Measurements and
Calculations Notes
Chapter 3
I. Scientific Method

The process researchers use to carry
out their investigations. It is a logical
approach to solving problems.
A. Steps
1.
2.
3.
4.
Ask a question
Observe and collect data
Formulate a hypothesis (a testable if-then
statement). The hypothesis serves as a basis
for making predictions and for carrying out
further experiments.
Test your hypothesis – Requires
experimentation that provides data to support
or refute your hypothesis.
B. Terms to Know
1.


Law vs. theory
Scientific (natural) Law: a general
statement based on the observed
behavior of matter to which no exceptions
are known.
Theory: a broad generalization that
explains a body of facts or phenomena.
1. Quantitative vs. qualitative data

Quantitative: numerical (mass, density)
Quantity - number + unit

Qualitative: descriptive (color, shape)
II. SI (System of International)
Units of Measurements
Adopted in 1960 by the General Conference
on Weights and Measures.
A. Metric System – must know this
 Mass is measured in kilograms (other mass
units: grams, milligrams)
 Volume in liters
 Length in meters

B. Prefixes are added to the stem or base unit to represent
quantities that are larger or smaller then the stem or base unit.
You must know the following:
Prefix
Value
Abbreviation
Pico
10-12
0.000000000001
Nano
10-9
0.000000001
Micro
10-6
0.000001
Milli
10-3
0.001
Centi
10-2
0.01
Deci
10-1
0.1
(stem: liter, meter, gram)
Deka
101
10
Hecto
102
100
Kilo
103
1000
Mega
106
1000000
Ex
p
n

m
c
d
pg
nm
g
mm
cl
dg
da
h
k
M
dal
hm
kg
Mm
Examples:
1Mm=1,000,000m
1km=1000m
1hm=100m
1dam=10m
1m=1m
1dm=0.1m
1cm=0.01m
1mm=0.001m
1μm=0.000001m
When solving
problems I will
always “put a 1
with the prefix.”
Starting from the largest value, mega, to the
smallest value, pico, a way to remember the
correct order is:











Miss (Mega)
Kathy (Kilo)
Hall (Hecto)
Drinks (Deka)
Gatorade, Milk, and Lemonade (Gram, Meter, Liter)
During (Deci)
Class on (Centi)
Monday (Milli)
Morning and (Micro)
Never (Nano)
Peed (Pico)

C. Derived Units: combinations of
quantities: area (m2), Density (g/cm3),
Volume (cm3 or mL) 1cm3 = 1mL
D. Temperature- Be able to convert between
degrees Celcius and Kelvin.
Absolute zero is 0 K, a temperature where all molecular
motion ceases to exist. Has not yet been attained, but
scientists are within thousandths of a degree of 0 K. No
degree sign is used for Kelvin temperatures.
Celcius to Kelvin: K = C + 273
Convert 98 ° C to Kelvin: 98° C + 273 = 371 K

Ex: New materials can act as superconductors at
temperatures above 250 K. Convert 250 K to degrees
Fahrenheit. (9/5C + 32 = F) or (5/9(F - 32)=C)
III. Density – relationship of mass to
volume D = m/V Density is a derived
unit (from both mass and volume)

For solids: D = grams/cm3
Liquids: D = grams/mL
Gases: D = grams/liter

Know these units


Density is a conversion factor. Water has a
density of 1g/mL which means 1g=1mL!!
When calculating density, 1) find out how many
grams you have, 2) determine the total volume
your mass occupies, divide the two numbers,
and report your answer in the correct units.
Example Problems:
 1. An unknown metal having a mass of 287.8 g
was added to a graduated cylinder that
contained 31.47 mL of water. After the addition
of the metal, the water level rose to 58.85 mL.
Calculate the density of the metal.




2. The density of mercury is 13.6 g/L. How
many grams would l.00 liter of mercury weigh?
3. A solid with dimensions of 3.0 cm X 4.0 cm X
2.0 cm has a mass of 28 g. Will this solid float in
water? (water has a density of l.00 g/ml)
4. A gas has a density of 0.824 g/L and
occupies a volume of 3.00 liters. How much
does the gas weigh?
I LOVE DIMENSIONAL
ANALYSIS!

