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PWISTA Math of Chemistry. Scientific measurement Accuracy and Precision Sig Figs Metric System. Types of measurement. Quantitative - use numbers to describe Qualitative - use description without numbers 4 feet extra large Hot 100ºF. Scientists prefer. Quantitative- easy check

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Pwista math of chemistry

PWISTAMath of Chemistry

Scientific measurement

Accuracy and Precision

Sig Figs

Metric System


Types of measurement
Types of measurement

  • Quantitative- use numbers to describe

  • Qualitative- use description without numbers

  • 4 feet

  • extra large

  • Hot

  • 100ºF


Scientists prefer
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
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
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



Accurate?

No

Precise?

Yes


Accurate?

Yes

Precise?

Yes


Precise?

No

Accurate?

Maybe?


Accurate?

Yes

Precise?

We cant say!


In terms of measurement
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?


Significant figures sig figs

1

2

3

4

5

Significant figures (sig figs)

  • How many numbers mean anything

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


Sig figs
Sig Figs

Significant digits, which are also called significant figures, are very important in Chemistry.

Each recorded measurement has a certain number of significant digits.

Calculations done on these measurements must follow the rules for significant digits.


Sig figs1
Sig Figs

The significance of a digit has to do with whether it represents a true measurement or not.

Any digit that is actually measured or estimated will be considered significant.

Placeholders, or digits that have not been measured or estimated, are not considered significant.

The rules for determining the significance of a digit will follow.


Significant figures sig figs1
Significant figures (sig figs)

  • The better marks the better we can estimate.

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

1

2

3

4

5


Sig figs2
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 do count


The standard rules
The Standard Rules

1) Digits from 1-9 are always significant.

2) Zeros between two other significant digits are always significant. (sandwich rule)

3) One or more additional zeros to the right of both the decimal place and another significant digit are significant.

4) Zeros used solely for spacing the decimal point (placeholders) are not significant.


Too confusing heads will spin
Too Confusing! Heads will spin!


Alternate rule for significant digits
Alternate Rule for Significant Digits

  • Rules Courtesy of Fordham Prep website.

  • When you look at the number in question, you must determine if it has a decimal point or not.  If it has a decimal, you should think of "P" for "Present".  If the number does not have a decimal place, you should think of "A" for "Absent".


Alternate rule for significant digits1
Alternate Rule for Significant Digits

  • Example, for  the number 35.700, think "P", because the decimal is present.

  • For the number 6500, you would think "A", because the decimal is absent.


Alternate rule for significant digits2
Alternate Rule for Significant Digits

  • Now, the letters "A" and "P" also correspond to the "Atlantic" and "Pacific" Oceans, respectively. 

  • Assume the top of the page is North, and imagine an arrow being drawn toward the number from the appropriate coast. 

  • Once the arrow hits a nonzero digit, it and all of the digits after it are significant.


Example 1
Example 1

  How many significant digits are shown in the number 20 400 ? 

(remember that we use spaces, rather than commas, when writing numbers in Science.

Well, there is no decimal, so we think of "A" for "Absent".  This means that we imagine an arrow coming in from the Atlantic ocean, as shown below;

20 400 


Example 11
Example 1

The first non-zero digit the arrow hits would be the 4, making it, and all digits to the left of it significant.There are 3 significant digits20400 


Example 2
Example 2

How many significant digits are shown in the number 0.090 ?


Example 21
Example 2

  • Well, there is a decimal, so we think of "P" for "Present".  This means that we imagine an arrow coming in from the Pacific ocean, as shown below;

  • 0.090


Example 2 0 090
Example 2 0.090

  • The first nonzero digit that the arrow will pass in the 9, making it, and any digit to the right of it significant.


Answer example 2
Answer Example 2

  • There are 2 significant digits in the number 0.090

  • Here are the significant digits, shown in boldface.  0.090


Which zeros count
Which zeros count?

  • Those at the end of a number before the decimal point don’t count

  • 12400

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

  • 0.045


Which zeros count1
Which zeros count?

  • Zeros between other sig figs do.

  • 1002

  • zeroes at the end of a number after the decimal point do count

  • 45.8300

  • If they are holding places, they don’t.

  • If they are measured (or estimated) they do


Sig figs3
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.


Sig figs4
Sig figs.

