# Lab 2 Conversion Factors in Calculations

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2

Conversion Factors in Calculations

Goals

•

•

Round off a calculated answer to the correct number of significant figures.

Determine the area of a rectangle and the volume of a solid by direct measurement.

•

Determine metric and metric-to-U.S.-unit equalities and corresponding conversion factors.

•

Use conversion factors in calculations to convert units of length, volume, and mass.

3

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Convert a Celsius temperature to Fahrenheit and Kelvin.

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Discussion

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As you begin to perform laboratory experiments, you will make measurements, collect data, and carry

out calculations. When you use measured numbers in calculations, the answers that you report must

reflect the precision of the original measurements. Thus it is often necessary to adjust the results you

see on the calculator display. Every time you use your calculator, you will need to assess the mathematical operations, count significant figures, and round off calculator results.

A.

Usually there are fewer significant figures in the measured numbers used in a calculation than there

are digits that appear in a calculator display. Therefore, we adjust the calculator result by rounding

off. If the first number to be dropped is less than 5, it and all following numbers are dropped. If the

first number to be dropped is 5 or greater, the numbers are dropped and the value of the last retained

digit is increased by 1. When you round a large number, the correct magnitude is retained by replacing the dropped digits with placeholder zeros. See Sample Problem 2.1. When a whole number

appears in the calculator display, significant zeros may be added.

Sample Problem 2.1

Round off each of the following calculator displays to report an answer with three significant figures

and another answer with two significant figures:

a. 75.6243

b. 0.528392

c. 387,600

d. 4

Solution:

a.

b.

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c.

d.

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Rounding Off

B.

Three Significant Figures

75.6

0.528

388000

4.00

Two Significant Figures

76

0.53

390000

4.0

Significant Figures in Calculations

When you carry out mathematical operations, the answer you report depends on the number of

significant figures in the data you used.

Multiplication/division When you multiply or divide numbers, report the answer with the same

number of significant figures as the measured number with the fewest significant figures. See Sample

Problem 2.2.

11

Laboratory 2

Sample Problem 2.2

Solve: 0.025 x 4.62 =

3.44

Solution:

On the calculator, the steps are

= 0.033575581 calculator display

0.025 x 4.62 + 3.44

final answer rounded to two significant figures

= 0.034

Addition/subtraction When you add or subtract numbers, the reported answer has the same number

of decimal places as the measured number with the fewest decimal places. See Sample Problem 2.3.

Sample Problem 2.3

Add: 2.11 + 104.056 + 0.1205

Solution:

C.

2.11

two decimal places

104.056

0.1205

106.2865 calculator display

106.29

final answer rounded to two decimal places

Conversion Factors for Length

Metric factors If a quantity is expressed in two different metric units, a metric equality can be stated.

For example, the length of 1 meter is the same as 100 cm, which gives the equality 1 m = 100 cm. When

the values in the equality are written as a fraction, the ratio is called a conversion factor. Two fractions

are always possible, and both are conversion factors for the equality.

Equality

1m=100cm

Conversion Factors

100 cm and 1m

100 cm

1m

Metric-U.S. system factors When a quantity measured in a metric unit is compared to its measured

quantity in a U.S. unit, a metric-U.S. conversion factor can be written. For example, 1 inch is the

same length as 2.54 cm, as seen in Figure 2.1.

1 in.

I

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1 em

Figure 2.1

I

2.54 em

Comparing centimeters and inches

For the length of 1 inch, two conversion factors can be written.

Equality

Conversion Factors

1 in. = 2.54 cm

2.54 cm

1 in.

—and – – I in.

2.54 cm

Conversion Factors in Calculations

D.

Conversion Factors for Volume

In the metric system, equalities for volume can be written along with their corresponding conversion

factors. A useful metric-U.S. equality is the relationship of 1 quart equaling 946 mL.

E.

