Chemistry 068, Chapter 2

Importance of Measurement

• Measurement and calculation of quantities is fundamental in all of the sciences.

• Scientists use the ideas of accuracy and precision, significant figures, and scientific notation in all calculations and measurements.

Exact and Inexact Numbers

• Scientists group numbers into two categories – exact and inexact.– Which category a number falls in affects how it is

used in calculations.• Exact numbers have no uncertainty in their

value.– Exact numbers include counted numbers (12 apples,

3 books), unit conversions (12 inches in a foot), and defined values.

• Most numbers are inexact, they have a degree of uncertainty in their value.– This includes almost every measured value.– For example, 5.5 inches, 12.034 lbs.

Exact and Inexact Number Problems

• Which of the following are exact numbers?– 3 feet in 1 yard.

– 12.36 inches.

– 12 inches.

– 3 cars.

Accuracy, Precision, and Error

• Real measurements always include some degree of inaccuracy and uncertainty.

• Precision is the measure of how close together multiple measurements of the same quantity are.

• Accuracy is the measure of how closely measurements agree to the true or correct value.

• Ideally, you want to have both high accuracy and high precision.

Accuracy vs. Precision

• It is possible to be precise but not accurate or to be accurate but not precise.

• For example, consider the following cases for a measurement, where the true value is 2.0

• Case A, high precision and high accuracy.– Values 1.9, 2.0, 2.0, 2.0, 2.1; average 2.0– This is the ideal case.

• Case B, high precision, low accuracy.– Values 1.5, 1.6, 1.6, 1.6, 1.7; average 1.6

• Case C, low precision, high accuracy.– Values 1.0, 1.5, 2.0, 2.5, 3.0; average 2.0

• Case D, low precision and low accuracy.– Values 0.1, 0.6, 1.1, 1.6, 2.1; average 1.1– This is the worst case scenario.

Accuracy and Precision Problems

• Which of the following data sets has the highest/lowest precision?

• Which has the highest/lowest accuracy if the true value is 1.0?

• Set 1: 0.5, 1.0, 1.5

• Set 2: 1.9, 2.0, 2.1

Types of Error

• Errors are classified as either random or systematic.– Random error is due to unknown or

uncontrollable variables.– Systematic error is a constant error due to a

controllable cause. It is usually due to either human error (limitations of the person) or instrument error (limitations of the equipment or technique).

Uncertainty in Measurements• Since there is always some error in any measurement, there will

always be some uncertainty.• Measured quantities are generally reported in such a way that only

the last digit is uncertain.– You “guess” or estimate one digit beyond the number you are certain of.– That last digit is the uncertain one.

• For example, a ruler with 1/10 inch divisions, you would estimate a value between 1.1 and 1.2 inches as 1.15 or 1.17, or any value between 1.1 and 1.2.– If it was exactly on 1.3, you would record the value as 1.30.

• Thus, the more digits/divisions in a measurement the better – there will be less uncertainty.– A ruler with 1/100 inch divisions is better than one with only 1/10 inch

divisions.• The uncertainty can be written as a ± value.

– For example, 3.1 ± 0.1; 47.89 ± 0.03; etc.– The ± is usually left off if it is ± a value of 1.

Significant Figures

• Measured quantities are generally reported in such a way that only the last digit is uncertain.– The number of significant figures is the number of

certain digits plus the one unknown digit.

• Significant figure rules tell you how to round uncertain quantities when they are added, subtracted, multiplied, or divided.

• You will need to know the number of significant figures in each quantity.

Determining # of Significant Figures

• How to find the number of significant figures – which numbers count and which don’t.– Pay close attention to where the decimal point is.– Always count non-zero digits.– Ignore zeroes before a non-zero digit.– Only count zeroes after a non-zero digit only if the

number contains a decimal point OR if there is a bar over the zeros.

• Find the number of significant digits for each number used in a calculation.

Determining # of Significant Figures Problems

• How many significant digits are there in each of the following values?

• 12340

• 12340.

• 12340.0

• 0.003089

• 1.230

Significant Figures and Math Operations

• When performing mathematical operations, the result of your calculation will have a specific number of significant figures (how many depends on the math operation and the numbers involved).

• Your answer will have to be rounded off to the correct number of significant figures.

• As a good rule of thumb, when doing a multiple step calculation, round your final answer rather than the result from each step.

