Chapter 3

The MetricSystem

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Vanessa N. Prasad-PermaulCHM 1025

Valencia Community College

The Metric System

The English system was used primarily in the British Empire and wasn’t very standardized.

The French organized a committee to devise a universal measuring system.

After about 10 years, the committee designed and agreed on the metric system.

The metric system offers simplicity with a single base unit for each measurement.

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Original Metric Unit Definitions

A meter was defined as 1/10,000,000 of the distance from the North Pole to the equator.

A kilogram (1000 grams) was equal to the mass of a cube of water measuring 0.1 m on each side.

A liter was set equal to the volume of one kilogram of water at 4 C.

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Metric System Basic Units

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Metric System Advantage

Another advantage of the metric system is that it is a decimal system.

It uses prefixes to enlarge or reduce the basic units.

For example:

A kilometer is 1000 meters.

A millimeter is 1/1000 of a meter.

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Metric System PrefixesThe following table lists the common

prefixes used in the metric system:

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Metric Prefixes, Continued For example, the prefix kilo-

increases a base unit by 1000:

1 kilogram is 1000 grams.

The prefix milli- decreases a base unit by a factor of 1000:

There are 1000 millimeters in 1 meter.

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EXAMPLE 3.1 Metric Basic Units and Prefixes

We compose the symbol for each unit by combining the prefix symbol and the basic unit symbol. If we refer to Tables 3.1 and 3.2, we have the following:(a) Gm, length(b) kg, mass(c) cL, volume(d) s, time

Solution

Give the symbol for each of the following metric units and state the quantity measured by each unit:(a) gigameter (b) kilogram(c) centiliter (d) microsecond

EXERCISE 3.1 Metric Basic Units and Prefixes

Give the symbol for each of the following metric units and state the quantity measured by each unit:(a) nanosecond (b)microiliter(c) kilogram (d)millimeter

Practice Exercise

What is the basic unit of length, mass, and volume in the metric system?

Concept Exercise

Metric Symbols The names of metric units are

abbreviated using symbols. Use the prefix symbol followed by the symbol for the base unit, so:

Nanometer is abbreviated nm.

Microgram is abbreviated g.

Deciliter is abbreviated dL.

Gigasecond is abbreviated Gs.

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Nanotechnology Nanotechnology refers to devices

and processes on the 1–100 nm scale. For reference, a human hair is about

100,000 nm thick! A DNA helix is a nanoscale substance,

with a diameter of about 1 nm. Nanoscale hollow tubes, called carbon

nanotubes, have slippery inner surfaces that allow for the easy flow of fluids.

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Metric Equivalents

We can write unit equations for the conversion between different metric units.

The prefix kilo- means 1000 basic units, so 1 kilometer is 1000 meters.

The unit equation is 1 km = 1000 m.

Similarly, a millimeter is 1/1000 of a meter, so the unit equation is 1000 mm = 1 m.

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Metric Unit Factors Since 1000 m = 1 km, we can write the

following unit factors for converting between meters and kilometers:

1 km or 1000 m 1000 m 1 km

Since 1000 mm = 1 m, we can write the following unit factors:

1000 mm or 1 m 1 m 1000 mm

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EXAMPLE 3.2 Metric Unit Equations

We can refer to Table 3.2 as necessary.(a) The prefix mega- means 1,000,000 basic units; thus, 1 Mm

= 1,000,000 m.1 Mm = 1 106 m

(b) The prefix kilo- means 1000 basic units; thus, 1 kg = 1000 g.1 kg = 1 103 g

(c) The prefix deci- means 0.1 of a basic unit, thus, 1 L = 10 dL.1 L = 1 101 dL

(d) The prefix nano- means 0.000 000 001 of a basic unit, thus; 1 s = 1,000,000,000 ns.1 s = 1 109 ns

Solution

Complete the unit equation for each of the following exact metric equivalents:(a) 1 Mm = ? m (b) 1 kg = ? g(c) 1 L = ? dL (d) 1 s = ? ns

EXERCISE 3.2 Metric Unit Equations

Complete the unit equation for each of the following

exact metric equivalents:

(a) 1 nm = ? m (b) 1 g = ? mg(c) 1 L = ? L (d) 1 s = ? Ms

Practice Exercise

EXAMPLE 3.3 Metric Unit Factors

We start by writing the unit equation to generate thetwo unit factors.(a)The prefix kilo- means 1000 basic units; thus, 1

km = 1000 m. The two unit factors are

(b)The prefix deci- means 0.1 basic unit; thus, 1 g = 10 dg. The two unit factors are

Solution

Write two unit factors for each of the following metric relationships:(a) kilometers and meters(b)grams and decigrams

EXERCISE 3.3 Metric Unit Factors

Write two unit factors for each of the following metric relationships:(a) liters and microliters(b)milliseconds and seconds

Practice Exercise

Metric–Metric Conversions We will use the unit analysis

method we learned in Chapter 2 to do metric–metric conversion problems.

Remember, there are three steps:

1. Write down the unit asked for in the answer.

2. Write down the given value related to the answer.

3. Apply unit factor(s) to convert the given unit to the units desired in the answer.

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Metric–Metric Conversion Problem

What is the mass in grams of a 325 mg aspirin tablet?

Step 1: We want grams.

Step 2: We write down the given: 325 mg.

Step 3: We apply a unit factor (1000 mg = 1 g) and round to three significant figures.

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325 mg x = 0.325 g1000 mg

1 g

Two Metric–Metric Conversions

A hospital has 125 deciliters of blood plasma. What is the volume

in milliliters?

Step 1: We want the answer in mL.

Step 2: We have 125 dL.

Step 3: We need to first convert dL to L and then convert L to mL:

1 L and 1000 mL 10 dL 1 L

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EXAMPLE 3.4 Metric Unit Factors

Continued… Apply both unit factors, and round the

answer to three significant digits.

Notice that both dL and L units cancel, leaving us with units of mL.

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125 dL x = 12,500 mLx10 dL 1 L 1000 mL

1 L

EXAMPLE 3.4 Metric Unit Factors

A dermatology patient is treated with ultraviolet light having a wavelength of 305 nm. What is the wavelength expressed in meters? In micrometers?

Practice Exercise

Express the volume of a cube 1 cm on a side in

milliliters.

Concept Exercise

EXERCISE 3.4 Metric-Metric Conversions

Another ExampleThe mass of the Earth’s moon is 7.35

× 1022 kg. What is the mass expressed in megagrams, Mg?

We want Mg; we have 7.35 x 1022 kg.

Convert kilograms to grams, and then grams to megagrams.

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7.35 x 1022 kg × = 5.98 x 1019 Mgx1 kg

1000 g 1 Mg

1000000 g

EXAMPLE EXERCISE 3.5 Metric–Metric Conversion

EXERCISE 3.5 Metric–Metric Conversion

Light travels through the universe at a velocity of 3.00 1010 cm/s. How many megameters does light travel in one second?

Practice Exercise

How many significant digits are in the following unit

factor?

1 g/1000 mg

Concept Exercise

Metric and English Units The English system is still very common

in the United States.

We often have to convert between English and metric units.

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Metric–English ConversionThe length of an American football field, including

the end zones, is 120 yards. What is the length in

meters?

Convert 120 yd to meters

(given that 1 yd = 0.914 m).

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120 yd x = 110 m1 yd

0.914 m

English–Metric ConversionA half-gallon carton contains 64.0 fl oz of milk.

How many milliliters of milk are in a carton?

We want mL; we have 64.0 fl oz. Use 1 qt = 32.0 fl oz, and 1 qt = 946

mL.

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64.0 fl oz x = 1,890 mLx32 fl oz

1 qt 946 mL1 qt

EXAMPLE EXERCISE 3.6 Metric–English Conversion

EXERCISE 3.6 Metric–English Conversion

A plastic bottle contains 4.00 gallons of distilled water. How many liters of distilled water are in the bottle (given that 1 gal = 4 qt)?

Practice Exercise

How many significant digits are in the following unit

factor?1 qt/946 mL

Concept Exercise

EXAMPLE 3.7 Metric–English Conversion

We apply the unit factor 1 lb/16 oz to cancel ounces , and 454 g/1 lb to cancel pounds .

The given value, 2.0 oz, limits the answer to two significant digits. Unit factor 1 has no effect as it is derived from an exact equivalent, and unit factor 2 has three significant digits.

EXAMPLE 3.7 Metric–English Conversion

If a tennis ball weighs 2.0 oz, what is the mass of the tennis ball in grams?