  • How many sig figs in the following measurements?

  • 458 g

  • 4085 g

  • 4850 g

  • 0.0485 g

  • 0.004085 g

  • 40.004085 g


Sig figs5
Sig Figs.

  • 405.0 g

  • 4050 g

  • 0.450 g

  • 4050.05 g

  • 0.0500060 g

  • Next we learn the rules for calculations


Problems same different sig figs
ProblemsSame # - Different # sig figs …

  • 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.


Adding and subtracting with sig figs
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.

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


27.93

+

6.4

27.93

27.93

+

6.4

6.4

For example

  • First line up the decimal places

Then do the adding

Find the estimated numbers in the problem

34.33

This answer must be rounded to the tenths place


Rounding rules
Rounding rules

  • 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


Adding and subtracting
Adding and Subtracting

  • RULE: When adding or subtracting your answer can only show as many decimal places as the measurement having the fewest number of decimal places.


Add 3 76 g 14 83 g 2 1 g 20 69 g

We look to the original problem to see the number of decimal places shown in each of the original measurements.

2.1 shows the least number of decimal places.

We must round our answer, 20.69, to one decimal place (the tenth place).

Our final answer is 20.7 g

add 3.76 g + 14.83 g + 2.1 g = 20.69 g


Practice
Practice places shown in each of the original measurements.

  • 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
Multiplication and Division places shown in each of the original measurements.

  • Rule is simpler

  • Same number of sig figs in the answer as the least in the question

  • 3.6 x 653

  • Calculated answer = 2350.8

  • 3.6 has 2 sig figs, 653 has 3 sig figs

  • answer can only have 2 sig figs

  • 2400


Multiplying and dividing
Multiplying and Dividing places shown in each of the original measurements.

  • RULE: When multiplying or dividing, your answer may only show as many significant digits as the multiplied or divided measurement showing the least number of significant digits.


Ex 22 37 cm x 3 10 cm x 85 75 cm 5946 50525 cm 3
EX: places shown in each of the original measurements. 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3

  • We look to the original problem and check the number of significant digits in each of the original measurements:

  • 22.37 shows 4 significant digits.

  • 3.10 shows 3 significant digits.

  • 85.75 shows 4 significant digits.


Ex 22 37 cm x 3 10 cm x 85 75 cm 5946 50525 cm 31
EX: places shown in each of the original measurements. 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3

  • Our answer can only show 3 significant digits because that is the least number of significant digits in the original problem.

  • 5946.50525 shows 9 significant digits, we must round to the tens place in order to show only 3 significant digits.

  • Our final answer becomes 5950 cm3.


Multiplication and division1
Multiplication and Division places shown in each of the original measurements.

  • 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
The Metric System places shown in each of the original measurements. An easy way to measure


Measuring
Measuring places shown in each of the original measurements.

  • The numbers are only half of a measurement

  • It is 10 long - - - 10 what?

  • Numbers without units are meaningless.

  • I’ll pay you 100 to mow the lawn … then give the person 100 cents. Won’t make friends that way.


The metric system
The Metric System places shown in each of the original measurements.

  • 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.


Base units
Base Units places shown in each of the original measurements.

  • Length - meter more than a yard - m

  • Mass - grams - a bout a raisin - g

  • Time - second - s

  • Temperature - Kelvin or ºCelsius K or C

  • Energy - Joules- J

  • Volume - Liter - half f a two liter bottle- L

  • Amount of substance - mole - mol


Prefixes
Prefixes places shown in each of the original measurements.

  • kilo k 1000 times

  • deci d 1/10

  • centi c 1/100

  • milli m 1/1000

  • kilometer - about 0.6 miles

  • centimeter - less than half an inch

  • millimeter - the width of a paper clip wire


Volume
Volume places shown in each of the original measurements.

  • calculated by multiplying L x W x H

  • Liter the volume of a cube 1 dm (10 cm) on a side

  • so 1 L = 10 cm x 10 cm x 10 cm

  • 1 L = 1000 cm3

  • 1/1000 L = 1 cm3

  • 1 mL = 1 cm3


Volume1
Volume places shown in each of the original measurements.

  • 1 L about 1/4 of a gallon - a quart

  • 1 mL is about 20 drops of water or 1 sugar cube


Mass places shown in each of the original measurements.