Rounding Off

• When you round off you drop extra digits beyond the allowed number of significant figures. You also change that last digit depending on the value of the first digit you drop.– You add 1 if it is higher than 5 or leave it alone if it is

less than 5.– If it is 5 followed by non zero values you add 1.– If it is 5 followed by all zeros (or no values) you add 1

if it is odd or add none if it is even.

Rounding Off Problems

• Round each of the following to three significant figures.

• 1.0340

• 0.06955

• 27559

Multiplication and Division

• The rules for multiplication and division are the same – your final answer will have the smallest number of significant figures of either quantity.

• You then round as usual.

Multiplication and Division Problems

• Calculate each of the following:

• 1.23 x 97.854 =

• 0.00910 / 9.8 =

Addition and Subtraction

• Addition and subtraction work differently.

• You only care about the number of digits after the decimal place.

• You’ll keep the smallest number or digits after the decimal point from among the quantities.

• You then round normally.

Addition and Subtraction Problems

• Calculate each of the following:

• 1.00 + 1.007 + 1.1 =

• 23.678 – 19.3 =

Calculations Using Exact Numbers

• Exact numbers do not count towards significant figure rules.

• They are considered to have an unlimited number of significant digits when performing calculations.

Calculations Using Exact Numbers Problems

• Calculate the number of inches in 3.58 feet.

Scientific Notation

• Scientific notation is the practice of expressing quantities as a number multiplied by a power of 10.

• Scientific notation is used because it is very easy to make mistakes when writing very small and very large quantities.– For example, 0.000000009 or 3000000000

Exponents

• Exponents are numbers with a superscript which indicates the power it is taken to – the number of times it is multiplied by itself.– For example, 34 = 3 x 3 x 3 x 3 = 81

• Note that exponents can have negative superscripts. In this case the value is one divided by the number to the power.– For example, 3-4 = 1 /(3 x 3 x 3 x 3) = 1/81

• The exponent of any number other than zero to the zero power is one.– For example, (-3)0 = 1

Converting Between Decimals and Scientific Notation

• Scientific notation uses exponents of 10.

• Decimal value are multiplied by exponents of ten to write all values as a single digit before the decimal place times an exponent of 10.– Only the significant digits are kept, all other

digits are dropped.

Converting Between Decimals and Scientific Notation Problems

• Write each of the following in scientific notation.

• 127.890

• 3.06

• 0.00369700

Converting Between Decimals and Scientific Notation Problems

(Cont’d)• Write each of the following in decimal

notation.

• 1.036 x 10-2

• 9.32 x 106

• 7.56 x 100

Uncertainty and Scientific Notation

• To determine the uncertainty of measurements written in scientific notation you multiply the uncertainty of the digits times the exponent value.

Uncertainty and Scientific Notation Problems

• Determine the uncertainty in each of the following measurements.

• 1.036 x 10-2

• 9.32 x 106

• 7.56 x 100

Multiplication and Division in Scientific Notation

• For both multiplication and division you perform the operator (multiplication or division) on the non exponent parts first.

• The exponent part works differently. Rather than multiplying or dividing, you add (multiplication) or subtract (division) the values of the superscripts of the exponents.– This can only be done if the exponents have the same

base (10 in the case of scientific notation).

• Adjust your answer, if needed, for significant figures.

Multiplication and Division Problems

• Calculate each of the following:

• (1.10 x 103) x (9.00 x 102)

• (3.67 x 109) / (1.00 x 102)

Addition and Subtraction in Scientific Notation

• In order to add or subtract in scientific notation, both values must have the same power of 10 (superscript).– This may mean you have to convert them to be the

same power.• Once they are in the same power you can add or

subtract the non exponent part as normal.• The exponent will be the same.• If necessary, you can convert your answer to a

different power of ten and then adjust your answer, if needed, for significant figures.

Addition and Subtraction Problems

• Calculate each of the following:

• (2.20 x 103) + (9.60 x 102)

• (7.7 x 102) – (1.00 x 101)

Unit Systems, Conversions, and Dimensional Analysis

• Scientists use the metric system, rather than the “English” system, to make measurements.

• It is thus necessary to be able to convert between metric units as well as to be able to convert between the “English” and metric unit systems.

• This conversion is called dimensional analysis.

The Metric System and SI Units

• The metric system was originally developed in 1791.

• It was revised in 1960, with units being redefined, leading to SI units.

• Metric units consist of two parts a prefix which indicated the power of 10, and a suffix which indicated the type of unit (mass, length, etc.)

