The Numerical Side of Chemistry

Chapter 2

Types of measurement

• Quantitative- use numbers to describe– 4 feet– 100 ْF

• Qualitative- use description without numbers

– extra large– Hot

Scientists prefer

• Quantitative- easy check

• Easy to agree upon, no personal bias

• The measuring instrument limits how good the measurement is

How good are the measurements?

• Scientists use two word to describe how good the measurements are

• Accuracy- how close the measurement is to the actual value

• Precision- how well can the measurement be repeated

Differences

• Accuracy can be true of an individual measurement or the average of several

• Precision requires several measurements before anything can be said about it

• examples

Let’s use a golf analogy

Accurate?

Precise?

Accurate?

Precise?

Accurate?

Precise?

In terms of measurement

• Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across.

• Were they precise?

• Were they accurate?

Summary of Precision Vs. Accuracy

• Precision– Grouping of

measurements– Need to have

several measurements• Repeatability• Can have precision

without accuracy

• Accuracy– How close to true

value– Can use one

measurement or many• Can have accuracy

without precision

Significant Figures

Significant figures (sig figs)

• How many numbers mean anything

• When we measure something, we can (and do) always estimate between the smallest marks.

21 3 4 5

Significant figures (sig figs)• The better marks, the better we can

estimate.

• Scientist always understand that the last number measured is actually an estimate

21 3 4 5

Sig Figs• What is the smallest mark on the ruler that

measures 142.15 cm?

– 142 cm?– 140 cm?

• Here there’s a problem does the zero count or not?

• They needed a set of rules to decide which zeroes count.

• All other numbers count

Which zeros count?• Those at the end of a number before the decimal

point don’t count

– 1000– 1000000000– 12400

• If the number is smaller than one, zeroes before the first number don’t count

– 0.045 – 0.123– 0.00006

Which zeros count?• Zeros between other sig figs COUNT.

– 1002– 1000000003

• zeroes at the end of a number after the decimal point COUNT

– 45.8300– 56.230000

• If they are holding places, they don’t.• If they are measured (or estimated) they do

Sig Figs• Only measurements have sig figs.

• Counted numbers are exact

• A dozen is exactly 12

• A a piece of paper is measured 11 inches tall.

• Being able to locate, and count significant figures is an important skill.

• YOU MUST KNOW ALL THE SIG FIG RULES !!!!

Summary of Significant FiguresA number is not significant if it is:

• A zero at the beginning of a decimal numberex. 0.0004lb, 0.075m

• A zero used as a placeholder in a number without a decimal pointex. 992,000,or 450,000,000

A number is a S.F. if it is:

• Any real number ( 1 thru 9)• A zero between nonzero

digitsex. 2002g or 1.809g

• A zero at the end of a number or decimal point

ex. 602.00ml or 0.0400g

Learning Check 2• How many sig figs in the following

measurements?

• 458 g_____• 4085 g_____• 4850 g______• 0.0485 g_____• 0.004085 g_____• 40.004085 g______

Learning Check 2 Con’t

• 405.0 g______• 4050 g_______• 0.450 g_______• 4050.05 g______• 0.0500060 g______

Scientific Notation

Problems• 50 is only 1 significant figure

• if it really has two, how can I write it?

• A zero at the end only counts after the decimal place

• Scientific notation

• 5.0 x 101

• now the zero counts.

Purposes of Scientific Notation• Express very small and very large

numbers in a compact notation.

– 2.0 x 108 instead of 200,000,000 – 3.5 x 10-7 instead of 0.00000035

• Express numbers in a notation that also indicates the precision of the number.

– What is meant if two cities are said to be separated by a “distance of 3,000 miles”?

What Do We Mean by 3,000 miles?

• A distance between 2,999 and 3,001 miles?

• A distance between 2,990 and 3,010 miles?

• A distance between 2,900 and 3,100 miles?

• A distance between 2,000 and 4,000 miles?

What Do We Mean by 3,000 miles?

• Without a context, we don’t know what is meant. In each case above, the colored digit is the largest one that is uncertain. As you ascend from bottom to top, the uncertainty decreases and the numbers become increasingly precise.

• Scientific notation will allow us to express these quantities (all are “three thousand”) with the precision or uncertainty being explicit.

First Things First…• Power-of-ten exponential notation is

central to scientific notation.

• To start, you should review powers of ten and make sure that you understand the exponential notation and can covert it to standard notation.

How to Handle Significant Figs and Scientific Notation

When Doing Math

Adding and subtracting with sig figs

• The last sig fig in a measurement is an estimate.