Unit Analysis Map

EXERCISE 3.7 Metric–English Conversion

If a tennis ball has a diameter of 2.5 inches, what is the diameter in millimeters?

Practice Exercise

How many significant digits are in the following unit

factor?

1 kg/2.20 lb

Concept Exercise

Compound Units Some measurements have a ratio of units.

For example, the speed limit on many highways is 55 miles per hour. How would you convert this to meters per second?

Convert one unit at a time using unit factors.

1. First, miles → meters

2. Next, hours → seconds

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Compound Unit ProblemA motorcycle is traveling at 75

km/hour. What isthe speed in meters per second?

We have km/h; we want m/s.

Use 1 km = 1000 m and 1 h = 3600 s.

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= 21 m/sx1 km

1000 m 1 hr3600 s

75 kmhr

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EXAMPLE 3.8 Conversion of a Unit RatioIf a Mazda Miata is traveling at 95 km/h, what is the speed in meters per second (given that 1 km = 1000 m, and 1 h = 3600 s)?

Unit Analysis Map

EXAMPLE 3.8 Conversion of a Unit Ratio

We apply the unit factor 1000 m/1 km to cancel kilometers , and 1 h/3600 s to cancel hours .

The given value has two significant digits, so the answer is limited to two digits. Since each unit factor is derived from an exact equivalent, neither affects the number of significant digits in the answer.

Solution

EXERCISE 3.8 Conversion of a Unit Ratio

If a runner completes a 10K race in 32.50 minutes (min), what is the 10.0 km pace in miles per hour (given that 1 mi = 1.61 km)?

Practice Exercise

Which speed is faster: 65 mi/h or 65 km/h?

Concept Exercise

Volume by Calculation

The volume of an object is calculated by multiplying the length (l) by the width (w) by the thickness (t).

volume = l x w x t

All three measurements must be in the same units.

If an object measures 3 cm by 2 cm by 1 cm, the volume is 6 cm3 (cm3 is cubic centimeters).

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EXAMPLE 3.9 Volume Calculation for a Rectangular Solid

We can calculate the volume of the rectangular solid by multiplying length times width times thickness: l w t.

5.55 cm 3.75 cm 2.25 cm = 46.8 cm3

The answer is rounded off to three significant digits because each given value has three significant digits.

Solution

If a stainless steel rectangular solid measures 5.55 cm long, 3.75 cm wide, and 2.25 cm thick, what is the volume in cubic centimeters?

EXERCISE 3.9 Volume Calculation for a Rectangular Solid

If a rectangular brass solid measures 52.0 mm by 25.0 mm by 15.0 mm, what is the volume in cubic millimeters?

Practice Exercise

Express the volume of a cube 10 cm on a side in liters.

Concept Exercise

EXAMPLE 3.10: Thickness Calculation for a Rectangular Solid

We can calculate the thickness of the foil by dividing the volume by length and width. Since the unit of volume is mm3, we obtain the thickness in mm by unit cancellation.

The answer is rounded off to three significant digits because each given value has three significant digits.

Solution

A sheet of aluminum foil measures 25.0 mm by 10.0 mm, and the volume is 3.75 mm3. What is the thickness of the foil in millimeters?

EXERCISE 3.10: Thickness Calculation for a Rectangular Solid

A sheet of aluminum foil measures 35.0 cm by 25.0 cm, and the volume is 1.36 cm3. What is the thickness of the foil in centimeters?

Practice Exercise

Which of the following is the greatest thickness?

1 mm, 0.1 cm, or 0.001 m

Concept Exercise

Aluminum foil :A thin sheet of aluminum foil.

Cubic Volume and Liquid Volume The liter (L) is the basic unit of volume in

the metric system.

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• One liter is defined as the volume occupied by a cube that is 10 cm on each side.

Cubic and Liquid Volume Units

1 liter is equal to 1000 cubic centimeters.

10 cm x 10 cm x 10 cm = 1000 cm3

1000 cm3 = 1 L = 1000 mL.

Therefore, 1 cm3 = 1 mL.

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Cubic–Liquid Volume ConversionAn automobile engine displaces a volume of498 cm3 in each cylinder. What is the

displacement of a cylinder in cubic inches, in3?

We want in3; we have 498 cm3.

Use 1 in = 2.54 cm three times.

