Unit 2
Measurements
Matter - anything that occupies space and has mass
mass – measure of the quantity of matter
SI unit of mass is the kilogram (kg)
1 kg = 1000 g = 1 x 103 g
weight – force that gravity exerts
on an object
weight = mass  g (F = ma)
on earth, g = 1.0
on moon, g ~ 0.1
La Grande K
1 Kg Pt/Ir alloy
World’s Roundest Object
https://www.youtube.com/watch?v=ZMByI4s-D-Y
A 1 kg bar will weigh
1 kg on earth
0.1 kg on moon
International System of Units (SI) Base Units
Used in this class and should have memorized
All other units are derived from these units
and are known as Derived Units
Velocity: m/s
Force: 1 Newton = 1 kg•m/s2
Volume: m3
Prefixes can be used to simplify for extremely large
or small quantities of base units
“mu”
Used most often in this class, be sure to memorize.
Prefix examples
Driving 321,000 meters to LR = 321 kilometers
Radio station 90.9 MHz = 90,900,000 Hz
A mosquito weighs 2.5 milligrams (mg) = 0.0025 grams (g)
A dust mite’s length 0.0002 meters = 200 micrometers (mm)
Conversion factors can be written/used 2 ways
-6
6
10 ML = 1 L
1 Mm = 10 L
-3
10
3
10
1km =
L
Liter (base)
-2
1 cL = 10 L
-3
1 mL = 10 L
-6
1 mL = 10 L
-9
1 nL = 10 L
Or
kL = 1 L
Liter (base)
2
10 cL = 1 L
3
10 mL = 1 L
6
10 mL = 1 L
9
10 nL = 1 L
I favor the forms using (+) exponents
Volume – SI derived unit for volume is cubic meter (m3)
(cm3 is commonly used)
1 cm3 = 1 mL
Density – SI derived unit for density is kg/m3
1 g/cm3 = 1 g/mL = 1000 kg/m3
*more commonly used
density =
mass
volume
m
d= V
2 L of Os
= 100 lbs
Example
1.1
Gold is a precious metal that is chemically unreactive.
It is used mainly in jewelry, dentistry, and electronic devices.
A piece of gold ingot with a mass of 301 g has a volume of
15.6 cm3. Calculate the density of gold.
gold ingots
Example
1.2
The density of mercury, the only metal that is a liquid at room
temperature, is 13.6 g/mL. Calculate the mass of 5.50 mL of the
liquid.
m
d= V
Chemistry In Action
On 9/23/99, $125M Mars Climate Orbiter entered
Mars’ atmosphere 100 km (62 miles) lower than
planned and was destroyed by heat.
1 lb = 1 N
1 lb = 4.45 N
Failed to convert
English to
metric units
“This is going to be the
cautionary tale that will be
embedded into introduction to the
metric system in elementary
school, high school, and college
science courses till the end of
time.”
Scientific Notation
The number of atoms in 12 g of carbon:
602,200,000,000,000,000,000,000
6.022 x 1023
The mass of a single carbon atom in grams:
0.0000000000000000000000199
1.99 x 10-23
We can factor out powers of 10 to simplify very large or small numbers
N is the base number
between 1 and 10
N x 10n
Exponent (n) is a positive
or
negative integer
Scientific Notation
• Base x 10exponent
• Base number ≥ 1 and < 10
5
320,000
=
3.2
x
10
.
0.000074
.