  • 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


Mass places shown in each of the original measurements.

  • 1 kg = 2.5 lbs

  • 1 g = 1 paper clip

  • 1 mg = 10 grains of salt or 2 drops of water.


Converting

k places shown in each of the original measurements.

h

D

d

c

m

Converting

  • how far you have to move on this chart, tells you how far, and which direction to move the decimal place.

  • The box is the base unit, meters, Liters, grams, etc.


Conversions

k places shown in each of the original measurements.

h

D

d

c

m

Conversions

  • Change 5.6 m to millimeters

  • starts at the base unit and move three to the right.

  • move the decimal point three to the right

5

6

0

0


Conversions1

k places shown in each of the original measurements.

h

D

d

c

m

Conversions

  • convert 25 mg to grams

  • convert 0.45 km to mm

  • convert 35 mL to liters

  • It works because the math works, we are dividing or multiplying by 10 the correct number of times


k places shown in each of the original measurements.

h

D

d

c

m

Conversions

  • Change 5.6 km to millimeters


Which is heavier

Which is heavier? places shown in each of the original measurements.

it depends


Density
Density places shown in each of the original measurements.

  • 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
Calculating places shown in each of the original measurements.

  • The formula tells you how (d=mass / volume)

  • BETTER YET … the UNITS tell 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 mass of 11.2 g and a volume of 23 mL what is the density?

  • We know the units for density are g/ml

  • The unit tells us that grams goes on top and ml goes on the bottom; so let’s do that.

  • 11.2g / 23ml = 0.49 g/ml


A piece of wood has a density of 0 93 g ml and a volume of 23 ml what is the mass
A piece of wood has a density of 0.93 g/mL and a volume of 23 mL what is the mass?

  • The Unit you want to find ALWAYS goes on top.

  • We have 0.93 g/ml. We want to get rid of the ml on the bottom. So multiply by ml on top. (ml on top and bottom cancel out)

  • 0.93g/ml x 23 ml / 1 = 21.39 g


Calculating1
Calculating 23 mL what is the mass?

  • 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
Floating 23 mL what is the mass?

  • 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
Density of water 23 mL what is the mass?

  • 1 g of water is 1 mL of water.

  • density of water is 1 g/mL

  • at 4ºC

  • otherwise it is less


Measuring temperature
Measuring Temperature 23 mL what is the mass?

0ºC

  • Celsius scale.

  • water freezes at 0ºC

  • water boils at 100ºC

  • body temperature 37ºC

  • room temperature 20 - 25ºC


Measuring temperature1
Measuring Temperature 23 mL what is the mass?

273 K

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

  • degrees are the same size

  • C = K -273

  • K = C + 273

  • Kelvin is always bigger.

  • Kelvin can never be negative.


Heat 23 mL what is the mass?

a form of energy


Temperature is different
Temperature is different 23 mL what is the mass?

  • than heat.

  • Temperature is which way heat will flow (from hot to cold)

  • Heat is energy, ability to do work.

  • A drop of boiling water hurts,

  • kilogram of boiling water kills


Units of heat are
Units of heat are 23 mL what is the mass?

  • calories or Joules

  • 1 calorie is the amount of heat needed to raise the temperature of 1 gram of water by 1ºC

  • a food Calorie is really a kilocalorie

  • How much energy is absorbed to heat 15 grams of water by 25ºC

  • 1 calorie = 4.18 J


Some things heat up easily
Some things heat up easily 23 mL what is the mass?

  • some take a great deal of energy to change their temperature.

  • The Specific Heat Capacity amount of heat to change the temperature of 1 g of a substance by 1ºC

  • specific heat SH

  • S.H. = heat (cal) mass(g) x change in temp(ºC)


Specific heat
Specific Heat 23 mL what is the mass?

  • Water has a high specific heat

  • 1 cal/gºC

  • units will always be cal/gºC

  • or J/gºC

  • the amount of heat it takes to heat something is the same as the amount of heat it gives off when it cools because...


Problems
Problems 23 mL what is the mass?

  • It takes 24.3 calories to heat 15.4 g of a metal from 22 ºC to 33ºC. What is the specific heat of the metal?

  • Iron has a specific heat of 0.11 cal/gºC. How much heat will it take to change the temperature of 48.3 g of iron by 32.4ºC?


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