Download Chapter 2 Notes - Data Analysis

2.1 – Data Analysis
Objectives:
1. Differentiate between SI base and derived units
2. Identify appropriate SI units for various measurements
3. Convert between common SI prefixes (change magnitudes)
4. Describe the Celsius and Kelvin temperature scales
2.1 – Units of Measurement
Systeme International d’Units (SI System)
‘The Metric System’
• Uses 7 ‘Base Units’ to measure everything that can be measured
• Base units reflect some natural constant condition (except kg)
• Rules of usage
•
•
•
•
•
•
•
•
An abbreviated metric unit is referred to as a ‘symbol’.
Symbols are never followed by a period unless they end a sentence.
Symbols should only follow numbers – they’re not abbreviations.
Leave a space between the number and the symbol.
Symbols are always written in the singular – no ‘s’ to make it plural.
Never use ‘p’ in an SI symbol to indicate ‘per’ (as in mph)– use a ‘/’
Capitalization matters – be careful.
SI units named after people are not capitalized, but the symbol is.
2.1 – Units of Measurement
Systeme International d’Units (SI System)
The Base Units
Quantity
Unit
Symbol
Length
meter
m
Mass
kilogram
kg
Time
second
s
Electric current
ampere
A
Temperature
kelvin
K
Amount of substance
mole
mol
Luminous intensity
candela
cd
2.1 – Units of Measurement
Systeme International d’Units (SI System)
Derived Units
• Units of measure made by combining the seven base units.
2.1 – Units of Measurement
Systeme International d’Units (SI System)
Move the decimal place to the right
Prefix
Symbol
Multiplier
giga-
G-
1 000 000 000
mega-
M-
1 000 000
kilo-
k-
1000
hecto-
h-
100
deka-
da-
10
unit
1
deci-
d-
1/10
centi-
c-
1/100
milli-
m-
1/1000
micro-
μ-
1 /1 000 000
nano-
n-
1/1 000 000 000
Move the decimal place to the left
Changing Magnitude - Prefixes
2.1 – Units of Measurement
Systeme International d’Units (SI System)
Changing Magnitude - Prefixes
0.001
kilok-
0.01
hectoh-
0.1
dekada-
1.0
unit
10
decid-
100
centic-
1000
millim-
KingHector
HenryDAnced
DAngledUnder
UnderDribbling
Dirty Capuchin
Monkeys
King
Chocolate
Milk
2.1 – Units of Measurement
Useful Non-SI Units in Chemistry
Liters (L)
• 1 liter (L) = 1 dm3
• 1 milliliter (mL) = 1 cm3
• 1 kiloliter = 1 m3
Celsius (oC)
• 0 oC = freezing point of pure water at standard
pressure
• 100 oC = boiling point of pure water at standard
pressure
• The Kelvin scale (K) is much better, because there are
no negative temperatures, but the Celsius scale is still
useful.
• 0 K = ‘absolute zero’ = -273 oC
• 0oC = 273 K and 100oC = 373 K
• Kelvins and oC are the same size, just offset by 273
2.2 – Scientific Notation
Objectives:
1. Convert between scientific notation and standard form
2.
3.
4.
5.
Use scientific notation correctly with your calculator
Solve chemistry calculations using an ordered process
Follow units through a calculation
Use dimensional analysis to convert between units
2.2 – Scientific Notation
Scientific Notation
Used
• For expressing very large or very small numbers
• For expressing level of precision (significance)
• Uses numbers (significand or coefficient) from 1 – 9.99 and a
multiplier (exponent) to show order of magnitude.
1 –used
9.99
x 10values greater than 1.
Positive exponents are
to show
x
850
= 8.5 x 10 x 10
= 8.5 x 102
8500.
= 8.500 x 10 x 10 x 10
= 8.500 x 103
4321
= 4.321 x 10 x 10 x 10
= 4.321 x 103
2 302 000
= 2.302 x 10 x 10 x 10 x 10 x 10 x 10
= 2.302 x 106
Negative exponents are used to show values less than 1.
0.0023
= 2.3  10  10  10
= 2.3 x 10-3
0.000023
= 2.3  10  10  10  10  10
= 2.3 x 10-5
2.2 – Scientific Notation
Scientific Notation
Samples
Put in standard form.
1.87 x 10–5 = 0.0000187
3.7 x 108
= 370 000 000
7.88 x 101
= 78.8
2.164 x 10–2 = 0.02164
Change to scientific notation.
12 340
= 1.234 x 104
0.369
= 3.69 x 10–1
0.008
= 8 x 10–3
1 000 000 000
= 1 x 109
6.02 x 1023 = 602 000 000 000 000 000 000 000
2.2 – Scientific Notation
Scientific Notation
Using your calculator
• Using scientific notation can cause problems with the order of
mathematical operations.
