 2-8

Calculating Percent Error

Juan calculated the density of aluminum three times.

Trial 1: 2.74 g/cm3

Trial 2: 2.68 g/cm3

Trial 3: 2.84 g/cm3

Aluminum has a density of 2.70 g/cm3. Calculate the percent error for each trial.

First, calculate the error for each trial by subtracting Juan’s measurement

from the accepted value (2.70 g/cm3).

Trial 1: error =2.70 g/cm3 - 2.74 g/ cm3 = 0.04 g/ cm3

Trial 2: error = 2.70 g/ cm3-2.68 g/ cm3 =0.02 g/ cm3

Trial 3: error= 2.70 g/v - 2.84 g/ cm3=0.14 g/v

Then, substitute each error and the accepted value into the percent

error equation. Ignore the plus and minus signs.

Trial 1: percent error 0.04/2.70 x 100 =1.48%

Trial 2: percent error 0.02/2.70 x 100 =0.741%

Trial 3: percent error 0.14/2.70 x 100 = 5.19%

Practice Problems

20. Suppose you calculate your semester grade in chemistry as

90.1, but you receive a grade of 89.4. What is your percent

error?

21. On a bathroom scale, a person always weighs 2.5 pounds less

than on the scale at the doctor’s office. What is the percent error

of the bathroom scale if the person’s actual weight is 125

pounds?

22. A length of wood has a labeled length value of 2.50 meters. You

measure its length three times. Each time you get the same

value: 2.35 meters.

a. What is the percent error of your measurements?

b. Are your measurements precise? Are they accurate?

Significant figures The number of digits reported in a measurement

indicates how precise the measurement is. The more digits

reported, the more precise the measurement. The digits reported in a

measurement are called significant figures. Significant figures

include all known digits plus one estimated digit.

These rules will help you recognize significant figures.

1. Nonzero numbers are always significant.

45.893421 min has eight significant figures

2. Zeros between nonzero numbers are always significant.

2001.5 km has five significant figures

3. All final zeros to the right of the decimal place are significant.

6.00 g has three significant figures

4. Zeros that act as placeholders are not significant. You can

convert quantities to scientific notation to remove placeholder

zeros.

0.0089 g and 290 g each have two significant figures

5. Counting numbers and defined constants have an infinite number

of significant figures.

Example Problem 2-9

Counting Significant Figures

How many significant figures are in the following measurements?

a. 0.002 849 kg

b. 40 030 kg

Apply rules 1–4 from above. Check your answers by writing the

quantities in scientific notation.

a. 0.002 849 kg has four significant figures; 2.849 x 10_3

b. 40 030 kg has four significant figures; 4.003 x 104

Practice Problems

23. Determine the number of significant figures in each

measurement.

a. 0.000 010 L c. 2.4050 x 10_4 kg

b. 907.0 km d. 300 100 000 g

Rounding off numbers When you report a calculation, your

answer should have no more significant figures than the piece of

data you used in your calculation with the fewest number of significant

figures. Thus, if you calculate the density of an object with a

mass of 12.33 g and a volume of 19.1 cm3, your answer should have

only three significant figures. However, when you divide these quantities

using your calculator, it will display 0.6455497—many more

figures than you can report in your answer. You will have to round

off the number to three significant figures, or 0.646.

Here are some rules to help you round off numbers.

1. If the digit to the immediate right of the last significant figure is

less than five, do not change the last significant figure.

2. If the digit to the immediate right of the last significant figure is

greater than five, round up the last significant figure.

3. If the digit to the immediate right of the last significant figure is

equal to five and is followed by a nonzero digit, round up the

last significant figure.

4. If the digit to the immediate right of the last significant figure is

equal to five and is not followed by a nonzero digit, look at the

last significant figure. If it is an odd digit, round it up. If it is an

even digit, do not round up.

Whether you are adding, subtracting, multiplying, or dividing, you

must always report your answer so that it has the same number of

significant figures as the measurement with the fewest significant

figures.

Example Problem 2-10

Rounding Off Numbers

Round the following number to three significant figures: 3.4650.

Rule 4 applies. The digit to the immediate right of the last significant

figure is a 5 followed by a zero. Because the last significant

figure is an even digit (6), do not round up. The answer is 3.46.

Practice Problems

24. Round each number to five significant figures. Write your

answers in scientific notation.

a. 0.000 249 950

b. 907.0759

c. 24 501 759

d. 300 100 500

25. Complete the following calculations. Round off your answers as

needed.

a. 52.6 g _ 309.1 g _ 77.214 g

b. 927.37 mL _ 231.458 mL

c. 245.01 km _ 2.1 km

d. 529.31 m _ 0.9000 s

2.4 Representing Data

A graph is a visual display of data. Representing your data in

graphs can reveal a pattern if one exists. You will encounter several

different kinds of graphs in your study of chemistry.

