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Chapter 5

Measurements and Calculations

Chapter 5

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Scientific Notation

• Section 5.1

• Objective:– To show how very large numbers or very small

numbers can be expressed as the product of a number between 1 and 10 and a power of 10.

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Very large/very

small numbers

93,000,000 miles to the sun!

Dictionary of Measurement

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Very large numbers can be awkward to write. For example, the approximate distance from

the earth to the sun is ninety three million miles. This is commonly written as the

number "93" followed by six zeros signifying that the "93" is actually 93 million miles and

not 93 thousand miles or 93 miles.

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.

Scientific notation (also called exponential notation) provides a more compact method for writing very large (or very small) numbers. In scientific notation, the distance from the earth to the sun is 9.3 x 107 miles.

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Using Scientific Notation

• When the decimal point is moved to the right, the exponent of 10 is negative. Ex. 0.000167 is 1.67 x 10-4

• If the decimal point is moved to the left, the exponent of 10 is positive. Ex. 238,000 is 2.38 x 105

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Very small numbers can be as awkward to write as large numbers. A paper clip weighs a bit more than one thousandth of a pound (0.0011 LB). This would be expressed in scientific notation as 1.1 x 10-3 lb. The negative sign indicates that the decimal point is moved to the left.

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Representing a number using scientific notation

The number 2,398,730,000,000 can be written in scientific notation as (the starred representation is most common):

0.239873 x 1013 ****2.39873 x 1012****

23.9873 x 1011 239.873 x 1010

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The number 0.003,483 can be written in scientific notation as

(the starred representation is most common):

0.3483x 10-2 ****3.483 x 10-3****

34.83x 10-4

348.3x 10-5 3483.x 10-6

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Multiplying numbers written in scientific notation

• To multiply 4 x 104 and 6 x 105: • Multiply the decimal parts together: • 4 x 6 = 24 • Add the two exponents: • 4 + 5 = 9 • Construct the result: • 24 x 109

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0011 0010 1010 1101 0001 0100 1011Adjust the result so only one digit is to the left of the

decimal point (if necessary):

24 x 109 = 2.4 x 1010

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General Rule(a x 10x)(b x 10y) = ab x 10x+y

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Dividing numbers written in scientific notation

To divide 6 x 105 and 4 x 104: 1. 6/4 = 1.5 2. Subtract the two exponents: 3. 6/4 = 1.5 4. Construct the result: 5. 1.5 x 101 6. Adjust the result so only one digit is to the

left of the decimal point (if necessary)1 7. 1.5 x 109 = 1.5 x 101 No adjustment

necessary

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General Rule (a x 10x)/(b x 10y) = a/b x 10x-y

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Now, neither the decimal part or the exponential part combine together in

any obvious manner (as they did with multiplication and division).

When adding or subtracting numbers written in exponential notation, the numbers must first be rewritten so the exponents are identical. Then,

the numbers can be added or subtracted normally

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To add 3.4 x 10-3 to 2.1 x 10-2:

1. 1. Adjust one of the numbers so that its exponent is equivalent

to the other number. In this case, change 2.1 x 10-2 into a number which has 10-3 as it's exponential part.

2.1 x 10-2 = 21 x 10-3

2. Add the decimal parts together:

21 + 3.4 = 24.4

3. The exponential part of the result is the same as the exponential parts of the two numbers, in this case, 10-3:

24.4 x 10-3

4. Adjust the result so only one digit is to the left of the decimal point (if necessary):

2.44 x 10-2

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Printable WorksheetComputer-Graded Quiz

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Survival Skills for

Quantitative Courses Algebra & Arithmetic

•http://www.brynmawr.edu/nsf/tutorial/ss/ssalg.html

•http://www.brynmawr.edu/nsf/tutorial/ps/pstips.html

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Units

• Section 5.2

• Objective: To compare the English, Metric and SI systems of measurement.

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Units

• Scientists became aware of the need for common units to describe quantities such as time, length, mass, and temperatures.

• Two systems: English System (U.S.) and the International System (the rest of the world).

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Fundamental SI Units

Quantity Name of Unit Abbreviation

Mass Kilogram Kg

Length Meter m

Time Second s

Temperature Kelvin K

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Table 5.2

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Measurements of Length, Volume and Mass

• Section 5.3

• Objective: To demonstrate the use of the metric system to measure, length, volume and mass.

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MeasuringMeasuring Volume Temperature Mass Length

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0011 0010 1010 1101 0001 0100 1011Reading the MeniscusReading the Meniscus

Always read volume from the bottom of the meniscus. The meniscus is the curved surface of a liquid in a narrow cylindrical container.

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Volume

• Amount of three-dimensional space occupied by a substance.