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= 30.4 in3x1 in

2.54 cm x498 cm3 x

1 in2.54 cm

1 in2.54 cm

EXAMPLE 3.11 Metric–English Volume Conversion

EXERCISE 3.11: Metric–English Volume Conversion

Given that an SUV has a 304 in.3 engine, express the engine volume in liters.

Practice Exercise

Which of the following is the greater volume? 500 mL or 500 cm3

Concept Exercise

Volume by Displacement If a solid has an irregular shape, its

volume cannot be determined by measuring its dimensions.

You can determine its volume indirectly by measuring the amount of water it displaces.

This technique is called volume by displacement.

Volume by displacement can also be used to determine the volume of a gas.

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Solid Volume by DisplacementYou want to measure the volume of an irregularly

shaped piece of jade. Partially fill a volumetric flask with water and

measure the volume of the water.

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• Add the jade, and measure the difference in volume.

• The volume of the jade is 10.5 mL.

Gas Volume by DisplacementYou want to measure the volume of gas given off

in a chemical reaction. The gas produced displaces the water in the flask

into the beaker. The volume of water displaced is equal to the volume of gas.

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EXAMPLE 3.12: Volume by Displacement

We can calculate the displaced volume in milliliters by subtracting the initial volume from the final volume.

36.5 mL – 25.0 mL = 11.5 mL

Solution

A quartz stone weighing 30.475 g is dropped into a graduated cylinder. If the water level increases from 25.0 mL to 36.5 mL, what is the volume of the quartz stone?

EXERCISE 3.12: Volume by Displacement

Hydrogen peroxide decomposes to give oxygen gas, which displaces a volume of water into a beaker. If the water level in the beaker increases from 50.0 mL to 105.5 mL, what is the volume of oxygen gas?

Practice Exercise

Which of the following has the greater volume?

1 mL or 1 cm3

Concept Exercise

The Density Concept

The density of an object is a measure of its concentration of mass.

Density is defined as the mass of an object divided by the volume of the object.

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= densityvolumemass

DensityDensity is expressed in different units. It is

usuallygrams per milliliter (g/mL) for liquids, grams

percubic centimeter (g/cm3) for solids, and grams

perliter (g/L) for gases.

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Densities of Common Substances

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Estimating Density We can estimate the density

of a substance by comparing it to another object.

A solid object will float on top of a liquid with a higher density.

Object S1 has a density less than that of water, but larger than that of L1.

Object S2 has a density less than that of L2, but larger than that of water.

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Calculating Density

What is the density of a platinum nugget that has a mass of 224.50 g and a volume of 10.0 cm3 ? Recall, density is mass/volume.

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= 22.5 g/cm3

10.0 cm3

224.50 g

EXAMPLE EXERCISE 3.13 Density Calculation

EXERCISE 3.13: Density Calculation

Carbon tetrachloride is a solvent used for degreasing electronic parts. If 40.0 mL of carbon tetrachloride has a mass of 39.75 g, what is the density of the liquid?

Practice Exercise

Which of the following has the greater density: ice or water?

Concept Exercise

Density as a Unit Factor We can use density as a unit factor for

conversions between mass and volume.

An automobile battery contains 1275 mL of acid. If the density of battery acid is 1.84 g/mL, how many grams of acid are in an automobile battery?

– We have 1275 mL; we want grams:

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1275 mL x = 2350 gmL

1.84 g

EXAMPLE EXERCISE 3.14 Density as a Unit Factor

EXERCISE 3.14: Density as a Unit Factor

The most abundant gases in our atmosphere are nitrogen, oxygen, and argon. What is the volume of 1.00 kg of air? (Assume the density of air is 1.29 g/L.)

Practice Exercise

Which of the following is the greater density?

1 g/mL or 1 kg/L

Concept Exercise

EXAMPLE 3.15: Density as a Unit Factor

A 1.00-in. cube of copper measures 2.54 cm on a side. What is the mass of the copper cube (given that d of copper = 8.96 g/cm3)?

Unit Analysis Map

EXAMPLE 3.15:Density as a Unit Factor

First, we find the volume of the copper cube. We obtain the volume of the cube, 16.4 cm3, by multiplying (2.54 cm) (2.54 cm) (2.54 cm). We use the given density, 8.96 g/1 cm3, as a unit factor to cancel cubic centimeters , which appears in the denominator.