IV. Dimensional Analysis - When you finish
this section, you will be able to: convert
between English and metric units; convert
values from one prefix to another.
Dimensional analysis is the single most valuable
mathematical technique that you will use in general
chemistry. The method involves using conversion
factors to cancel units until you have the proper unit
in the proper place. A conversion factor is a ratio of
equivalent measurements, so a conversion factor is
equal to one. For example, how many quarters are in
$5.85? You know there are 4 quarters in one dollar.
4 quarters = $1.00 is a conversion factor.


$5.85
4 quarters = 23.40 quarters
$1.00
Notice that the dollar sign cancels out and
your answer is in “quarters.”
When you are setting up problems using
dimensional analysis, you are more
concerned with units than with numbers.
Let’s illustrate this by finding out the mass
of a 125 pound box (1 kg = 2.2046 pounds).
PROBLEM SOLVING STEPS
1. List the relevant conversion factors
2. Set up the problem as follows*
125 pounds
1 kg
2.2046 pounds
= 56.7 kg

3. Multiply all the values in the numerator
and divide by all those in the denominator.

4. Double check that your units cancel
properly. If they do, your numerical
answer is probably correct. if they don’t,
your answer is certainly wrong.
UNITS ARE THE KEY TO
PROBLEM SOLVING
Practice with Dimensional Analysis
1. It takes exactly one egg to make 8
pancakes, including other ingredients.
A pancake eating contest was held at
which the winner ate 74 pancakes in 6
minutes. At this rate, how many eggs
(in the pancake) would be eaten by the
winner in 1.0 hour?

Following our problem solving procedure,
Step 1. Conversion Factors:
1 egg = 8 pancakes. Keep in mind that this is exactly the
same as 8 pancakes = 1 egg. You can therefore either use
1 egg/ 8 pancakes or 8 pancakes/ 1 egg
However, it is NOT CORRECT to use 8 eggs/1 pancake or 1
pancake/ 8 eggs
When you flip units, the numbers must flip with them.
b. Although it is not stated in the problem, you need a
conversion factor from minutes to hours. 60 minutes/
1 hour or 1 hour /60 minutes
c. 74 pancakes per 6 minutes can be expressed as 74
pancakes/ 6 minutes or 6 minutes/ 74 pancakes
Step 2. Setting up the Problem

Dimensional analysis will often involve
converting between prefixes of the same
unit. You must be very careful to think
about if your final value makes sense.
Complete the following using dimensional
analysis:
1.
a.
b.
c.
d.
Convert the following metric units:
42 µm to m
62.9 kg to g
49.8 mL to L
33.9 pm to m
2. Convert the following units:
a. 7.51 miles o meters
b. 38 feet to cm


3. Your heart pumps 2,000 gallons of blood per
day. How long (in years) would your heart have
been pumping if it pumped 1,500,000 gallons of
blood?
4. Eggs are shipped from a poultry farm in
trucks. The eggs are packed in cartons of one
dozen eggs each; the cartons are placed in
crates that hold 20.cartons each. The crates are
stacked in the trucks, 5 crates across, 25 crates
deep, and 25 crates high. How many eggs are
in 5.0 truckloads?
V. Using Scientific Measurements
ACCURATE = CORRECT
PRECISE = CONSISTENT
A. Precision and Accuracy
1. Precision – the closeness of a set of
measurements of the same quantities made in
the same way (how well repeated
measurements of a value agree with one
another).
2. Accuracy – is determined by the agreement
between the measured quantity and the correct
value.
Ex: Throwing Darts
B. Percent Error-is calculated by
subtracting the experimental value from
the accepted value, then dividing the
difference by the accepted value. Multiply
this number by 100. Accuracy can be
compared quantitatively with the accepted
value using percent error.
Percent error =
Accepted value - Experimental value X 100
Accepted value
C. Counting Significant Figures


When you report a measured value, it is
assumed that all the figures are correct except
for the last one, where there is an uncertainty of
±1. If your value is expressed in proper
exponential notation, all of the figures in the preexponential value are significant, with the last
digit being the least significant figure (LSF).
“7.143 x 10-3 grams” contains 4 significant
figures
If that value is expressed as 0.007143, it still has 4
significant figures. Zeros, in this case, are
placeholders. If you are ever in doubt about the
number of significant figures in a value, write it in
exponential notation.
Example of nail on page 46: the nail is 6.36cm
long. The 6.3 are certain values and the final 6
is uncertain! There are 3 significant figures in
6.36cm (2 certain and 1 uncertain). The reader
can see that the 6.3 are certain values because
they appear on the ruler, but the reader has to
estimate the final 6.
Significant Figures

Indicate precision of a measurement.