Equality

1L=1000mL

Conversion Factors

1000 mL

1 L

– – – and

I L

1000 mL

I qt = 946 mL

946 mL

I qt

– – – and

1 qt

946 mL

Conversion Factors for Mass

In the metric system, equalities for mass can be written along with their corresponding conversion

factors. A useful metric-U.S. equality is the relationship of 454 g equaling I pound.

F.

Equality

1 kg = 1000 g

Conversion Factors

1000 g and 1 kg

1 kg

1000 g

lib = 454 g

454 g and ~

1 lb

454 g

Percent by Mass

A percent (%) by mass gives the number of grams of each component in 100 grams of the mixture. It

is calculated by dividing the mass of each component by the mass of the mixture and multiplying by

100.

G.

Mass (g) of component 1

Mass (g) of the mixture

x

100

=

% of component 1

Mass (g) of component 2

Mass (g) of the mixture

x

100

=

% of component 2

Converting Temperature

Temperature measures the intensity of heat in a substance. A substance with little heat feels cold.

Where the heat intensity is great, a substance feels hot. The temperature of our bodies is an indication

of the heat produced. An infection may cause body temperature to deviate from normal. On the

Celsius scale, water freezes at O°C; on the Fahrenheit scale, water freezes at 32°F. A Celsius temperature is converted to its corresponding Fahrenheit temperature by using the following equation:

=

TF

1.8 (Tc ) + 32

When the Fahrenheit temperature is known, the Celsius temperature is determined by rearranging the

equation. Be sure you subtract 32 from the T F , then divide by 1.8.

T

– (T F -32)

1.8

C –

A Celsius temperature can be converted to a Kelvin temperature by using the following equation:

TK

=

Tc + 273

13

Laboratory 2

Lab Information

Time:

Comments:

3 hr

Tear out the report sheets and place them beside the procedures.

Determine what the smallest lines of measurement are on each measuring tool you use.

Include an estimated digit for each measurement.

Round off the calculator answers to the correct number of significant figures.

Related Topics: Conversion factors, significant figures in mathematical operations, calculator use

Experimental Procedures

A.

Rounding Off

Materials: Meterstick, solid

–

—

—

Al Rounding A student has rounded off some numbers. Determine whether the rounding was done

correctly. If it is not correct, write the correctly rounded number.

A2 Area Determine the length (cm) and width (cm) of the sides of the rectangle on the report sheet.

Obtain a second set of measurements from another student. Record. Calculate the area (cm 2 ) of the

rectangle using your measurements and this formula: Area = Lx W. Obtain the area calculated by

the other student. Compare the calculated areas from both sets of measurements.

A3 Volume of a solid by direct measurement Obtain a solid object that has a regular shape, such as a

cube, rectangular solid, or cylinder. Record its shape. Use a ruler to determine the dimensions of

the solid in centimeters (cm). Use the appropriate formula from the following list to calculate the

volume in cm3 .

Shape

Dimensions to Measure

Cube

Length (L)

Rectangular solid

Length (L), width (W), height (H)

Cylinder

Diameter (D), height (H)

Formulas for Volume

V=LxWxH

v

B.

=

Significant Figures in Calculations

B.I Solve the multiplication and division problems. Report your answers with the correct number of

significant figures.

B.2 Solve the addition and subtraction problems. Report your answers with the correct number of

significant figures.

C.

Conversion Factors for Length

Materials: Meterstick

C.l Metric factors Observe the markings for millimeters on a meterstick. Write an equality that states

the number of millimeters in 1 meter. Write two metric conversion factors for the relationship.

Observe the number of millimeters in a centimeter. Write equality and corresponding conversion

factors for the relationship between centimeters and millimeters.

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Conversion Factors in Calculations

C.2 Metric-U.S. system factors Measure the length of the dark line on the report sheet in centimeters

and in inches. Convert any fraction to a decimal number. Divide the number of centimeters by the

number of inches to give a relationship. Round off correctly for your reported answer. This is your

experimental value for the number of centimeters in 1 inch.

C.3 Your metric height Record your height in inches. Or use a yardstick to measure. Using the appropriate conversion factors, calculate your height in centimeters and meters. Show your setup for

each calculation.

