Measurements and Calcuations

Chapter 2: Measurements and Calculations

A measurement must always consist of a number and a unit.

Scientific Notation simply expresses a number as a product of a number between 1 and 10 and the appropriate power of 10.

Rules for Scientific Notation1. Keep one digit to the left of the decimal

point; that digit should be between 1 and 9. That is, no zeros or tens.

2. Moving the decimal point to the left requires a positive exponent.

3. Moving the decimal point to the right requires a negative exponent.

Using Scientific Notation Any number can be represented as the

product of a number between 1 and 10 and a power of 10 (either positive or negative).

The power of 10 depends on the number of places the decimal point is moved and in which direction. The number of places the decimal point is moved determines the power of 10. The direction of the move determines whether the power of 10 is positive or negative. If the decimal point is moved to the left, the power of 10 is positive; if the decimal point is moved to the right, the power of 10 is negative.

A number that is greater than 1 will always have a positive exponent when written in scientific notation.

A number that is less than 1 will always have a negative exponent when written in scientific notation.

Left Is Positive; remember LIP

1. 12,5002. 2473. 104. 3,500,0005. 14306. 0.1357. 0.00248. 0.1049. 0.030610. 0.00000072

1. 1.25 x 104

2. 2.47 x 102

3. 1 x 101

4. 3.5 x 106

5. 1.43 x 103

6. 1.35 x 10-1

7. 2.4 x 10-3

8. 1.04 x 10-1

9. 3.06 x 10-2

10. 7.2 x 10-7

Chapter 2: Measurements and Calculations

The units part of a measurement tells us what scale or standard is being used to represent the results of the measurement.

Two widely used systems of measurements used today are the English system used in the United States and the metric system used in most of the rest of the industrialized world.

The metric system has long been preferred for scientific work.

In 1960 an international agreement set up a comprehensive system of units called the International System (le Système Internationale in French), or SI.

The SI units are based on the metric system and units derived from the metric system.

Physical Quantity

Name of Unit Abbreviation

Mass Kilogram kg

Length Meter m

Time Second s

Temperature Kelvin K

Because the fundamental units are not always a convenient size, the SI system uses prefixes to change the size of the unit.

Prefix Symbol Meaning Power of 10 for scientific notation

Mega M 1,000,000 106

Kilo k 1000 103

Deci d 0.1 10-1

Centi c 0.01 10-2

Milli m 0.001 10-3

Micro µ 0.000001 10-6

Nano n 0.000000001 10-9

Chapter 2: Measurements and Calculations

The fundamental SI unit of length is the meter, which is a little longer than a yard (1 meter = 39.37 inches).

The meter was originally defined, in the eighteenth century, as one ten-millionth of the distance from the equator to the North Pole and then, in the late nineteenth century, as the distance between the two parallel marks on a special metal bar stored in a vault in Paris. Most recently, for accuracy and convenience, a definition expressed in terms of light waves has been adopted.

In the metric system fractions of a meter or multiples of a meter can be expressed by powers of 10.

Unit Symbol Meter Equivalent

Kilometer km 1000 m or 103 m

Meter m 1 m

Decimeter dm 0.1 m or 10-1 m

Centimeter cm 0.01 m or 10-2 m

Millimeter mm 0.001 m or 10-3 m

Micrometer µm 0.000001 m or 10-

6m

Nanometer nm 0.000000001 or 10-9 m

Volume is the amount of three-dimensional space occupied by a substance.

The fundamental SI unit of volume is based on the volume of a cube that measures 1 meter in each of the three directions. That is, each edge of the cube is 1 meter in length. The volume of this cube is:

1 m x 1 m x 1 m = (1m)3 = 1m3

The cube is then divided into 1000 smaller cubes. Each cube have a volume of 1 dm3, which is commonly called the liter and abbreviated L.

The cube with the volume of 1 dm3 can in turn be broken into 1000 smaller cubs each representing a volume of 1 cm3. This means that each liter contains 1000 cm3. One cubic centimeter is called a milliliter (abbreviated mL), a unit of volume used very commonly in chemistry.