Metric Prefixes

Prefix Symbol Meaning

Giga G 1x109

Mega M 1x106

Kilo k 1x103

Deci d 1x10-1

Centi c 1x10-2

Milli m 1x10-3

Micro μ 1x10-6

Nano n 1x10-9

Pico p 1x10-12

Femto f 1x10-15

Metric/SI Units of Measurement

Quantity Metric Unit Symbol SI Unit Symbol

Mass Gram g Kilogram kg

Length Meter m Meter m

Time Second s Second s

Temperature Celsius oC Kelvin K

Amount Mole mol Mole mol

Electric Current Ampere A Ampere A

Luminous Intensity Candela cd Candela cd

Density gram per mL g/mL gram per cc g/cm3

Volume Liter l Cubic Meter m3

Comparison of Metric and English Units

• A meter is a little bit longer than a yard – it is about 39 1/3 inches.

• A kilogram is about 2.2 pounds.

• A liter (a cubic decimeter) is just a little bit larger than a quart.

Units and Math Operators

• Units can have exponents just like numbers.– For example, ft2, cm3, etc.

• When you are working problems that involve the multiplication of units you have to multiply both the unit and the values.– For example, 4ft x 2ft = 8ft2

Dimensional Analysis

• A fancy name for unit conversion calculations.

• Chain rule – carry out conversions by cancelling out units as you go.

• Always make sure your final units are correct and that your final answer makes sense.

Metric to Metric Conversion

• Metric to metric conversions consist of changing the power of 10 as you go from one prefix to another.

Metric to Metric Conversion Problems

• Convert 100.0g to kg.

• Convert 25.0km to m.

• Convert 12.3mL to kL.

English to English Conversion

• English to English unit conversion are a little more complicated as the differences between units are not simple factors of 10.

English Unit Conversion Values

• The following are conversion factors for some common units.

• Length– 12 inches = 1 foot– 36 inches = 1 yard– 3 feet = 1 yard– 5280 feet = 1 mile

• Mass– 16 ounces = 1 pound– 2000 pounds = 1 ton

• Volume– 32 fluid ounces = 1 quart– 2 pints = 1 quart– 4 quarts = 1 gallon

English to English Conversion Problems

• Convert 30.0 inches to feet.

• Convert 6.2 pounds to ounces.

• Convert 27 inches to yards.

Metric to English and English to Metric Conversion

• These work in a similar way but involve a different set of conversion factors.

English/Metric Unit Conversion Values

• The following are conversion factors for some common units.

• Length– 2.540cm = 1 inch– 1.609km = 1 mile– 39.37 inch = 1m– 1.094 yard = 1m

• Mass– 453.6g = 1 pound– 2.205pound = 1kg

• Volume– 0.9463L = 1 quart– 3.785L = 1 gallon

Metric to English and English to Metric Conversion Problems

• Convert 3 gallons to L.

• Convert 10. pounds to kg.

• Convert 33.0cm to feet.

Derived Unit Conversions

• For conversions involving a squared or cubed unit you also need to square or cube the conversion factor.

• These calculations are easiest to handle using the chain rule.

• Area• Special note on volume.

– Volume can use several different units in the metric system.

– 1mL = 1cm3 = 1 cc– When working with liters you do not need to square or

cube your results.

Derived Unit Conversion Problems

• Convert 15cm3 to m3.

• Convert 25m2 to yards2.

• Convert 10.0 inches2 to m2.

Multiple Unit Conversion

• Sometime you need to convert unusual units.– For example kg/hour to pounds/day.

• This works similarly to derived unit calculations.– Again, follow the chain rule.

Multiple Unit Conversion Problems

• Convert 27.6mL/hour to L/week.

• Convert 250.0 pounds/inch2 to kg/m2.

• Convert 65 miles/hour to km/minute.

Density

• Density is the ratio of mass to volume.• Density can be calculated by dividing mass by a

volume.

d = mass/volume• A higher density means that an object has

greater mass for the same unit of volume.• Densities are usually given as g/mL or g/cm3;

except for gasses, which are usually g/L (as they have such low densities).

Calculation of Density Problems

• A 20.0g sample of an unknown liquid has a volume of 1.5mL. What is the unknown’s density?

• A cube of metal 2.0cm on a side has a mass of 10.0g. What is its density?