• Your answer when you add or subtract can not be better than your worst estimate.

• You have to round it to the least place of the measurement in the problem

For example

27.93 6.4+ First line up the decimal places

27.936.4+

Then do the adding

34.33 Find the estimated numbers in the problem.

27.936.4

This answer must be rounded to the tenths place

What About Rounding? • look at the number behind the one you’re rounding.

– If it is 0 to 4 don’t change it

– If it is 5 to 9 make it one bigger

• round 45.462 to four sig figs

• to three sig figs

• to two sig figs

• to one sig fig

Practice• 4.8 + 6.8765• 520 + 94.98• 0.0045 + 2.113• 6.0 x 102 - 3.8 x 103 • 5.4 - 3.28• 6.7 - .542• 500 -126• 6.0 x 10-2 - 3.8 x 10-3

Multiplication and Division

• Rule is simpler• Same number of sig figs in the answer as

the least in the question• 3.6 x 653• 2350.8• 3.6 has 2 s.f. 653 has 3 s.f.• answer can only have 2 s.f.• 2400

Multiplication and Division• Same rules for division• practice • 4.5 / 6.245• 4.5 x 6.245• 9.8764 x .043• 3.876 / 1983• 16547 / 714

The Metric System

An easy way to measure

Measuring• The numbers are only half of a

measurement

– It is 10 long

– 10 what.

• Numbers without units are meaningless.

The Metric System

• Easier to use because it is a decimal system

• Every conversion is by some power of 10.

• A metric unit has two parts

• A prefix and a base unit.

• prefix tells you how many times to divide or multiply by 10.

The SI System

• The SI system has seven base units from which all others are derived. Five of them are showed here

Physical Quantities

Name of Unit Abbreviation

Mass Kilogram kg

Length Meter m

Time Second s

Temperature Kelvin K

Amount of Substance

Mole mol

SI Units (Con’t)

• These prefixes indicate decimal fractions or

multiples of various units

Prefix Abbreviation Meaning

Mega- M 106

Kilo- k 103

Deci- d 10-1

Centi- c 10-2

Milli- m 10-3

Micro- 10-6

Nano- n 10-9

Pico- p 10-12

Femto- f 10-15

Derived Units

Derived Units• SI units are used to derive the units of

other quantities.

• Some of these units express speed, velocity, area and volume….

• They are either base units squared or cubed, or they define different base units

Volume

• calculated by multiplying

– L x W x H (for a square)

– π x r2 x H (for a cylinder)

– (for a sphere)

• Basic SI unit of volume is the cubic meter (m3 ).

• Smaller units are sometimes employed ex. cm3, dm3 ….

• Volume is more commonly defined by liter (L).

Mass

• weight is a force, is the amount of matter.• 1gram is defined as the mass of 1 cm3 of

water at 4 ºC.• 1000 g = 1000 cm3 of water• 1 kg = 1 L of water

Temperature Scales

Measuring Temperature

• Celsius scale.

• water freezes at 0ºC

• water boils at 100ºC

• body temperature 37ºC

• room temperature 20 - 25ºC

0ºC

Measuring Temperature

• Kelvin starts at absolute zero (-273 º C)

• degrees are the same size

• C = K -273.15

• K = C + 273.15

• Kelvin is always bigger.

• Kelvin can never be negative.

273 K

Temperature Conversions

At home you like to keepthe thermostat at 72 F. Whiletraveling in Canada, you findthe room thermostat calibratedin degrees Celsius. To whatCelsius temperature would

youneed to set the thermostat toget the same temperature youenjoy at home ?

• °C = 5/9 ( °F-32)

• °F = 9/5 (°C ) +32

• K = °C + 273.15

Which is heavier?

it depends

Density

• how heavy something is for its size• the ratio of mass to volume for a

substance• D = M / V• Independent of how much of it you have• gold - high density• air low density.

Calculating

• The formula tells you how• units will be g/mL or g/cm3 • A piece of wood has a mass of 11.2 g and

a volume of 23 mL what is the density?• A piece of wood has a density of 0.93

g/mL and a volume of 23 mL what is the mass?

Calculating

• A piece of wood has a density of 0.93 g/mL and a mass of 23 g what is the volume?

• The units must always work out.• Algebra 1• Get the thing you want by itself, on the

top.• What ever you do to onside, do to the

other

Floating

• Lower density floats on higher density.• Ice is less dense than water.• Most wood is less dense than water• Helium is less dense than air.• A ship is less dense than water

Density of water

• 1 g of water is 1 mL of water.• density of water is 1 g/mL• at 4ºC• otherwise it is less

Problem Solving

Word Problems

• The laboratory does not give you numbers already plugged into a formula.