Circle graphs A circle graph is used to show the parts of a fixed

whole. This kind of graph is sometimes called a pie chart because it

is a circle divided into wedges that look like pieces of pie. Each

wedge represents a percentage of the whole. The entire graph represents

100 percent.

Bar graphs A bar graph is often used to show how a quantity

varies with time, location, or temperature. In this situation, the quantity

being measured appears on the vertical axis. The independent

variable—time, for example—appears on the horizontal axis.

Line graphs The points on a line graph represent the intersection

of data for two variables. The independent variable is plotted on the

horizontal axis. The dependent variable is plotted on the vertical

axis. The points on a line graph are connected by a best fit line,

which is a line drawn so that as many points fall above the line as below it.

If a best fit line is straight, there is a linear relationship between

the variables. This relationship can be described by the steepness, or

slope, of the line. If the line rises to the right, the slope is positive. A

positive slope indicates that the dependent variable increases as the

independent variable increases. If the line falls to the right, the slope

is negative. A negative slope indicates that the dependent variable

decreases as the independent variable increases. You can use two

data points to calculate the slope of a line.

Example Problem 2-11

Calculating the Slope of a Line from Data Points

Calculate the slope of a line that contains these data points:

(3.0 cm3, 6.0 g) and (12 cm3, 24 g).

To calculate the slope of a line from data points, substitute the

values into the following equation.

slope _ __y/_x

slope _ _ _ 2.0 g/cm3

Practice Problems

26. Calculate the slope of each line using the points given.

a. (24 cm3, 36 g), (12 cm3, 18 g)

b. (25.6 cm3, 28.16 g), (17.3 cm3, 19.03 g)

c. (15s, 147 m), (21 s, 205.8 m)

d. (55 kJ, 18.75°C), (75 kJ, 75.00°C)

Interpreting data When you are asked to read the information

from a graph, first identify the dependent variable and the independent

variable. Look at the ranges of the data and think about what

measurements were taken to obtain the data. Determine whether the

relationship between the variables is linear or nonlinear. If the relationship

is linear, determine if the slope is positive or negative.

▲

18 g

_

9.0 cm3

If the points on the graph are connected, they are considered

continuous. You can read data that falls between the measured

points. This process is called interpolation. You can extend the line

on a graph beyond the plotted points and estimate values for the

variables. This process is called extrapolation. Extrapolation is less

reliable than interpolation because you are going beyond the range

of the data collected.

Chapter 2 Review

27. Which SI units would you use to measure the following

quantities?

a. the amount of water you drink in one day

b. the distance from New York to San Francisco

c. the mass of an apple

28. How does adding the prefix kilo- to an SI unit affect the

quantity being described?

29. What units are used for density in the SI system?

Are these base units or derived units? Explain your answer.

30. Is it more important for a quarterback on a football team to be

accurate or precise when throwing the football? Explain.

31. A student takes three mass measurements. The measurements

have errors of 0.42 g, 0.38 g, and 0.47 g. What information

would you need to determine whether these measurements are

accurate or precise?

32. What conversion factor is needed to convert minutes to hours?

33. What kind of graph would you use to represent the following

data?

a. the segments of the population who plan to vote for a

certain candidate

b. the average monthly temperatures of two cities

c. the amount of fat in three different kinds of potato chips

d. the percent by mass of elements in Earth’s atmosphere

e. your scores on math quizzes during a year

f. the effect of a hormone on tadpole growth

12.4 Phase Changes

Most substances can exist in three states—solid, liquid, and gas—

depending on the temperature and pressure. States of substances are

called phases when they coexist as physically distinct parts of a mixture,

such as ice water. When energy is added to or taken away from

a system, one phase can change into another.

Phase changes that require energy You know the three phases

of water: ice, liquid water, and water vapor. When you add ice to

water, heat flows from the water to the ice and disrupts the hydrogen

bonds that hold the water molecules in the ice together. The ice

melts and becomes liquid. The amount of energy required to melt

one mole of a solid depends on the strength of the forces keeping

the particles together. The melting point of a crystalline solid is the

temperature at which the forces holding the crystal lattice together

are broken and the solid becomes a liquid. Because amorphous

solids tend to act like liquids when they are in the solid state, it’s

hard to specify their melting points.

When liquid water is heated, some molecules escape from the

liquid and enter the gas phase. If a substance is usually a liquid at

room temperature (as water is), the gas phase is called a vapor.

Vaporization is the process by which a liquid changes into a gas or

vapor. When vaporization occurs only at the surface of a liquid, the

process is called evaporation.

Vapor pressure is the pressure exerted by a vapor over a liquid.

As temperature increases, water molecules gain kinetic energy and

vapor pressure increases. When the vapor pressure of a liquid equals

atmospheric pressure, the liquid has reached its boiling point, which

is 100°C for water at sea level. At this point, molecules throughout

the liquid have the energy to enter the gas or vapor phase.