Sample Density Values

Substance Density in g/cm3

Gold 19.3Lead 11.3Aluminum 2.70Table Salt 2.16Wood (Varies) 0.30 to 0.50Water 0.997Air 1.29*10-3

Methane 6.6*10-4

Density and Floating

• An object will “float” in another substance if it has a lower denisty.

• For example, wood (density 0.30 to 0.50 g/cm3) will float on water (0.997g/cm3); but lead (density 11.3g/cm3) will not.

Using Density in Dimensional Analysis

• Densities units can be converted just like any other multiple unit conversions.

• Additionally, you can use a density value to go from mass to volume and from volume to mass.

Density Dimensional Analysis Problems

• Convert 10.0g/mL to kg/L.

• A substance has a density of 2.6g/mL.– What is the volume of a 26.0g sample?– What is the mass of a 5.2L sample?

Other Equivalence Conversions

• Other equivalence conversions exist in addition to density and are used in a similar way.

• The conversion factor can be used to freely switch between the two units that make up the factor.

• The most commonly used are concentration, dosage relationships, rate relationships, and cost relationships.

• Many others exist and work the same way.

Other Equivalence Conversions (Cont’d)

• Concentrations are an amount of substance (usually mass) per unit volume of a solution.– For example, mg salt/mL solution.

• Dosage relationships work the same way but with mass over mass units.– For example, mg Tylenol/kg body mass.

• Rate relations are a quantity over time.– For example miles/hour.

• Cost relations work the same way.– For example dollars/hour, dollars/ounce gold.

Other Equivalence Conversion Problems

• How many grams of salt must be dissolved into 5.0L of water if the solution is to have a concentration of 1.5g/L?

• How many grams of antibiotic should be administered to a patient weighing 100.0kg if the safe dosage is 150mg antibiotic/kg patient?

Other Equivalence Conversion Problems (Cont’d)

• How long will it take you to drive 300miles if you are traveling an average of 75miles/hour?

• If you make 15dollars/hour how long will it take you to earn $1000.00?

Percentage and Percent Error

• The concept of percentage (%) is fairly well known.

• It is often useful to use % as a way to express fractional quantities.

• When working % problems always remember that the total of all % must add up to 100%.

• Also, when doing conversions with % it is often better to use the % as a fraction and then convert back if needed.

Percentage and Percent Error (Cont’d)

• %Error is a way to describe the amount of deviation from an accepted value. It is often used as a guage of experimental accuracy.

• % Error is defined as:100*(Measured Value – Accepted Value)/Accepted Value

• Note that % error can be positive or negative. A negative % indicates that the measured value is lower than the accepted value.

Percentage and Percent Error Problems

• A 200.0mL sample of dry air is found to be 20.9% oxygen by volume. What volume of oxygen is there in the sample?

• A population of 450 fruit flies is found to be 55% male. 45% of the flies are found to have red eyes, 25% have green eyes, and 30% have blue eyes. How many female fruit flies have red eyes in this population?

Percentage and Percent Error Problems (Cont’d)

• A scientist determines that a 52.0g sample consists of three chemicals A, B, and C. If it is 25% A and 25% B (both by mass), what mass of C is in the sample?– If the accepted value for the mass of C is

30.0g in the sample, what is the % error in the experiment?

Temperature Scales

• There are two temperature scales are commonly used for everyday measurements.

• Fahrenheit, oF, used in the English system and Celsius, oC, used in the metric system.

• Scientists often use a third temperature scale called Kelvin, K, which is now the SI unit for temperature.

Comparisons Between the Three Temperature Scales.

Scale oF oC K

Water Boils 212 100 373.15

Body Temp. 98.6 37 310.15

Water Freezes 32 0 273.15

Absolute Zero -459.7 -273.15 0

Temperature Scale Conversions

• To convert to/from Celsius and Fahrenheit:oC = 5/9(oF – 32)oF = 9/5(oC) + 32

• To convert to/from Celsius and Kelvin:oC = K – 273.15

K = oC + 273.15K

Significant Figure Conventions for Temperature Readings

• For temperature measurements, by convention it is assumed that, unless otherwise noted, that all digits after a nonzero digit are significant figures.– Even if they are trailing zeros without a

decimal point.– For example 10oF is assumed to have 2

significant figures even though it does not contain a decimal point.

Temperature Scale Conversion Problems

• Convert each of the following.

• 100oC to oF.

• 27.5oF to K.

• -250oC to K.