Recording Significant Figures (sig figs)

Sig figs in a measurement include the known
digits plus a final estimated digit
2.35 cm
The rules for counting
significant figures are:
1. Leading zeros do not count.
2. Captive zeros always count.
3. Trailing zeros count only if there is a
decimal.
Give the number of significant
figures in the following values:
a. 38.4703 mL
c. 0.05700 s
b. 0.00052 g
d. 6.19 x 101 years

Helpful Hint :Convert to exponential form if you
are not certain as to the proper number of
significant figures.

A very important idea is that you DO NOT
ROUND OFF YOUR ANSWER UNTIL THE
VERY END OF THE PROBLEM.
D. Significant Figures in
Calculations

Perform the indicated calculations on the
following measured values, giving the final
answer with the correct number of
significant figures.
1. In addition and subtraction, your answer should
have the same number of decimal places as the
measurement with the least number of decimal
places. EX: find the answer for 12.734-3.0
Solution: 12.734 has 3 figures past the decimal
point. 3.0 has only 1 figure past the decimal
point. Therefore, your final result, where only
addition or subtraction is involved, should round
off to one figure past the decimal point.
12.734
- 3.0
9.734 -------- 9.7
Add/Subtract – additional example
3.75 mL
+ 4.1 mL
7.85 mL  7.9 mL



a.
b.
c.
32.3 – 25.993
84 + 34.99
43.222 – 38.12834
2. In multiplication and division, your
answer should have the same number of
significant figures as the least precise
measurement.
61 x 0.00745 = 0.45445 = 0.45 2SF
 a. 32 x 0.00003987
 b. 5 x 1.882
 c. 47. 8823 X 9.322
3. There is no uncertainty in a
conversion factor; therefore they do not
affect the degree of certainty of your
answer. The answer should have the
same number of SF as the initial value.
a. Convert 25. meters to millimeters.

b. Convert 0.12 L to mL.
E. Use of the Calculator in Chemistry. You need to
know how to enter the following functions on your
calculator:
A. exp or ee button – to enter scientific notation
B. yx or xy button – to enter powers of numbers
C. take the square root of a number
We’ll practice!!!
F. Scientific Notation-used to express very
large or very small numbers
1 X 10-2
 Convert to scientific notation:
a. 1760
b. 0.00135
c. 10.2
d. –0.00000673
e. 301.0
f. 0.000000532
Convert to scientific notation:
a. 1760
b. 0.00135
c. 10.2
Expand each number (or convert to
regular notation):
a. 4.78 x l02
c. –9.3 x l0-3
b. 5.50 x l04
G. Correct scientific notation only has
ONE number (1-9) on the left side of the
decimal. Correct these:
a. 36.7 x l0-2
b. 0.l5 x l0-3
c. 176.4 x l0-1
H. Real World Connections :
 Information
from the website
“Medication Math for the
Nursing Student” at
http://www.alysion.org/dime
nsional/analysis.htm#proble
ms
A shocking number of patients die every year in United
States hospitals as the result of medication errors, and
many more are harmed. One widely cited estimate
(Institute of Medicine, 2000) places the toll at 44,000
to 98,000 deaths, making death by medication
"misadventure" greater than all highway accidents,
breast cancer, or AIDS. If this estimate is in the
ballpark, then nurses (and patients) beware:
Medication errors are the forth to sixth leading cause
of death in America.
Actual problems encountered in nursing
practice (others posted on website):
 You
are to give "grain 5 FeSO4"
but the available bottle gives only
the milligrams of iron sulfate per
tablet (325 mg/tab). How many
milligrams is the order for?