• 1cm3 = 1 mL

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Try to avoid parallax errors.Try to avoid parallax errors.ParallaxParallax errorserrors arise when a meniscus or arise when a meniscus or needle is viewed from an angle rather than needle is viewed from an angle rather than from straight-on at eye level.from straight-on at eye level.

Correct: Viewing the meniscus

at eye level

Incorrect: viewing the meniscus

from an angle

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The glass cylinder has etched marks to indicate volumes, a pouring lip, and quite often, a plastic bumper to prevent breakage.

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0011 0010 1010 1101 0001 0100 1011Measuring VolumeMeasuring Volume

Determine the volume contained in a graduated cylinder by reading the bottom of the meniscus at eye level. Read the volume using all certain digits and one uncertain digit.

Certain digits are determined from the calibration marks on the cylinder. The uncertain digit (the last digit of the reading) is estimated.

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Use the graduations to find all Use the graduations to find all certain digitscertain digits

There are two unlabeled graduations below the meniscus, and each graduation represents 1 mL, so the certain digits of the reading are…

52 mL.

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0011 0010 1010 1101 0001 0100 1011Estimate the uncertain digit and Estimate the uncertain digit and take a readingtake a reading

The meniscus is about eight tenths of the way to the next graduation, so the final digit in the reading is .

The volume in the graduated cylinder is

0.8 mL

52.8 mL.

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10 mL Graduate10 mL GraduateWhat is the volume of liquid in the graduate?

_ . _ _ mL6 26

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25mL graduated cylinder 25mL graduated cylinder What is the volume of liquid in the graduate?

_ _ . _ mL1 1 5

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0011 0010 1010 1101 0001 0100 1011100mL graduated cylinder 100mL graduated cylinder

What is the volume of liquid in the graduate?

_ _ . _ mL5 2 7

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Self TestSelf TestExamine the meniscus below and determine the volume of liquid contained in the graduated cylinder.

The cylinder contains:

_ _ . _ mL7 6 0

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The The ThermometerThermometer

o Determine the temperature by reading the scale on the thermometer at eye level. o Read the temperature by using all certain digits and one uncertain digit. o Certain digits are determined from the

calibration marks on the thermometer. o The uncertain digit (the last digit of the reading) is estimated. o On most thermometers encountered in a general chemistry lab, the tenths place is the uncertain digit.

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Do not allow the tip to touch the Do not allow the tip to touch the walls or the bottom of the flask.walls or the bottom of the flask.

If the thermometer bulb touches the flask, the temperature of the glass will be measured instead of the temperature of the solution. Readings may be incorrect, particularly if the flask is on a hotplate or in an ice bath.

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Reading the ThermometerReading the ThermometerDetermine the readings as shown below on Celsius thermometers:

_ _ . _ C _ _ . _ C 8 7 4 3 5 0

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Measuring MassMeasuring Mass - - The Beam Balance

Our balances have 4 beams – the uncertain digit is the thousandths place ( _ _ _ . _ _ X)

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In order to protect the balances and ensure accurate results, a number of rules should be followed:

Always check that the balance is level and zeroed before using it. Never weigh directly on the balance pan. Always use a piece of weighing paper to protect it. Do not weigh hot or cold objects. Clean up any spills around the balance immediately.

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0011 0010 1010 1101 0001 0100 1011Mass and Significant FiguresMass and Significant Figures

o Determine the mass by reading the riders on the beams at eye level. o Read the mass by using all certain digits and one uncertain digit.

oThe uncertain digit (the last digit of the reading) is estimated. o On our balances, the thousandths place is uncertain.

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Determining Determining MassMass

1. Place object on pan

2. Move riders along beam, starting with the largest, until the pointer is at the zero mark

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Check to see that the Check to see that the balance scale is at zerobalance scale is at zero

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Read Mass

_ _ _ . _ _ _1 1 4 ? ? ?

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Read Mass More Closely

_ _ _ . _ _ _1 1 4 4 9 7

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Measuring Length

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1 inch = 2.54 cm

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Read the length by using all certain digits and one uncertain digit.

Read it as 2.85cm.

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Uncertainty in Measurement and Significant Figures

• Section 5.4-5.5

• Objectives: To observe how uncertainty can arise and to use significant figures to indicate a measurement’s uncertainty.

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Rules for Significant Figures• Nonzero integers always count as significant. Ex.

1457 has 4 nonzero integers, so it has 4 sig. figs.• Zeros: 3 classes

– Leading zeros are not significant and place holding zeros are not significant.

– Captive zeros (between nonzero digits) are significant. Ex. 1.008 has 4 sig. figs.

– Trailing zeros are significant if the number is written with a decimal point. Ex. 100. (has 3).

