Lab 2 Conversion Factors in Calculations

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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.
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1 em
Figure 2.1
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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.