The given value and unit factor each has three significant digits, so the answer is rounded offto three significant digits.

Solution

Copper Metal A 1.00-inch cube of copper metal.

EXERCISE 3.15: Density as a Unit Factor

A cube of silver is 8.00 cm on a side and has a mass of 1312.5 g. What is the density of silver?

Practice Exercise

If some humans float in water and other sink, what is

the approximate density of the human body?

Concept Exercise

Critical Thinking: GasolineThe density of gasoline is 730. g/L at 0 ºC

(32 ºF) and 713 g/L at 25 ºC (77 ºF). What is the mass difference of 1.00 gallon of gasoline at these two temperatures (1 gal = 3.784L)?

The difference is about 60 grams (about 2 %).

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= 2760 gxAt 0 ºC: 1.00 gal x730. g

L3.784 L1 gal

= 2700 gxAt 25 ºC: 1.00 gal x713 g

L3.784 L1 gal

Temperature Temperature is a measure of the

average kinetic energy of the individual particles in a sample.

There are three temperature scales:

1. Celsius

2. Fahrenheit

3. Kelvin

Kelvin is the absolute temperature scale.

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Temperature Scales

On the Fahrenheit scale, water freezes at 32 °F and boils at 212 °F.

On the Celsius scale, water freezes at 0 °C and

boils at 100 °C. These are the reference points

for the Celsius scale.

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Water freezes at 273 K and boils at 373 K on the Kelvin scale.

Temperature Scales

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Temperature Conversions

This is the equation for converting °C to °F.

This is the equation for converting °F to °C.

To convert from °C to K, add 273.

°C + 273 = K or K – 273 = oC 66

= °F°C x100°C180°F( )

( )180°F100°C

= °C(°F - 32°F) x

Fahrenheit–Celsius Conversions Body temperature is 98.6 °F. What is

body temperature in degrees Celsius? In Kelvin?

K = °C + 273 = 37.0 °C + 273 = 310 K

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( )180°F100°C

= 37.0°C(98.6°F - 32°F) x

EXAMPLE 3.16 °F and °C Temperature Conversions

The average surface temperature of Mars is –55 °C. What is the average temperature in degrees Fahrenheit?

Practice Exercise

What is the relationship between the Celsius and

centigrade temperature scales?

Concept Exercise

Australian Stamp The cartoon illustrates that 38 °C is approximately equal to 100 °F

EXAMPLE 3.16: °F and °C Temperature Conversions

EXAMPLE 3.17: °C and K Temperature Conversions

Given the Celsius temperature, we add 273 units to find the corresponding Kelvin temperature.

–1.96 ° C + 273 = 77 k

Solution

Dermatologists use liquid nitrogen to freeze skin tissue. If the Celsius temperature of liquid nitrogen is –196 °C, what is the Kelvin temperature?

Liquid Nitrogen Although nitrogen is normally a gas, it liquefies at –196 °C. When liquid nitrogen is poured from a Thermos, it is cold enough to freeze the moisture in air and form a white mist.

EXERCISE 3.17: °C and K Temperature Conversions

Which of the following temperatures does not exist?

–100 °F, –100 °C, –100 K

Concept Exercise

The secret to “fire-walking” is to first walk barefoot through damp grass and then step lively on the red-hot coals. If the bed of coals is 1475 K, what is the Celsius temperature?

Practice Exercise

Heat

Heat is the flow of energy from an object of higher temperature to an object of lower temperature.

Heat measures the total energy of a system.

Temperature measures the average energy of particles in a system.

Heat is often expressed in terms of joules (J) or calories (cal).

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Heat Versus Temperature Although both beakers below have the

same temperature (100 ºC), the beaker on the right has twice the amount of heat because it has twice the amount of water.

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EXAMPLE 3.18 Energy Conversion

Burning one liter of natural gas produces 9.46 kcal of heat energy. Express the energy in kilojoules (given that 1 kcal = 4.184 kJ).

Unit Analysis Map

We apply the unit factor 4.184 kJ /1 kcal to cancel kilocalories , which appears in the denominator.

The given value has three significant digits, and the unit factor has four digits. Thus, we round off the answer to three significant digits.

Solution

Bunsen Burner A laboratory burner that uses natural gas for fuel.