• You have to decide how to get the answer.

• Like word problems in math.

• The chemistry book gives you word problems.

Problem solving1 Identify the unknown.

1 Both in words and what units it will be measured in.

1 May need to read the question several times.

2 Identify what is given

2 Write it down if necessary. 2 Unnecessary information may also be given

Problem solving

3 Plan a solution

3 The “heart” of problem solving

3 Break it down into steps.

3 Look up needed information.

3 Tables3 Formulas3 Constants3 Equations

Problem solving

4 Do the calculations – algebra

5 Finish up Sig Figs Units Check your work Reread the question, did you answer it Is it reasonable? Estimate

Conversion factors

• “A ratio of equivalent measurements.”• Start with two things that are the same. One meter is one hundred centimeters• Write it as an equation. 1 m = 100 cm• Can divide by each side to come up with

two ways of writing the number 1.

Conversion factors

• A unique way of writing the number 1.• In the same system they are defined

quantities so they have unlimited significant figures.

• Equivalence statements always have this relationship.

• 1000 mm = 1 m

Write the conversion factors for the following

• kilograms to grams• feet to inches• 1.096 qt. = 1.00 L

What are they good for?

• We can multiply by one creatively to change the units .

• 13 inches is how many yards?• 36 inches = 1 yard.• 1 yard = 1

36 inches• 13 inches x 1 yard =

36 inches

What are they good for? We can multiply by one creatively to

change the units . 13 inches is how many yards? 36 inches = 1 yard. 1 yard = 1

36 inches 13 inches x 1 yard =

36 inches

Dimensional Analysis

• Dimension = unit• Analyze = solve• Using the units to solve the problems.• If the units of your answer are right,

chances are you did the math right.

Dimensional Analysis

• A ruler is 12.0 inches long. How long is it in cm? ( 1 inch is 2.54 cm)

• in meters?• A race is 10.0 km long. How far is this in

miles? – 1 mile = 1760 yds– 1 meter = 1.094 yds

• Pikes peak is 14,110 ft above sea level. What is this in meters?

Multiple units• The speed limit is 65 mi/hr. What is this in

m/s?– 1 mile = 1760 yds– 1 meter = 1.094 yds

65 mihr

1760 yd1 mi 1.094 yd

1 m 1 hr60 min

1 min60 s

Units to a Power

• How many m3 is 1500 cm3?

1500 cm3 1 m100 cm

1 m100 cm

1 m100 cm

1500 cm3 1 m

100 cm

3

Dimensional Analysis

• Another measuring system has different units of measure.

6 ft = 1 fathom 100 fathoms = 1 cable length10 cable lengths = 1 nautical mile 3 nautical miles = 1 league

• Jules Verne wrote a book 20,000 leagues under the sea. How far is this in feet?

Quantifying Energy

Recall Energy

• Capacity to do work

• Work causes an object to move (F x d)

• Potential Energy: Energy due to position

• Kinetic Energy: Energy due to the motion of the object

EnergyKinetic Energy – energy of motion

KE = ½ m v 2

Potential Energy – stored energy Batteries (chemical potential energy) Spring in a watch (mechanical potential energy) Water trapped above a dam (gravitational potential energy)

mass velocity (speed)

B

AC

The Joule

The unit of heat used in modern thermochemistry is the Joule

2

211smkgjoule

Non SI unit calorie1Cal=1000cal4.184J =1cal or 4.184kJ=Cal

CalorimetryThe amount of heat absorbed or released during a physical or chemical change can be measured…

…usually by the change in temperature of a known quantity of water

1 calorie is the heat required to raise the temperature of 1 gram of water by 1 C

Coffee Cup Calorimeter Bomb Calorimeter

Specific heat

• Amount of heat energy needed to warm 1 g of that substance by 1oC

• Units are J/goC or cal/goC

Specific heats of some common substances

SubstanceWaterIronAluminumEthanol

(cal/g° C) (J/g ° C)• 1.000 4.184• 0.107 0.449• 0.215 0.901• 0.581 2.43

Calculations Involving Specific Heat

Tmsq

s = Specific Heat Capacityq = Heat lost or gainedT = Temperature

change (Tfinal-Tinital)

ExampleExampleCalculate the energy required to raise the temperatureof a 387.0g bar of iron metal from 25oC to 40oC. Thespecific heat of iron is 0.449 J/goC