Another important measurable quantity is mass, which can be defined as the quantity of matter present in an object.

The fundamental SI unit of mass is the kilogram

Because the metric system, which existed before the SI system, used the gram as the fundamental unit, the prefixes for the various mass units are based on the gram.

In a laboratory we determine the mass of an object by using a balance.

Difference between mass and weight? Mass is the measurement of the amount

of matter in an object. Weight is the measurement of the

amount of force acting on that matter. Mass is measured on a balance, weight

is measured on a scale. The SI unit for mass it the kilogram, the

SI unit for weight is the Newton. The English unit for mass is the slug,

while the English unit for weight is the pound.

Chapter 2: Measurements and Calculations

Every measurement has some degree of uncertainty.

It is very important to realize that a measurement always has some degree of uncertainty.

The uncertainty of a measurement depends on the measuring device.

The numbers recorded in a measurement (all the certain numbers plus the first uncertain number) are called significant figures.

The number of significant figures for a given measurement is determined by the inherent uncertainty of the measuring device.

Chapter 2: Measurements and Calculations

1. Nonzero integers. Nonzero integers always count as significant figures. Ex. 1457 has four significant figures

2. Zeros. There are three classes of zeros.a. Leading Zeros are zeros that precede all of the nonzero digits. They never count as significant figures. Ex. 0.0025 has two significant figures, the 2 and the 5.b. Captive Zeros are zeros that fall between nonzero digits. They always count as significant figures. Ex. 1.008 has four significant figures.c. Trailing zeros are zeros at the right end of the number. They are significant only if the number is written with a decimal point. Ex. 100 has one significant figure, but 100. has three significant figures, because of the decimal point.

3. Exact Numbers. Often calculations involve numbers that were not obtained by using measuring devices but were determined by counting. Such numbers are exact numbers. They can be assumed to have a unlimited number of significant figures.

Give the number of significant figures for each measurement. The mass of a single eyelash is 0.000304 g.

3 The length of a skid mark is 1.270 x 102 m.

4 A 125 g sample of chocolate chip cookie contains 10

g of chocolate. 3 1

The volume of soda remaining in a can after a spill is 0.09020 L.

4 A dose of antibiotic is 4.0 x 10-1 cm3.

2

Leading zeros are never significant figures.

Captive zeros are always significant figures.

Trailing zeros are only significant if the number contains a decimal point.

Exact numbers never limit the number of significant figures in a calculation.

Significant figures are easily indicated by scientific notation.

When you perform a calculation on your calculator, the number of digits displayed is usually greater than the number of significant figures that the result should possess.

One must “round off” the number (reduce it to fewer digits).

1. If the digit to be removeda. Is less than 5, the preceding digit stays the

same. For example, 1.33 rounds to 1.3b. Is equal or greater to 5, the preceding digit

is increased by 1. For example, 1.36 rounds to 1.4, and 3.15 rounds to 3.2

2. In a series of calculations, carry the extra digits through to the final result and then round off. This means that you should carry all of the digits that show on your calculator (within reason) until you arrive at the final number (the answer) and then round off, using the procedures in Rule 1.

For multiplication or division, the number of significant figures in the result is the same as that in the measurement with the smallest number of significant figures. We say this measurement is limiting, because it limits the number of significant figures in the result.

Ex: 4.56 x 1.4 = 6.384 = 6.4 Ex: 8.315 ÷ 298 = 0.0279027 = 2.79 x

10-2

For addition and subtraction, the limiting term is the one with the smallest number of decimal places.

Ex: 12.11 + 18.0 + 1.013 = 31.123 = 31.1

Ex: 0.6875 – 0.1 = 0.5875 = 0.6

Note that for multiplication and division, significant figures are counted. For addition and subtraction, the decimal places are counted.

1. 17.1 + 0.77 + 241 =

2. 47.2 – 9 = 3. 1.27 x 3.1416

= 4. 0.072 ÷ 4.365. The cost of 2

tickets to a concert at $27.50 per ticket.

