3.1

Chapter 3: Scientific Measurement

A measurement is a quantity that has both a number and a unit.

Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct.

Scientific NotationHow important is a change in the power of 10?Diameter of Earth’s orbit around the sun

≈ 100,000,000,000 m = 1.0*1011 m

Diameter of an atom

≈ 0.0000000001 = 1.0*10-10 m

1. Move the decimal point in the original number so that it is located to the right of the first nonzero digit.

2. Multiply the new number by 10 raised to the proper power that is equal to the number of places the decimal moved.

3. If the decimal point moves: To the left, the power of 10 is positive. To the right, the power of 10 is negative.

Writing in scientific notation

3.1

Accuracy, Precision, and Error

Accuracy is a measure of how close a measurement comes to the actual, true, or accepted value of whatever is measured.

Precision is a measure of how close a series of measurements are to one another.

• How closely individual measurements compare with each other• The “fineness” of a measurementTo evaluate the accuracy of a measurement, the measured value must be

compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Accurate or Precise?

Precise!(but not accurate)

What is the temperature at which water boils?

•Measurements: 95.0°C, 95.1°C, 95.3°C

•True value: 100°C

Accurate or Precise?

Accurate!(it’s hard to be accurate without being precise)

What is the temperature at which water freezes?

•Measurements: 1.0°C, 1.2°C, -5.0°C

•True value: 0.0°C

Accurate or Precise?

Not Accurate & Not Precise(don’t quit your day job)

What is the atmospheric pressure at sea level?

•Measurements: 10.01 atm, 0.25 atm, 234.5 atm

•True value: 1.00 atm

Accurate or Precise?

Accurate & Precise(it’s time to go pro)

What is the mass of one Liter of water?

•Measurements: 1.000 kg, 0.999 kg, 1.002 kg

•True value: 1.000 kg

A graduated cylinder:

50

100 mL Beaker

50 mL Graduated cylinder

A beaker:

41.0

41.2 mL (3 sig figs = very precise)

40. mL (2 sig figs = not as precise)

Accuracy or Precision?

When deciding on accuracy, precision, both, or neither….it is quantitative data (numerical), not qualitative (descriptive)

1) The recipe calls for 25 chocolate chips per cookie. The cookies analyzed have 34, 35, and 32 chips respectively.

2) The percent NaCl is 99%, 99%, and 98%.3) The number of grams of KF required is 0.04 g. The amounts

used were 0.038, 0.039, 0.041, and 0.040 g.4) To win, Henry must earn 500 points. In his three trials, he

earned 400, 480, and 395 points.

3.1

Accuracy, Precision, and ErrorDetermining ErrorThe accepted value is the correct value based on reliable

references. The experimental value is the value measured in the lab. The difference between the experimental value and the

accepted value is called the error.

The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

Accuracy, Precision, and Error3.1

Ex. Find the percent error if your data shows 1.23 g Mg and the true answer is 1.40 g Mg.

%1.1210040.1

40.123.1=•

−

3.1

Accuracy, Precision, and Error

Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.

Significant FiguresEvery measurement has some degree of uncertainty.Significant figures (“sig figs”): the digits in a

measurement that are reliable (or precise). The greater the number of sig figs, the more precise that measurement is.

A more precise instrument will give more sig. figs. in its measurements.The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

Ex. Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures.

Uncertainty examples: To measure the time for a pencil to fall…

compare a stopwatch and a wall clock. To measure the volume of a liquid…compare

a graduated cylinder and a beaker.

The stopwatch & graduated cylinder are more precise instruments…so the readings

they produce will have more sig figs.

Significant Figures in Measurements3.1

a. 3

c. 5

b. 5

d. unlimited

e. 4

f. 2

Significant Figures in Calculations

In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.

To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

3.1

“PACIFIC”Decimal point is PRESENT. Count digits from left side, starting with the first nonzero digit.

40603.23 ft2

0.01586 mL= 7 sig figs= 4 sig figs

PACIFIC

PACIFIC

The “Atlantic-Pacific” Rule

When are digits “significant”?“ATLANTIC”

Decimal point is ABSENT. Count digits from right

side, starting with the first

nonzero digit.

40600 ft2

1000 mL3 sig figs =1 sig fig =

ATLANTIC

ATLANTIC

• 0.00932Decimal point present → “Pacific” → count digits from

left, starting with first nonzero digit= 3 sig figs

• 4035Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit= 4 sig figs

• 27510Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit= 4 sig figs

Examples

Write the following measurements in scientific notation, then record the number of sig figs.