• Every calculator has a method to enter scientific notation as a
single expression so you don’t need ( ).
• Never use “^”!
These calculators all
show the same number
6.02 x 1023.
2.2 – Scientific Calculations
Chemistry Math
Follow the same steps every time!!
• List the data values given with their correct variable.
• List the response variable (the variable you are solving for)
• Select the correct formula.
• Isolate the response variable
• Substitute the values and solve.
• Find the mass of an object with a density of 5 g/mL and a volume of
10 mL.
ρ = m/V
m=?
ρ = 5 g/mL
(V) ρ = m/V (V)
V = 10 mL
(V) ρ = m/V (V)
Vρ=m
10 mL  5 g/mL = m
10 mL  5 g/mL = m
50 g = m
2.2 – Scientific Calculations
Chemistry Math
Isolating Variables
• A key step in solving chemistry problems is isolating a variable
of interest.
• The solution of a given equation does not change if we:
• Add or subtract the same number from both sides of the equation
Solve for x,
x+y=z
-y + x + y = z – y
-y + x + y = z – y
x=z–y
Solve for x,
x-y=z
+y + x - y = z + y
+y + x - y = z + y
x=z+y
2.2 – Scientific Calculations
Chemistry Math
Isolating Variables
• The solution of a given equation does not change if we:
• Multiply both the sides with the same number or divide both the
sides with the same non-zero number.
Solve for x,
xy = z
xy = z
y
y
xy = z
y
y
x = z
y
Solve for x,
(y) xy
x = z
y
= z (y)
(y) xy = z (y)
x = zy
2.2 – Scientific Calculations
Chemistry Math
Dealing with Units
• Units must always be carried through the calculation into the
answer:
5.2 kg (2.9 m) = 0.64 kg•m
(18 s)(1.3 s)
s2
4.8 kg (23 s)
(18 s)(37 s)
= 0.57 kg
s
2.2 – Data Analysis
Dimensional Analysis
Converting Units
• Method of converting units using equivalent measures
• Conversion factors
• Any two equivalent measures may be used to write a ratio with a
value of 1. This ratio can be used as a conversion factor.
1 hour
60 min
= 1
60 min
1 hour
= 1
1 hour = 6 hour
360 min  60
min
1000 mL = 1
1L
1L
1000 mL = 1
1L
430 mL  1000
mL = .430 L
2.2 – Data Analysis
Dimensional Analysis
Converting Units
• The same unit will always be written diagonally in the expression so
that the unit cancels out in the calculation.
1 hour = 6 hour
360 min  60
min
min = 21 600 min2/hr
360 min  60
1 hour
• Write as many conversion factors as necessary to get to answer
• Example: Convert 2 300 000 seconds into days
seconds
2 300 000 sec
minutes

1 min
60 sec
1 hour
 60
min
hours

1 day
24 hour
days
= 26.62037 days
• Multiply across the top, then divide all the way across the bottom
• Round the answer appropriately
= 27 days
2.2 – Data Analysis
Dimensional Analysis
• The Box Method
Method of converting units similar to dimensional analysis using a
series of boxes.
• Place the measurement you are converting from in the top left
corner of your boxes.
• In the next set of boxes, place a ratio that relates your unit to the
unit you want to convert to, so that the unit you are converting
from cancels out.
• Example: Convert 2 300 000 seconds into days
2 300 000 sec
1 min
60 sec
1 hour 1 day
60 min 24 hour
= 26.62037 days
• Multiply across the top, then divide all the way across the bottom
• Round the answer appropriately
= 27 days
2.3 – Reliability of Measurements
Objectives:
1. Differentiate between accuracy & precision
2. Calculate percent errors for data
3. Identify and use significant digits in calculations
2.3 – Reliability of Measurements
Accuracy & Precision
• Accuracy = comparison of a measure against a known standard
• If you get the number that is expected when compared to a known
standard, the measure is accurate (we calibrate scales to make sure
they are accurate)
• Precision = ability to reproduce the measurement
• If you can reproduce over and over, the measure is said to be
precise
• If a measure is precise and accurate, then it is said to be
reliable.
2.3 – Reliability of Measurements
Calculating Percent Error
• Comparison of a measurement to its accepted value.
• Percent error is calculated using the formula:
Percent Error =
•
measured value – accepted value
accepted value
x 100
Example: A cookie recipe anticipates a yield of 60 cookies. When
the baking is finished only 54 cookies have been produced.
Calculate the percent error.
measured value = 54 cookies
accepted value = 60 cookies
54 cookies – 60 cookies
Percent Error =
x 100 = -10%*
60 cookies
* While some people suggest taking the absolute value of this
calculation, I prefer using the sign of the calculation.