1. 259

2. 38

3. 3.99

4. 0.017

5. $55.00

Chapter 2: Measurements and Calculations

VIDEO

Step 1: To convert from one unit to another, use the equivalence statement that relates the two units. The conversion factor needed is a ratio of the two parts of the equivalence statement.

Step 2: Choose the appropriate conversion factor by looking at the direction of the required change (make sure the unwanted units cancel).

Step 3: Multiply the quantity to be converted by the conversion factor to give the quantity with the desired units.

Step 4: Check that you have the correct number of significant figures.

Step 5: Does your answer make sense?

An Italian bicycle has its frame size given as 62 cm. What is the frame size in inches?

incm

incmin 24

54.2

162?

The length of the marathon race is approximately 26.2 mi. What is this distance in kilometers?

kmm

km

yd

m

mi

ydmikm 1.42

1000

1

094.1

1

1

17602.26?

A new baby weighs 7.8 lb. What is its mass in kilograms?

kglb

kglbkg 5.3

205.2

18.7?

A piece of lumber is 88.4 cm long. What is its length in millimeters? In inches?

incm

incmin

mmm

mm

cm

mcmmm

8.3454.2

14.88?

8841

1000

100

14.88?

A bottle of soda contains 2.0 L. What is its volume in quarts?

qtL

qtLqt 1.2

1

06.10.2?

Whenever you work problems, remember the following points:

1. Always include the units (a measurement always has two parts: a number and a unit).

2. Cancel units as you carry out the calculations.

3. Check that your final answer has the correct units. If it does not, you have done something wrong.

4. Check that your final answer has the correct number of significant figures.

5. Think about whether your answer makes sense.

Chapter 2: Measurements and Calculations

Quantity Fahrenheit (0F)

Celsius (0C) Kelvin (K)

Freezing Point of Water

32 0F 0 0C 273.15 K

Boiling Point of Water

212 0F 100 0C 373.15 K

There are several important facts about the three temperature scales:

1. The size of each temperature unit (each degree) is the same for the Celsius and Kelvin scales. This follows from the fact that the difference between the boiling and freezing points of water is 100 units on both of these scales.

2. The Fahrenheit degree is smaller than the Celsius and Kelvin unit. Note that on the Fahrenheit scale there are 180 Fahrenheit degrees between the boiling and freezing points of water, as compared with 100 units on the other two scales.

3. The zero points are different on all three scales.

Celsius Kelvin

CelsiusKelvin

Add 273.15

Subtract 273.15

Fahrenheit Celsius

FahrenheitCelsius=1.80(Tc) +

32

80.1

32Tf

Chapter 2: Measurements and Calculations

Which is heavier, a pound of lead or a pound of feathers?

Density can be defined as the amount of matter present in a given volume of substance.

That is, density if mass per unit volume, the ratio of the mass of an object to its volume.

volume

massDensity

A block has a volume of 25.3 cm3. Its mass is 21.7 g. Calculate the density of the block.

3

3

/858.0

3.25

7.21

cmgD

cm

gD

volume

massDensity

A student fills a graduated cylinder to 25.0 mL with liquid. She then immerses a solid in the liquid. The volume of the liquid rises to 33.9 mL. The mass of the solid is 63.5 g. What is the density?

mLgDmL

gD

mLmL

gD

volume

massDensity

/79

5.63

)0.259.33(

5.63

Isopropyl alcohol has a density of 0.785 g/mL. What volume should be measured to obtain 20.0 g of the liquid?

mLvolume

mLg

gvolume

Density

massvolume

volume

massDensity

5.25

/785.0

0.20

A beaker contains 725 mL of water. The density of water is 1.00 g/mL. Find the mass of the water.

gmass

mLgmLmass

Densityvolumemassvolume

massDensity

725

/00.1725

In certain situations, the term specific gravity is used to describe the density of a liquid.

Specific gravity is defined as the ratio of the density of a given liquid to the density of water at 4 0C.

Because it is a ratio of densities, specific gravity has no units.