1. 789 g2. 96,875 mL3. 0.0000133 J4. 8.915 atm5. 0.94°C

7.89*102 g9.6875*104 mL1.33*10-5 J

8.915 atm or 8.915 x 100

9.4*10-1 °C

3 sig figs5 sig figs3 sig figs4 sig figs2 sig figs

Practice problems

Sig. Figs. In Calculations And Scientific Notation

• In this class, use scientific notation for all numbers greater than 1000 and smaller than 0.001

• In this class, we will not follow the sig. fig. rules for operations (see text if interested)• In this class, we will carry all digits in our calculations and then round the final answer to 3 significant digitsException…in very large or very small numbers, you may use 4 digits in intermediate steps (i.e. 0.0034567432 can be rounded to 0.003457)

3.1 Section Quiz

1. In which of the following expressions is the number on the left NOT equal to the number on the right?

a) 0.00456 10–8 = 4.56 10–11

b) 454 10–8 = 4.54 10–6

c) 842.6 104 = 8.426 106

d) 0.00452 106 = 4.52 109

3.1 Section Quiz

2. Which set of measurements of a 2.00-g standard is the most precise?

a) 2.00 g, 2.01 g, 1.98 gb) 2.10 g, 2.00 g, 2.20 gc) 2.02 g, 2.03 g, 2.04 gd) 1.50 g, 2.00 g, 2.50 g

3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement?

a) 2b) 3c) 4d) 5

3.1 Section Quiz

3.23.2 The International System of Units

In the signs shown here, the distances are listed as numbers with no units attached. Without the units, it is impossible to communicate the measurement to others. When you make a measurement, you must assign the correct units to the numerical value.

3.2

Measuring with SI Units

All measurements depend on units that serve as reference standards. The standards of measurement used in science are those of the metric system. The International System of Units (abbreviated SI, after the French name, Le Système International d’Unités) is a revised version of the metric system.

3.2

Measuring with SI Units

The five SI base units commonly used by chemists are the meter, the kilogram, the kelvin, the second, and the mole.

3.2

Units and QuantitiesUnits of Length

In SI, the basic unit of length, or linear measure, is the meter (m). For very large or and very small lengths, it may be more convenient to use a unit of length that has a prefix.

History of SI• The metric system or Système International d'Unités (S.I.),

was first organized in Paris as part of the French Revolution and adopted by France in 1795. At that time, the meter and kilogram were standardized.

• Why was it organized? It is simple, being based on powers of 10. (like our number system)

• Every country in the world uses the metric system except the USA, Myanmar, and Liberia.

• By 2009, all products sold in Europe must use the metric system. No dual-labelling will be permitted.

• Visit U.S. Metric Association (USMA), Inc. for more info.

3.2

Units and Quantities

Common metric units of length include the centimeter, meter, and kilometer.

3.2 Units and QuantitiesUnits of Volume

• The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge. This volume is the cubic meter (m)3. A more convenient unit of volume for everyday use is the liter, a non-SI unit.

• A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm 10 cm 10 cm = 1000 cm3 = 1 L).

• Common metric units of volume include the liter, milliliter, cubic centimeter, and microliter.

3.2

Units and QuantitiesThe volume of 20 drops of liquid from a medicine dropper is approximately 1 mL.

A sugar cube has a volume of 1 cm3. 1 mL is the same as 1 cm3.

3.2

Units and QuantitiesUnits of Mass

• The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass.

• A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g.

• Common metric units of mass include kilogram, gram, milligram, and microgram.

3.2

Units and Quantities

Weight is a force that measures the pull on a given mass by gravity.

The astronaut shown on the surface of the moon weighs one sixth of what he weighs on Earth.

3.2

Units and QuantitiesUnits of Temperature

• Temperature is a measure of how hot or cold an object is.

• Thermometers are used to measure temperature.

3.2

Units and Quantities

• Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin.

• On the Celsius scale, the freezing point of water is 0°C and the boiling point is 100°C.

• On the Kelvin scale, the freezing point of water is 273.15 kelvins (K), and the boiling point is 373.15 K.

• The zero point on the Kelvin scale, 0 K, or absolute zero, is equal to 273.15 °C.