2.3 – Reliability of Measurements
Certainty
• Measurements always include a number of digits that are
“certain” and a final digit that is “uncertain”.
•
The final digit is uncertain because:
• some measuring instruments require estimation
• some measuring instruments automatically round
Require Estimation
Automatically Round
2.3 – Reliability of Measurements
Making Valid Measurements
• When using a scaled instrument (ruler, meter stick, graduated
cylinder, pipet, buret, etc.) always estimate one place beyond the
certainty of the instrument.
25.45 cm
The human eye is very capable of estimating tenths on most scaled
instruments used in the lab. So ALWAYS estimate to the nearest 1/10 of
the smallest graduation.
2.3 – Reliability of Measurements
Significant Digits
• The certain digit(s) plus the final uncertain digit of a
measurement are considered “significant”.
• Rules for identifying sig figs.
•
•
•
•
•
Non-zero digits are always significant.
3010 g
1 902.51 g
0.01380 g
0.00103 g
7.123 g
Zeros between non-zero digits are always significant.
3010 g
1 902.51 g
1.0138 g
0.00103 g
100 033 g
Left hand zeroes are never significant.
0.301 g
0.0091 g
0.01380 g
0.00103 g
0.6703 g
Right hand zeroes are significant when a decimal is present.
300.0 g
190 g
12 000 g
12 000. g
0.01030 g
Counts and defined constants have an infinite number of significant
digits. Don’t use them for rounding.
2.3 – Reliability of Measurements
Rounding Answers
• At the end of calculations, the answer must be rounded to the
correct number of significant digits.
• Deciding where to round.
•
In addition and subtraction, round the answer to match the
measurement with the fewest decimal places.
3010 g
+ 30.6 g
= 3040.6 g
3040 g
•
1 902.51 g
+ 30.6 g
= 1933.11 g
1933.1 g
1.01 g
- 0.0306 g
= 0.9794 g
0.98 g
13 g
-6g
=7g
7g
In multiplication and division, round the answer to match the
measurement with the fewest significant digits.
3.6 cm
x 30.6 cm
= 110.16 cm2
110 cm2
2.51 cm
x 0.6 cm
= 1.506 cm2
2 cm2
1.01 cm3
 0.0306 cm
= 33.00653 cm2
33.0 cm2
13 cm3
 6 cm
= 2.166667 cm2
2 cm2
2.3 – Reliability of Measurements
Rounding Answers
• Deciding how to round.
•
•
After finding the correct rounding position, look at the digits that
follow it.
If the digits are less than 5, do not change the final digit.
313.251 g  313 g
•
If the digits are greater than 5, round the final digit up.
313.651 g  314 g
•
0.01380 g  0.01 g
0.01580 g  0.02 g
If the digits exactly equal 5, round the final digit to be even.
313.500 g  314 g
0.04500 g  0.04 g
2.4 – Representing Data
Objectives:
1. Select appropriate graph types for representing data
2. Determine how different trends appear on graphs
3. Understand why we use best fit lines
4. Calculate slopes for data lines
2.4 – Representing Data
Graphing
• Pie charts
Used to represent categorical data as parts of a whole. Pie charts
typically depict percentages (fractions). A pie chart is constructed
by converting the share of each component part into a percentage
of 360o.
2.4 – Representing Data
Graphing
• Bar Graphs
Used to represent categorical or numeric values within certain
intervals. Bars may be drawn horizontally or vertically. Each bar
represents a category, value or range of values. They are often used
to show changes over time and clearly illustrate differences in
magnitude.
2.4 – Representing Data
Graphing
• Line Graphs
•
Most common graph used in science. They show the relationship
between two variables. Line graphs are useful for showing trends.
•
Relationships are typically shown using best-fit lines which may not
intersect any of the data points.
2.4 – Representing Data
Graphing
• Trends
Reliable data will generally show one of a limited number of
trends or relationships:
• Direct Relationships (Positive Correlation)
2.4 – Representing Data
Graphing
• Trends
• Inverse Relationships (Negative Correlation)
2.4 – Representing Data
Graphing
• Trends
• Exponential Relationships
Decay
Growth
2.4 – Representing Data
Best Fit Lines & Slope
•
•
Best fit lines are used to show the underlying trend of the data
in a scatterplot. The trend can then be used to make
predictions.
Drawing a best fit line simply requires drawing a line through
the center of the plotted data so that data points are evenly
distributed above and below the line.
Making predictions
beyond the data
points is called
extrapolation.
Making predictions
between the data
points is called
interpolation.
2.4 – Representing Data
Slope
• The slope of the line may be calculated using the formula
slope = ∆y / ∆x
where ∆ means “change in”.