Transcript Chapter 2 -measuring and sig dig

Chapter 2
Measuring,
Significant Digits,
Scientific Notation
Metric System
Why we use it?

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International system (SI) of measuring since
1790
Standard system of measuring around the
world, everyone speaks the same language.
(US only industrialized nation not using it. Yet
in 1893 the US adopted it as a standard of
measuring)
Conversions from smaller to larger units
easier
–
–
based on 10’s, move decimal
no conversion factors to memorize (like 12in=1ft)
Video- Metric vs English
Conversions
Base Units
 Length= meter (m)
 Mass = gram (g)
 Volume = liter (l)
 Time = seconds (s)
kilo(k)
hecto(h)
deka(D)
Base
Unit
g,m,s,l deci(d)
centi(c)
milli(m)
micro(µ)
Metric Prefixes & Symbols
(staircase)
 Move
the decimal the number of
“stairs” it takes to get to the new unit
 The direction you move on the
staircase is the direction you move
the decimal

Conversion by moving decimal video
45 mm =
m
(move decimal 3 places to the left)
45 mm= .045
m
Measuring
Length
-Standard for Length = meter
-distance light travels in 1/300
millionth of a second
-Ruler Uncertainty = + .05 cm
(means -> 4.15 cm measurement is somewhere
in between 4.10 cm – 4.20 cm)
1.
Each ruler line = .1 cm
-read to the hundredths place
11
11.45 cm
12
2. Volume
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Amount of space an object takes up
V=lxwxh
Measured with a ruler the unit label is cm3
Using a graduated cylinder the unit label is
milliliters (ml) or liters (l)
1 cm3 = 1 ml
1dm3 = 1 L
Large graduated cylinder (50 ml)
Each line = 1 ml
Read to tenths place
Uncertainty ±.5 ml
Small graduated cylinder ( 10 ml)
Each line = .2 ml
Read to tenths place
Uncertainty ± .1 ml
Rules for Reading a Graduated
Cylinder
1.
2.
3.
Read at eye level
Place cylinder on solid surface
Read from the bottom of the meniscus
- curvature of the liquid
3. Temperature
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Measure of the average kinetic energy of the
particles
Measured in Celsius or Kelvin
0 ° C = freezing pt. water
37 ° C = body temp.
100 ° C = boiling pt. water
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Every mark = 1 ° C
Uncertainty of Thermometer ± .5 ° C
Read to tenths place
4. Mass

Standard for MassInternational Prototype
Kilogram (a piece of
metal kept at the
International Bureau of
Weights and Measures
in France)
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Use a balance (triple beam or electric)
Reads out to the hundredths place
Each line on front bar of triple beam is .1 g
–
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(reads like a ruler)
Uncertainty Triple Beam ±.05 g
Uncertainty Electric Balance ±.01 g
Rules for Using a Balance
1.
2.
3.
4.
5.
Always weighing paper when using chemicals
No liquids on electric balance
Re-zero triple beam with side knob
Tare/Re-zero/on Button on Electric balance
will set it to zero (make sure it is in grams)
To turn off electric – press and hold OFF key
5. Time
 Unit
for time = seconds
 Standard for time = ticks on a
cesium atomic clock (measures the
electron transition in an atom)
Density
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Volume = amount of space an object takes up
Mass = amount of matter in an object
Density =
-how closely molecules are packed together
-objects are more or less dense (not heavier or
lighter)
Density Formula
Density = mass
volume
D=m
v
Units –> g/ml or g/cm3
Derived Unit Conversions:
Squared Conversions
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Measuring area (flat surface)
Length (cm) x width (cm) = cm2
Ex. Piece of paper
Length = 27.95 cm
Width = 21.60 cm
Area= 603.72 cm2
Convert to m2=
Steps:
1.
2.
3.
Ignore the square symbol for now
Convert like normal ( cm ->m)
Then double the # of places you would
move. (two places for the length, two for
the width)
Ex. 603.72 cm 2 =
cm-> m
cm2 ->m2
m2
move 2 place
move 4 places
603.72 cm2 = .060372 m2
Cubed Conversions
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Measuring volume
Units are cubic centimeter (cm3)
Steps:
1. Ignore the cubed unit
2. Convert like normal
3. Triple the number of places you would move
Ex.
1503.5 cm 3 =
cm -> mm
cm3 -> mm3
mm3
-move 1 place
- move 3 places
1503.5 cm 3 =
1503500 mm3
Complex Unit Conversions