= 7.4 x 10-5
Decimal moved left so (+)
Decimal moved right so (–)
Scientific Notation Practice
Write these in scientific
notation
• 0.00578
• 579
• 96,000
• 0.0140
Write these in long
notation
• 2.0 x 103
• 3.58 x 10-4
• 4.651 x 107
• 9.87 x 10-2
Mathematics in Scientific Notation
Addition or Subtraction: Must have same exponent
1. Write each quantity with the
same exponent n
2. Combine N1 and N2
4.31 x 104 + 3.9 x 103 =
4.31 x 104 + 0.39 x 104 =
3. The exponent, n, remains the
same
4.70 x 104
Tip: change smaller number to match larger exponent
1.36 x 10-1 – 4 x 10-3 =
1.36 x 10-1 – 0.04 x 10-1 =
= 1.32 x 10-1
More practice:
1.45 x 10601 + 2.4 x 10600
Mathematics in Scientific Notation
Multiplication: Add exponents
1. Multiply N1 and N2
2. Add exponents n1 and n2
Practice: (3 x 10250) (7.2 x 10200 )
(4.0 x 10-5) x (7.0 x 103) =
(4.0 x 7.0) x (10-5+3) =
28 x 10-2 =
2.8 x 10-1
Division: Subtract exponents
1. Divide N1 and N2
2. Subtract exponents n1 and n2
Practice: 9.5 x 1045) ÷ (3.7 x 1050)
8.5 x 104 ÷ 5.0 x 109 =
(8.5 ÷ 5.0) x 104-9 =
1.7 x 10-5
Bell Ringer
a) Write in scientific notation
• 8,705,000 m
• 0.0000045 L
• 0.00237 sec
• 9,300 g
b) Rewrite above numbers using the nearest SI prefix
c) Perform the below mathematics in Sci. Notation
• (9.01 x 103 g) + (3.8 x 102 g)
• (2.61 x 107 m) x (9.87 x 10-2 m)
• (3.98 x 10-2 m) – (8.2 x 10-3 m)
• (8.4 x 109 g) ÷ (2.0 x 104 mL)
Precision indicates to what degree we
know our measurement. (Arithmetic precision)
A measurement of 18.0 grams could be made on an
average countertop food scale (balance). (~$20)
A high-precision milligram scale could
weigh the same sample with a much higher
precision (18.0235 grams) (~$1,500)
Every measurement is limited by the
equipment’s level of precision. (Never exact)
e.g. mass of hydrogen atom:
0.0000000000000000000000017 grams
Significant Figures: Used to prevent uncertainty from rounding of
various measured quantities with various levels of precision.
1) Any digit that is not zero is significant
1.234 kg
34,000 mm
4 significant figures
2 significant figures
2) Zeros between nonzero digits are significant
606 cm
50,050 s
3 significant figures
4 significant figures
3) Zeros to the left of the first nonzero digit are not significant
0.08 mL
0.00054 ML
1 significant figure
2 significant figures
4) If a number is greater than 1, then all zeros to the right of the decimal point are
significant
2.0 mg
20.000 g
2 significant figures
5 significant figures
5) If a number is less than 1, then only the zeros at the end are significant
0.00420 g 3 significant figures
0.1000 g 4 significant figures
Significant Figures
Every significant figure is shown when using
Scientific notation.
____ m
0.001400
4 significant figures
1.400 x
-3
10
Not 1.4 x
-3
10
__ mL
500
2 significant figures
5.0 x
2
10
Not 5 x
2
10
Exampl 1.4 Unit Conversions
e
Determine the number of significant figures in the following
measurements:
(a) 478 cm
(d) 0.0430 kg
22
10
(b) 600,001 g
(e) 1.310 ×
(c) 0.85 m
(f) 7000 mL
atoms
Example
1.4 Solution
(a) 478 cm -- Three, because each digit is a nonzero digit.
(b) 600,001- Six, because zeros between nonzero digits are significant.
(c) 0.825 m -- Three, because zeros to the left of the first nonzero
digit do not count as significant figures.
(d) 0.0430 kg -- Three. The zero after the nonzero is significant
because the number is less than 1.
(e) 1.310 × 1022 atoms -- Four, because the number is greater than one
so all the zeros written to the right of the decimal point count as
significant figures.
Example
1.4 solution
(f)7000 mL -- This is an ambiguous case. The number of significant
figures may be four (7.000 × 103), three (7.00 × 103), two (7.0 × 103),
or one (7 × 103).
This example illustrates why scientific notation must be used to
show the proper number of significant figures.
If no decimal is present it is usually assumed only non-zeros are
significant. If a decimal is present, than all zero’s are significant.
7,000 mL ≠ 7,000. mL
They display differing degrees of precision.
Significant Figures
Addition or Subtraction
The answer cannot have more digits to the right of
the decimal point than any of the original numbers.
Use the least precise number.
89.392 L
+ 1.1XX
90.492
± 50 mL
one significant figure after decimal point
round off to 90.5
± 1.0 mL
3.70XX
-2.9133
0.7867
two significant figures after decimal point
round off to 0.79
Significant Figures
Multiplication or Division
The number of significant figures in the result is set by the original
number that has the smallest number of significant figures.