°F = (1.8 * °C) + 32°C = (°F-32) 1.8

3.2

Units and Quantities

Units of EnergyEnergy is the capacity to do work or to produce heat.• The joule (J) is the SI unit of energy. • One calorie (cal) is the quantity of heat that raises the

temperature of 1 g of pure water by 1°C.• One Calorie (Cal) equals 1000 calories

3.2 Section Quiz.

1. Which of the following is not a base SI unit?a) meterb) gramc) secondd) mole

3.2 Section Quiz.

2. If you measured both the mass and weight of an object on Earth and on the moon, you would find that

a) both the mass and the weight do not change.

b) both the mass and the weight change.

c) the mass remains the same, but the weight changes.

d) the mass changes, but the weight remains the same.

3.2 Section Quiz.3. A temperature of 30 degrees Celsius is equivalent

toa) 303 K.b) 300 K.c) 243 K.d) 247 K.

3.33 3.3 Conversion Problems

Because each country’s currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problem-solving approach called dimensional analysis.

Practice Problems before moving on

1) Round to 3 sig. fig.A) 21.8704B) 0.0021504

C) 6,299,7432) Write answers to #1 in scientific notation3) Convert 8 km to inches.

Use railroad tracks, write in scientific notation, round to 3 digits!

3.3

Conversion Factors

A conversion factor is a ratio of equivalent measurements.

The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors.

3.3 Conversion FactorsWhen a measurement is multiplied by a conversion factor,

the numerical value is generally changed, but the actual size of the quantity measured remains the same.

The scale of the micrograph is in nanometers. Using the relationship 109 nm = 1 m, you can write the following conversion factors.

3.3

Dimensional Analysis

Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements.Problems in which a measurement with one unit is converted to an equivalent measurement with another unit are easily solved using dimensional analysis.

Otherwise known as factor label method

Dimensional AnalysisUse “railroad tracks”Show unitsBox answer or write answer in statement unless

part of a larger problemYOU MUST DO IT MY WAY AND SHOW WORK IF

YOU WANT CREDIT Ex. Convert 140,500,000 mL to kL140,500,000 mL L kL = 140.5 kL (141 kL) 1000 mL 1000L

Practice problems• 1) Convert 55°F to °C and K

• 2) How many seconds in a year?

• 3) Convert 1.23 km to m and then to um

• 4) Convert 42.8 kg to oz.

.sec10x15.3.min1.sec60

hour1.min60

day1hours24

year1days365

1year1 7=

m10x23.1m1

m10km

m10001km23.1 9

6

μ=μ

K286C138.128.132F

C =°==−°=°

.oz10x51.156.1506.lb_1.oz16

kg1.lb20.2

1kg8.42 3==

1. 1 Mg = 1000 kg. Which of the following would be a correct conversion factor for this relationship?

a) 1000.b) 1/1000.c) ÷ 1000.d)1000 kg/1Mg.

3.3 Section Quiz

3.3 Section Quiz

2. The conversion factor used to convert joules to calories changes

a) the quantity of energy measured but not the numerical value of the measurement.

b) neither the numerical value of the measurement nor the quantity of energy measured.

c) the numerical value of the measurement but not the quantity of energy measured.

d) both the numerical value of the measurement and the quantity of energy measured.

3.3 Section Quiz

3. How many μ g are in 0.0134 g?a) 1.34 10–4

b) 1.34 10–6

c) 1.34 106

d) 1.34 104

3.3 Section Quiz

4. Express the density 5.6 g/cm3 in kg/m3.a) 5.6 106kg/m3

b) 5.6 103kg/m3

c) 0.56 kg/m3

d) 0.0056 kg/m3

3.4 Density

If you think that these lily pads float because they are lightweight, you are only partially correct. The ratio of the mass of an object to its volume can be used to determine whether an object floats or sinks in water.

3.4

Determining DensityDensity is the ratio of the mass of an object to

its volume.

Density is an intensive property that depends only on the composition of a substance, not on the size of the sample.

3.4

Determining DensityThe density of corn oil is less than the density of corn syrup. For that reason, the oil floats on top of the syrup.

3.4

Each of these 10-g samples has a different volume because the densities vary.

Experiments show that the volume of most substances increases as the temperature increases. Meanwhile, the mass remains the same. Thus, the density must change. The density of a substance generally decreases as its temperature increases.

What is density?• The density of water is 1.0

g/mL (At 4.0°C).• The relative density of any substance

can be determined by adding it to water.