Use for converting derived units
Derived Units = made from a division of 2
or more base (measured) units
Ex.
Speed -> m/s
Density-> g/ml or g/cm3
Steps for Converting:
Ex. 3.4 m/s =
1.
2.
km/hr
Set up problem as two separate conversion
problems.
Straighten the diagonal line and re-write
problem
3.4 m ->
1s
km
hr
Convert top line to top line, bottom line to
bottom line.
3.4 m ->
.0034km
1s
1/3600 hr
4. Divide out problem, so bottom is equal to
one.
.0034km ->
12.2 km/hr
3.
1/3600 hr
Reading Instruments and
Uncertainty
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Human-built instruments have built in error
A measurement is the comparison of an object
to a standard
Measurements always have a numerical part
and a unit label
When measuring your last digit that you write is
an estimated or uncertain digit
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The # of places each measurement goes out
depends on the measuring device you choose
Each device has its own ± value
Ex. Ruler ± .05 cm
Accuracy and Precision
 Because
every measurement is a
comparison with a standard, one
needs to describe the reliability of
that measurement
Accuracy
-
refers to how close a measurement is to the
true or correct value for the quantity
-
Ex. You measure waters density = 1.2 g/ml
- Actual measurement = 1.0 g/ml
Precision
-refers to how close a set of measurements for a
quantity are to one another (regardless if they are
correct)
-this is why we take averages and have uncertainty
or sources of error in measurements
-something can be very precise but not accurate
(keep hitting 20’s instead of bulls eyes)
-something that is accurate is also precise
Percent Error
-a comparison of the experimental or observed value to
the real accepted value
-use percent error to compare your results, how far off
you are
Formula:
|O-A|x 100 = % Error
(use absolute value)
A
(O= experimental value, A = accepted value)
Significant Figures/Digits
(video)
All #’s are based on measurements
 Your calculations can only be as accurate as
your measuring device
Ex. D = m/v
m=3.41 g v= 24.2 m l
D = .14090909 g/ml (on calculator)
Round off to .141 g/ml
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Rules for Assigning Sig. Dig’s.
Non-zero numbers are always significant
7.5 L
26.45 g
2 sig. dig.
4 sig. dig.
2. Any zeroes between 2 sig. dig’s are significant
4.106 g
10.5 ml
20
4 sig. dig.
3 sig. dig.
1.
10
3.
A final zero or trailing zeroes in the decimal
portion ONLY are significant
.340 g
3 sig dig
10
10.00 cm (4 sig. dig.)
11
Special Zero Rules
(when zeros don’t count)
1.
Space holding zeros on #’s less than one are
not significant
Ex.
Convert 54 ml to L
54 ml = .054 L
.054 L has 2 sig dig
(the zero is a place holder)
Hint: put number in scientific notation, if the
zeros disappear they were not significant
2.
Zeros to the left of the decimal on #’s less
than one are not significant
Ex.
0.14 ml
2 sig dig
(this number is the same as .14 ml)
3.
Trailing zeros in a whole # are not significant
(they are place holders)
Ex.
200 g
25000 m
1 sig dig
2 sig dig
Exception: when zeros are identified as significant (with a
line over last sig. dig. Or placed into scientific
notation)
Ex. 200 m = 2.00 x 102 = 20Ō m
makes it so it has 3 sig dig
Atlantic – Pacific Rule
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When a decimal is Present, start on the Pacific side
(which is the left-west) and the first non-zero number
and so forth are significant.
 0.0980 (3 sig dig)
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When a decimal is Absent, start on the Atlantic side
(which is the right) and the first non-zero number and
so forth are significant.
245000  (3 sig dig)
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So, remember:
Present = pacific
Absent = atlantic
Counting numbers
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counting numbers and defined constants have
an infinite number of significant figures
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Ex. 25 books
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Infinite sig. dig.
60 sec = 1 min
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60= infinite, 1= infinite
Number of Sig. Dig. Problems
.049450 g
.0004802 m