• Exact numbers are significant. One inch = 2.54 cm or 4 apples

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Rounding Rules

• Less than 5, stays the same

• = to or greater than 5, round up

• In a series of calculations wait until the last step to round off.

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Rounding with Significant Figures

• When multiplying or dividing – go with the smallest number of significant figures.– Ex. 4.56 x 1.4 = 6.384 or 6.4 (2 sig. figs.)

• When adding and subtracting use the smallest number of decimal places.– Ex. 0.6875 – 0.1 = 0.5875 or 0.6

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Practice- How many significant figures should the following contain?

• 5.19 + 1.9 + 0.842 =

• 2.3 x 3.14 =

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Practice- How many significant figures should the following contain?

• 5.19 + 1.9 + 0.842 = (2 sig. figs.)

• 2.3 x 3.14 = (2 sig. figs.)

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Calculations using Significant Figures

• Give each result to the correct number of significant figures: 5.18 x 0.0208 =

(3.60 x 10-3)(8.123)/4.3 =

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Calculations using Significant Figures

• Give each result to the correct number of significant figures: 5.18 x 0.0208 = 0.107744 is 0.108

(3.60 x 10-3)(8.123)/4.3 = 6.8006 x 10-3 is 6.8 x10-3

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Problem Solving & Dimensional Analysis

• Section 5.6

• Objective: to practice using dimensional analysis to solve problems.

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Dimensional AnalysisDimensional Analysis refers to the physical nature of the quantity and the type of unit (Dimension) used to specify it.

•Distance has dimension L.

•Area has dimension L2.

•Volume has dimension L3.

•Time has dimension T.

•Speed has dimension L/T

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To change a measurement from one unit to another, we use a conversion factor, which is a number that shows the relationship between two different units of measure.

For example: metric units and English units

2.54cm or 1 inch

1 inch 2.54cm

2.85cm = ? In.

•We choose a conversion factor that cancels units we want to get rid of and leaves the units we want.

•2.85 cm x 1 inch = 1.12 in. 2.54cm

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Convert 7.00 in. to cm

•7.00 in. x 2.54cm = 17.8 cm 1 in.

The length of a marathon is 26.2 mi. What is this distance in kilometers?

•Miles --- Yards --- Meters ----Kilometers

•26.2 mi x 1760 yd x 1 m x 1 km = 42.1km 1 mi 1.094yd 103 m

Racing cars at the Indianapolis Motor Speedway now routinely travel at an average of 225mi/h. What is this speed in kilometers per hour?

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How did you do?

• 225mi x 1760yd x 1 m x 1 km = 362 km hr 1 mi 1.094yd 1000m hr

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Sec. 5.7 Temperature Conversions

• Objectives: – To identify the three temperature scales.– To practice converting from one scale to

another.

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Figure 5.6: The three major temperature scales.

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Comparison of Scales

• The zero point is different on all three scales.• The Fahrenheit degree is smaller than the Celsius

or Kelvin. There are 180 units between freezing and boiling.

• The size of the unit (degree) is the same for the Kelvin and Celsius scale. There are 100 units between freezing and boiling on both scales.

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Figure 5.7: Converting 70 degrees Celsius to Kelvin units.

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Converting between Kelvin & Celsius Scales

• Add 273 to the Celsius temperature to convert to Kelvin

• 70 degrees Celsius is 343 Kelvin

• Subtract 273 to go from Kelvin to Celsius

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Converting between Fahrenheit & Celsius Scales

• Must adjust for the size of the units and the zero points

• 1.80 Fahrenheit degrees= 1.00 Celsius degree

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Figure 5.8: Comparison of the Celsius and Fahrenheit scales.

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Converting between Fahrenheit & Celsius

• Convert 28 degrees Celsius to Fahrenheit

• T°F = 1.80(T°C) + 32

• T°F = 1.80(28) + 32

• T°F = 82

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Converting from Fahrenheit to Celsius

• T°C = T°F – 32

1.80

Convert 101 degrees Fahrenheit to Celsius

• °C = 101-32

1.80

• °C = 38

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Density

• Section 5.8

• Objective: to define density and its units

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Density

• Ratio of the mass of an object to its volume

• Density = Mass/Volume

• 1ml = 1cm3

• Often determined by submerging an object in water and measuring the volume of water displaced

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Figure 5.9: Tank of water.

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Figure 5.9: Person submerged in the tank.

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Density

• Can rearrange equation to isolate desired quantity

• Mercury has a density of 13.6g/mL. What volume of mercury must be taken to obtain 225g of the metal?

• Volume = mass/density• 225 g = 16.5mL 13.6g/mL

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Specific Gravity

• The ratio of the density of a given liquid to the density of water at 4°C.