4.51 x 3.0006 = 13.532706 = 13.5
round to 3 sig figs
3 sig figs
6.8 ÷ 112.04 = 0.0606926 = 0.061
2 sig figs
round to 2 sig figs
Example
1.5
Carry out the following arithmetic operations to the correct number of
significant figures:
(a) 11,254.1 g + 0.1983 g
(b) 66.59 L − 3.113 L
(c) 8.16 m × 5.1355 kg
(d) 0.0154 kg ÷ 88.3 mL
(e) (2.64 × 103 cm) + (3.27 × 102 cm)
Example
1.5 Solution
Solution In addition and subtraction, the number of decimal places in
the answer is determined by the number having the lowest number of
decimal places.
(a)
(b)
Example
1.5 Solution
In multiplication and division, the significant number of the
answer is determined by the number having the smallest
number of significant figures.
(c)
(d)
(e) First we change 3.27 × 102 cm to 0.327 × 103 cm and then carry
out the addition (2.64 cm + 0.327 cm) × 103. Following the procedure
in (a), we find the answer is 2.97 × 103 cm.
Bell Ringer
a) Perform the below mathematics in Sci. Notation. using
Significant Figures in your answer.
1. (9.8 x 105 g) + (6.75 x 104 g)
2. (5.98 x 10-6 m) – (7 x 10-8 m)
b) Rewrite the first 2 solutions
using the nearest SI prefix
3. (2.612 x 1010 m) x (9.87 x 10-3 m)
4. (7 x 102 mg) ÷ (1.875 x 104 mL)
Significant Figures
Exact Numbers
Numbers from definitions or numbers of objects are
considered to have an infinite number of significant figures.
•The average of three measured lengths: 6.64, 6.68 and 6.70?
6.64 + 6.68 + 6.70
3
= 6.67333 = 7
= 6.673
Because 3 is an exact number, not a
measured number; It is not used for sigfigs.
• How many feet are in 6.82 yards?
6.82 yards x 3 ft/yard = 20.5 ft = 20 ft
1 yard = exactly 3 ft by definition
Accuracy – how close a measurement is to the
true value
Precision – how close a set of measurements are
to each other
accurate
&
precise
precise
but
not accurate
not accurate
&
not precise
Percent Error
A way to determine how accurate your measurements are
to a known value.
|Obtained value – Actual value| x 100%
Actual Value
Ranges between 0 and 100%
Ex. I weigh a 3 kg block on three different scales:
3.2 kg, 3.0 kg, 3.1 kg = 3.1 kg average
3.1 – 3.0 x 100% = 3.3% error
3.0
Dimensional Analysis of Solving Problems
(Train-Tracks)
1. Determine which unit conversion factors are needed
2. Carry units through calculation
3. If all units cancel except for the desired unit(s), then the
problem was solved correctly.
given quantity x conversion factor = desired quantity
given unit x
desired unit
given unit
= desired unit
Train Track Example
How many inches are in 3.0 miles?
Identify beginning information
Draw a train track
3 miles
Write measurement as a fraction
Train Track Example
How many inches are in 3.0 miles?
• We are going from a larger measurement to a smaller one.
• Find a conversion factor you know that changes miles into
something smaller.
Conversion Factor: 1 mile = 5,280 feet
• Write your conversion factor on the track so that miles
cancels out and you are left with the unit feet.
3 miles
5280 feet
1 mile
Always need same units on
opposite sides to cancel out
Train Track Example
How many inches are in 3.0 miles?
We now need another conversion factor between
Feet and Inches: 1 foot = 12 inches
Again, place conversion factor so that the previous unit
cancels out.
3.0 miles
5280 feet
1 mile
12 inches
1 foot
Train Track Unit Conversions
How many inches are in 3.0 miles?