Height (in.) X 2.54 cm

1 in.

Height (cm) x

D.

= your height (cm)

1m

= your height (m)

100cm

Conversion Factors for Volume

Materials: I-L graduated cylinder, I-quart measure (or two I-pint measures)

DJ Observe the markings on a I-liter graduated cylinder. Write an equality that states the number of

milliliters in 1 liter. Write two conversion factors for the equality.

0.2 Using a I-pint or I-quart measure, transfer 1 quart of water to a I-liter graduated cylinder. Record

the number of milliliters in 1 quart. Write the equality that states the number of milliliters in a

quart. Write two conversion factors for the equality.

E.

Conversion Factors for Mass

Materials: Commercial product with mass (weight) of contents given on label

E.I Grams and pounds Labels on commercial products list the amount of the contents in both metric

and U.S. units. Obtain a commercial product. Record the mass (weight) of the contents stated on

the product label. Do l1J2t. weigh. If the weight is given in ounces, convert it to pounds (lib =

I6oz).

_ _ _ _ _ oz x

1 lb

16 oz

=

_ _ _ _ lb

Divide the grams of the product by its weight in pounds. (Be sure to use the correct number of

significant figures.) This is your value for grams in one pound (gnb).

E.2 Pounds and kilograms State the mass on the label in kilograms. If necessary, convert the number

of grams to kilograms.

_____ g x

lkg

1000 g

=

_ _ _ _ kg

Divide the number of pounds by the number of kilograms. Report the ratio as lblkg. (Be sure to

use the correct number of significant figures.)

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Laboratory 2

F.

F.l

Percent by Mass

Materials: 100- or 250-mL beaker, graduated cylinder

Sucrose (sugar), water

–

Weigh a 100-mL or 250-mL beaker or tare the beaker. Record.

–

Taring a container on an electronic balance: The mass of a container on an electronic balance can be set to 0 by pressing the tare bar. As a substance is added to the container, the

mass shown on the readout is for the substance only. (When a container is tared, it is not

necessary to subtract the mass of the beaker.)

F.2

Add about 5 g of sugar. Record the mass.

F.3

With the beaker and sugar still on the balance, add 15-20 mL of water to the sugar in the

beaker. Record the mass of the sugar-water mixture.

FA

Calculate the % sugar and % water by mass in the sugar-water mixture.

G.

Converting Temperature

Materials: Thermometer (0C), a 150- or 250-mL beaker, ice, and rock salt

G.1 Observe the markings on a thermometer. Indicate the lowest and highest temperatures that can be

read using that thermometer. Caution: Never shake down a laboratory thermometer. Shaking a

laboratory thermometer can cause breakage and senous accidents.

G.2 To measure the temperature of a liquid, place the bulb of the thermometer in the center of the solution. Keep it immersed while you read the temperature scale. When the temperature becomes constant, record the temperature (OC). On most thermometers, you can estimate the tenths of a degree

(O.l°C). A set of beakers with the following contents may be set up in the lab; otherwise fill the

beakers as instructed. Determine the temperature of each of the following:

a. Room temperature: Place the thermometer on the lab bench.

b. Tap water: Fill a 250-mL beaker about 1/3 full of water.

c. lee-water mixture: Add enough ice to the water in part b to double (approximately) the volume.

Allow 5 minutes for the temperature to change.

d. A salted ice mixture: Add rock salt to the ice-water mixture in part c. Stir and allow a few

minutes for the temperature to change.

G.3 Convert the Celsius temperatures to corresponding temperatures on the Fahrenheit and Kelvin

scales.

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Print out this page to meaure the area of the rectangle

Do not do A.3, skip this part only!!!!

Print this page out in order to measure the line in inches and cm

Use and any commerical food product from home and include a copy

of the food label by taking a picture of it. The label should have mass in

grams and in pounds or ounces.

convert grams to kilograms to then do.

In calculations with measured numbers, significant figures or decimal places are

counted to determine the number of figures in the final answer.