The process by which a solid changes directly into a gas without

first becoming a liquid is called sublimation. Solid air fresheners

and dry ice are examples of solids that sublime. At very low temperatures,

ice will sublime in a short amount of time. This property of

ice is used to preserve freeze-dried foods.

Practice Problems

10. Classify each of the following phase changes.

a. dry ice (solid carbon dioxide) to carbon dioxide gas

b. ice to liquid water

c. liquid bromine to bromine vapor

d. moth balls giving off a pungent odor

Phase changes that release energy Some phase changes release

energy into their surroundings. For example, when a vapor loses

energy, it may change into a liquid. Condensation is the process by

which a gas or vapor becomes a liquid. It is the reverse of vaporization.

Water vapor undergoes condensation when its molecules lose

energy, their velocity decreases, and hydrogen bonds begin to form

between them. When hydrogen bonds form, energy is released.

When water is placed in a freezer, heat is removed from the

water. The water molecules lose kinetic energy, and their velocity

decreases. When enough energy has been removed, the hydrogen

bonds keep the molecules frozen in set positions. The freezing point

is the temperature at which a liquid becomes a crystalline solid.

When a substance changes from a gas or vapor directly into

a solid without first becoming a liquid, the process is called

deposition. Deposition is the reverse of sublimation. Frost is an

example of water deposition.

Practice Problems

11. Classify each of the following phase changes.

a. liquid water to ice

b. water vapor to liquid water

c. dew forming on grass

d. water vapor to ice crystals

e. beads of water forming on the outside of a glass containing a cold drink

substance. A phase diagram is a graph of pressure versus temperature

that shows in which phase a substance exists under different

conditions of temperature and pressure. A phase diagram typically

has three regions, each representing a different phase and three

curves that separate each phase. The points on the curves indicate

conditions under which two phases coexist. The phase diagram for

each substance is different because the normal boiling and freezing

points of substances are different.

The triple point is the point on a phase diagram that represents

the temperature and pressure at which three phases of a substance

can coexist. All six phase changes can occur at the triple point:

freezing and melting, evaporation and condensation, sublimation

and deposition. The critical point indicates the critical pressure and

the critical temperature above which a substance cannot exist as a

liquid.

Practice Problems

12. Answer the following questions about the phase diagram for

water in chapter 12 in your textbook.

a. List the phase changes a sample of ice would go through if

heated to its critical temperature at 1 atm pressure.

b. At what range of pressure will water be a liquid at temperatures

above its normal boiling point?

c. In what phase does water exist at its triple point?

13. Answer the following questions about the phase diagram for

carbon dioxide in chapter 12 of your textbook.

a. What happens to solid carbon dioxide at room temperature

at 1 atm pressure?

b. The triple point for carbon dioxide is 5 atm and -57°C. List

the phase changes a sample of dry ice would go through if

heated from -100°C to 10°C at 6 atm pressure.

Chapter 2

Practice Problems

Solving Problems: A Chemistry Handbook Answer Key

1. 100 centigrams

2. 1000 liters

3. 1 000 000 000 nanoseconds

4. 1000 meters

5. 1.9 g/cm3

6. 1.6 g/mL, or 1.6 g/cm3

7. 0.862 g/mL, or 0.862 g/cm3

8. silver

9. 2.2 cm3

10. 17 g

11. a. 327 K

b. 219 K

c. 288 K

12. a. -241°C

b. -273°C

c. 8°C

13. a. 5 x 104 m/s2

b. 6.2 x 10–10 kg

c. 2.3 x 10–5 s

d. 2.13 x 107 mL

e. 9.909 x 108 m/s

f. 4 x 10–9 L

16. 884 700 cm

17. 1560 mm

18. 11 L

19. 168 hr; 10 080 min

20. 0.783%

21. 2.00%

22. a. 6.00%

b. The measurements are

extremely precise but

not accurate.

23. a. 2

b. 4

c. 5

d. 4

24. a. 2.4995 x10_4

b. 9.0708 x102

c. 2.4502 x107

d. 3.0010 x108

25. a. 439 g

b. 695.91 mL

c. 510 km2

d. 588.1 m/s

26. a. 1.5 g/cm3 c. 9.8 m/s

b. 1.10 g/cm3 d. 2.8°C/kJ

Chapter 2 Review

27. a. mL or L b. km c. g

28. It multiplies the quantity by 1000.

29. g/cm3 or g/mL; they are derived units because they involve a

combination of base units.

30. accurate, because the target changes with each throw

31. The accepted value; for a large value, the measurements might

be precise. For a small value, they would not be.

32. 1 hour/60 min

33. a. circle or bar graph d. circle graph

b. bar or line graph e. line graph

c. bar graph f. line graph

chapter 12.4

12. a. The ice would melt (become liquid water) and vaporize

(become water vapor).

b. 1.00–217.75 atm

c. Ice, liquid water, and water vapor coexist at the triple point.

13. a. The dry ice sublimes to carbon dioxide gas.

b. The dry ice would melt (become a liquid) and vaporize

(become a gas.)

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