3.0 miles
5,280 feet
1 mile
12 inches
1 foot
Inches are the only
remaining unit ✔
Multiply all numbers on the top
Divide all numbers on the bottom
3.0 x 5,280 x 12
1x1
= 190,080 inches
= 1.9 x 105 inches
(2 sig figs)
More practice:
Convert 1.40 x 10-6 g to mg
Example
Metric to Metric Conversion Problems
Convert
2.79 x 105 mm to km
Don’t try to convert directly from mm to km. Go to the base unit (m) first
Conversion factors: 1,000 mm = 1 m; 1,000 m = 1 km
2.79 x 105 mm
1m
103 mm
1 km
103 m
Kilometers are the only remaining units ✔
2.79 x 105 = 2.79 x 10(5-3-3)
103 x 103
= 2.79 x 10-1 km
(3 sig figs)
More practice:
Convert 3.4 x 103 cg to mg
Example
A person’s average daily intake of glucose (a form of sugar) is 0.0833
pound (lb). What is this mass in milligrams (mg)? (1 lb = 453.6 g.)
A metric conversion is needed to convert grams
to milligrams (1 mg = 1 × 10−3 g)
(Or we could write: 1,000 mg = 1 g)
Either Conversion factor will work
0.0833 lb
453.6 g
1 lb
103 mg
1g
= 37,784.88 mg
= 3.78 x 104 mg
(3 sig figs)
Example
2-D Conversion Problems (Unit1/Unit2)
Convert
70.0 miles/hour to m/s
We convert one unit at a time, followed by the other
Conversion factors: 1 mile = 1,609 meters; 1 hour = 60 min; 1 min = 60 sec
*Note: to cancel out hours (on bottom) it must appear again on the top
70.0 miles
1 hour
1609 meter
1 mile
1 hour
60 min
1 min
60 sec
Meter/sec are the only remaining units ✔
70.0 x 1,609 x 1 x 1
1 x 1 x 60 x 60
= 31.286 m/s
= 31.3 m/s
More practice:
Convert 3.4 kg/L to g/mL
2-D Conversion Problems (Unit#)
Example
Convert
2.5 x 10-5 m3 to mm3
Conversion factors: 1 m = 1,000 mm
1 m3 ≠ 1,000 mm3
1 m3 = (1,000)3 mm3 ✔
1. Write the 1-D units first (1m = 103m)
2. Add exponent to entire conversion factor
3
2.5 x 10-5 m3
1000 mm
1m
=
2.5 x 10-5 m3 109 mm3
1 m3
2.5 x 10-5 x 109 = 2.5 x 104 mm3
More practice:
Convert 34 yd2 to ft2
Example
Convert
2-D Conversion Problems
#
(Unit )
6.70 x 103 ft2 to inches2
Conversion factors: 1 ft = 12 in
1 ft2 ≠ 12 in2
1. Write the 1-D units first (1ft = 12 in)
1 ft2 = 122 in2✔
6.70 x 103 ft2
12 in.
1 ft
Alternate 2nd Method
2. Write the same conversion factor again
until they cancel
12 in.
1 ft
= 9.65 x 105 in2
More practice:
Convert 34 yd3 to ft3
Example
An average adult has 5.2 L of blood. What is the
3
volume of blood in m ?
1 mL = 1 cm3
5.2 L
103
mL
1L
1 cm3
1 mL
1 m3
1003 cm3
= 5.2 x
-3
10
3
m
Example
2-D Conversion Problem
The circumference of the earth is approximately 2.49 x 104 miles long.
If the speed of sound travels at 760 mph, how
many days would it take a sound wave to
circulate Earth’s circumference.
Starting with length and velocity; needing time
Conversion factor: 760 miles = 1 hour
2.49 x 104 miles
1 hour
760 miles
1 day
24 hours
= 1.4 days
More Practice
Conversion problems
• Convert 3.0 mL to
ounces (33.8 oz = 1 L)
• 42.0 km/h to ft/ms
• 1.67 Mm to mm
• 0.55 Acres to m2 (247
acre = 1 km2)
• 2.35 x 1012 inches to cm
(1 ft = 0.305 m)
• 10.6 g/mm3 to kg/m3
• 3.50 x 104 mL to cL
Review
Unit Conversion & Significant
Figures: Crash Course Chemistry #2
www.youtube.com/watch?v=hQpQ0hxVNTg
Sample of Topics to Study
• Metric base units
• Derived unit
• Using Prefixes
• Significant Figures (+ math)
• Scientific Notation (+ math)
• % Error calculation
•Accuracy
• Precision
• Dimensional Analysis