• If the substance floats, it’s density is less than the density of water:d < 1.0 g/mL

• If the substance sinks, it’s density is greater than the density of water:d > 1.0 g/mL

1.0 g/mL

0.80 g/mL

1.2 g/mL

Practice problems

• What is the density of water?• What is the density of ice?• Calculate the density of a liquid if 40.0 mL of

the liquid has a mass of 90.2 g. Is the liquid more or less dense than water?

• If the liquid were added to a beaker of water, what would happen?

1.0 g/mL at 4°C

0.917 g/mL

2.26 g/mL; more

Don’t know—could sink to the bottom or it could mixwith the water (polar vs. nonpolar solutions)

3.4 Section Quiz

1. If 50.0 mL of corn syrup have a mass of 68.7 g, the density of the corn syrup is

a) 0.737 g/mL.b) 0.727 g/mL.c) 1.36 g/mL.d) 1.37 g/mL.

3.4 Section Quiz

2. What is the volume of a pure gold coin that has a mass of 38.6 g? The density of gold is 19.3 g/cm3.

a) 0.500 cm3

b) 2.00 cm3

c) 38.6 cm3

d) 745 cm3

3.4 Section Quiz

3. As the temperature increases, the density of most substances

a) increases.b) decreases.c) remains the same.d) increases at first and then decreases.

What you need to know: Chapter 33.1scientific notationprecision vs. accuracy% errorsignificant figures (identifying)rounding (3 sig. fig.)3.2SI base units (meter, kilogram, kelvin,

second, mole)metric prefixes—know values (kilo,

centi, milli)metric prefixes—recognize relative size

(giga, mega, deci, micro, nano, pico)temperature (K, °C, °F); absolute zero

3.3 conversion factordimensional analysisrailroad tracks3.4density—definitiondensity—water

Last modified 9-07

Practice Problems

• Write problems in your notebook. Head the top of the page with “Ch. 3 HW” followed by the page and numbers assigned. Write in complete sentences/answers for credit.

• pp. 96-97• (57-59, 63-64, 66, 68-72, 74-75, 78, 79, 82, 85,

87)

The Metric Prefixes

Prefix Symbol Value Power Use

mega M 1,000,000 106 megaton

kilo k 1,000 103 kilometer

deci d 0.1 10-1 decimate

centi c 0.01 10-2 centipede

milli m 0.001 10-3 millimeter

micro μ 0.000001 10-6 microscope

nano n 0.000000001 10-9 nanotechnology

giga G 1,000,000,000 109 gigabyte

Extra Slides….

The Standard Units

Quantity Unit name Abbreviation

length meter m

mass kilogram kg

temperature kelvin K

time second s (or sec)

amount mole mol

What is a meter?

• *In the 1790’s, a meter was originally defined as “1/10,000,000th of the distance from the Equator to the North Pole measured along the meridian that runs through Dunkirk, France.”

• *It is currently defined as the “distance through which a beam of light travels in a vacuum in 1/299,792,458 of a sec.”

• (It’s also exactly 39.37 inches.)

What is a Liter?

• *The S.I. unit for volume is actually the cubic meter (m3) but that’s pretty big.

• The Liter is defined as a cube measuring 1 decimeter on each side, or 1 dm3, or 1000 cm3.

• *How many Liters are there in 1 m3?• *So, the Liter is based on the meter,

which is based on the Earth.

1 dm

1 dm

1dm

*What is a kilogram?

• The S.I. unit for mass is the kilogram, and is defined as the mass of 1 dm3 of water at 4°C.

• Why water?• So, the kilogram is based on the Liter,

which is really based on the meter, which is really based on the Earth.

1 dm

1 dm

1dm

*What is a second?• The S.I. unit for time is the second. It

was originally defined as 1/86,400th of the average solar day.

• It is presently defined with some high level chemistry/physics concepts. (1 second is the time that elapses during 9.192631770 x 109 cycles of the radiation produced by the transition between two levels of the Cs-133 atom.)

• So, the second is based on the time it takes the Earth to rotate.

What is a kelvin?• *The S.I. unit for temperature is the kelvin, and is

defined as 1/100 of the temperature difference between the boiling point & freezing point of water at one atmosphere of pressure.

• The kelvin (K) and the degree Celsius (°C) are exactly the same size, although 1 degree Fahrenheit (°F) is equal to about 1.8°C.

• To convert:

• K = °C + 273.15• °F = (1.8 * °C) + 32• °C = (°F-32) 1.8

• *So, the kelvin is based on water and absolute zero (the coldest temperature possible.)