EXAMPLE 3.18 Energy Conversion

EXERCISE 3.18 Energy Conversion

Burning one gram of gasoline produces 51.3 kJ of energy. Express the heat energy in kilocalories (given that 1 kcal = 4.184 kJ).

Practice Exercise

If an aerosol can feels cold after releasing the spray, is heat flowing from the can or from your hand?

Concept Exercise

Specific Heat

The specific heat of a substance is the amount of heat required to bring about a change in temperature.

It is expressed with units of calories per gram per degree Celsius.

The larger the specific heat, the more heat is required to raise the temperature of the substance.

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EXAMPLE 3.19 Specific Heat

An energy-efficient home may have solar panels for heating water. If 350,000 cal heat water from 20.0 °C to 35.0 °C, what is the mass of water (specific heat = 1.00 cal/g × °C)?

Unit Analysis Map

EXAMPLE 3.19 Specific Heat

We apply the unit factor to cancel calories , and (35.0 – 20.0) °C to cancel degrees Celsius .

Solution

Solar Panels on Rooftop Solar panels collect light energy from the Sun, which is converted to heat energy for the home.

EXERCISE 3.19 Specific Heat

A 725 g steel horseshoe is heated to 425 °C and dropped into a bucket of cold water. If the horseshoe cools to 20 °C and the specific heat of steel is 0.11 cal/g °C, how much heat is released?

Practice Exercise

If the crust of an apple pie cooks faster than the filling, which has the higher specific heat: the crust or apple filling?

Concept Exercise

Chapter Summary

The basic units in the metric system are grams for mass, liters for volume, and meters for distance.

The base units are modified using prefixes to reduce or enlarge the base units by factors of ten.

We can use unit factors to convert between metric units.

We can convert between metric and English units using unit factors.

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Chapter Summary, Continued Volume is defined as length x width x

thickness.

Volume can also be determined by displacement of water.

Density is mass divided by volume.81

Chapter Summary, Continued

Temperature is a measure of the average energy of the particles in a sample.

Heat is a measure of the total energy of a substance.

Specific heat is a measure of how much heat is required to raise the temperature of a substance.

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Give the symbol for microliter.

a. mLb. miLc. MLd. μL

Give the symbol for microliter.

a. mLb. miLc. MLd. μL

2.00 microliters = ? liters

a. 2.00 × 10–9 Lb. 2.00 × 10–6 Lc. 2.00 × 10–3 Ld. 2.00 × 103 L

2.00 microliters = ? liters

a. 2.00 × 10–9 Lb. 2.00 × 10–6 Lc. 2.00 × 10–3 Ld. 2.00 × 103 L

Given that 1 yd = 0.914 m and 1 mi = 1760 yd, which Olympic track event covers a distance that is closest to a half mile?a. 200 m

b. 400 mc. 800 md. 1000 m

Given that 1 yd = 0.914 m and 1 mi = 1760 yd, which Olympic track event covers a distance that is closest to a half mile?a. 200 m

b. 400 mc. 800 md. 1000 m

Flu vaccines are usually administered using a hypodermic needle. A typical dose is 0.500cc (0.500cubiccentimeters). Given that 2.54 cm = 1 in, what is 0.500 cubic centimeters in cubic inches?

a. 0.0122 cubic inchesb. 0.0305 cubic inchesc. 0.0775 cubic inchesd. 0.197 cubic inches

Flu vaccines are usually administered using a hypodermic needle. A typical dose is 0.500cc (0.500 cubic centimeters). Given that 2.54 cm = 1 in, what is 0.500 cubic centimeters in cubic inches?a. 0.0122 cubic inches

b. 0.0305 cubic inchesc. 0.0775 cubic inchesd. 0.197 cubic inches

When added to a graduated cylinder containing 4.00 mL of water, 5.00 g of which metal will displace the greatest volume of water?

a. iron, d = 7.87 g/cm3

b. lead, d = 11.3 g/cm3

c. gold d = 19.3 g/cm3

d. 5.00 g of any of these metals will displace the same volume of water

When added to a graduated cylinder containing 4.00 mL of water, 5.00 g of which metal will displace the greatest volume of water?a. iron, d = 7.87 g/cm3

b. lead, d = 11.3 g/cm3

c. gold d = 19.3 g/cm3

d. 5.00 g of any of these metals will displace the same volume of water