3.1500 s
4900 mg
490Ō ml
.049450 g
.000482 m
3.1500 s
4900 mg
490Ō ml
5 sig dig
3
5
2
4
Rule for Multiplying & Dividing
-
your answers in calculations can not be more
precise than the least precise measurement
Rules:
The LEAST number of significant digits in any
number of the problem determines the # of sig.
dig’s in the answer
-Count up the sig. dig.’s in all the measurements
-choose the smallest # of sig. dig.’s
-round the answer to that # of sig. dig.’s
Ex. 2.5 cm x 3.42 cm
= 8.55 (on calculator)
Actual Answer = 8.6 cm2
(answer can only have 2 sig dig’s –least #)
651 cm
(3 sig dig)
x
75 cm = 48825 cm2
(2 sig dig) = answer 2 sig dig
= 49000 cm2
( round off to 2 sig dig, fill in with place holder
zeros, to decimal)
7.835 kg / 2.5 L =
14.75 L / 1.20 s =
360 cm x 51 cm x 9.07 cm =
7.835 kg / 2.5 L =
(4)
(2) = 3.1 kg/L
14.75 L / 1.20 s =
(4)
(3) = 12.3 L/s
360 cm x 51 cm x 9.07 cm =
(2)
(2)
(3)
= 170000 cm3
Adding and Subtracting Rules:
-
-
look at the decimal portion only ( you are
looking at the number of places each
measurement goes out – NO
COUNTING)
DIFFERENT RULES THAN X & 
Steps:
1.
2.
3.
Count the number of places that the decimal
goes out to in each of the measurements. If
there is no decimal look at the last sig. dig. in
the whole number. (find least accurate
measurement (LAM))
Add or subtract as normal.
Round and record the answer so it is round to
the # of places in the LEAST accurate
measurement. (Hint: draw a line after LAM)
Ex.
23.1
+4.77
(Hint: line up decimal & draw a line after the last
sig dig in LAM)
23.1
+4.77
23.1
+4.77
27.87
-round off to the right of the line
27.9
22.101
- .9307
564000
+ 3142
.04216
-.0004134
22.101
- .9307
21.1703
564000
+ 3142
567142
.04216
-.0004134
.0417466
21.170
567000
.04175
Scientific Notation
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Convenient for writing extremely large or small
numbers
Don’t have to worry about placeholder zeros
with sig. dig.
Ex. 92,900,000 miles to the sun
- move the decimal to the left so that one
number is to the left of the decimal, number of
places you move becomes the exponent
9.29 x 10 7 miles to the sun
Ex. Resolution of the electron scanning
microscope = .00000002 inches
 Move the decimal to the right so that one
number is to the left of the decimal, the
number of places you move becomes a (-)
exponent
2 x 10 –8 in
Scientific Notation on your
Calculator
Use ee or exp key, never use x 10
Graphing Calculator
6.02 x 10 23
6 . 0 2 2nd ee
2
3
Regular Calculator
6
.
0
2
ee
2
3
Sig. Dig. Rules w/ Scientific
Notation
Multiplying and dividing:
 Count the number of significant digits in each
number (do not count the (x 10 exponent )part)
 Choose the smallest number of sig. dig. and
round off answer to that number of sig. dig.
4.56 x 10 5
x
(3 sig dig)
1.368 x 10 8
(need 2 sig dig)
1.4 x 10 8
3.0 x 10 2
(2 sig dig)
Adding and Subtracting Sig Dig
Rules w/ Scientific Notation
Need to make the exponents match first
( can
not add mismatched exponents)
 Change the smaller exponent to match the larger by
moving the decimal point to the left
7.5 x 104
+ 3.21 x 103

7.5
x 104
+ .321 x 104
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Draw your line after the last sig. dig. in the least
accurate number
Add or subtract, carry down the exponent
Round off to the line
7.5
x 104
+ .321 x 104
7.821 x 104
7.8 x 104
Graphing
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Visual display of data
Types: bar & circle (for
comparing relative
amounts), line (used
most often in chemistry)
Line Graph
Independent variable (x-axis)- one you are
deliberately changing
Dependent variable (y-axis) – changes because
of the independent variable
Best fit line or curve – when points are scattered,
draw a line that goes through as many points a
possible, with equal number of points above
and below the line
Setting up a graph:
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Has a title
Axis’ are labeled w/ units (ex.mass (g))
Number the axis so that they are equally space
Numbering does not have to start at zero
Interpreting a graph
Slope:
change in y
change in x
y2-y1
x2-x1
Interpolationreading between plotted points
Extrapolationreading beyond plotted data
extrapolation
interpolation