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•Quantitative observation = measurement
Number
Unit
Both must be present for measurement to have meaning!!
•Two major systems: English system (used in the US) and
the metric system (used by most of rest of the world).
•Scientists worldwide use the metric system.
•In 1960, the International System of units was created and
is known as the SI system.
Physical Quantity
Name of Unit
Abbreviation
Mass
kilogram
kg
Length
meter
m
Time
second
s
Temperature
kelvin
K
Electric current
ampere
A
Amount of substance
mole
mol
Luminous intensity
candela
cd
•Prefixes are used to change the size of fundamental SI
units.
•Volume has the dimensions of (distance)3.
•Example: length x width x height
•Derived SI unit = cubic meter (m3).
•Traditional metric unit of volume is the liter (L).
•In SI terms, 1 L = 1 dm3.
•Normally we measure volumes in lab in units of milliliters
(mL).
1000 mL = 1L
Common types of laboratory equipment to
measure volume.
•Mass – the quantity of matter in an object
•Measured using a balance.
•Weight – the force exerted on an object by gravity.
•Measured using a spring scale.
•When a person uses a measuring device such as a buret or
a ruler, there is always uncertainty in the measurement.
•In other words, the last digit in a measurement is always
estimated.
•The uncertainty of a measurement depends on the
precision of the measuring device.
•Consider the measurement of volume
from the buret.
•Suppose 5 different people read this
buret and the following measurements
are obtained.
Person
1
2
3
4
5
Measurement
20.15 mL
20.14 mL
20.16 mL
20.17 mL
20.16 mL
•The first 3 digits are all the same;
these are digits read with certainty.
•The last digit is estimated and is called
the uncertain digit.
•Consider the two
centimeter rulers at the
left.
•Each ruler is measuring
the same pencil.
•The best measurement obtained from the first ruler would be 9.5 cm.
•The best measurement obtained from the second ruler would be 9.51
cm.
•Why two different measurements?
•The second measurement is more precise, because you used a
smaller unit to measure with.
•Digits that result from measurement such that only the digit farthest
to the right is not known with certainty are called significant figures.
•Precision – the degree of agreement among general
measurements of the same quantity (how reproducible your
measurements are).
•Accuracy – the agreement of a particular value with the
true value.
•Random errors mean that a measurement has an equal
probability of being high or low. Occurs in estimating the
value of the uncertain digit.
•Systematic errors occur in the same direction each time;
they are always high or always low.
•Figure (a) indicates large random errors (poor technique).
•Figure (b) indicates small random errors but a large
systematic error.
•Figure (c) indicates small random errors and no systematic
error.
•Each of 4 general chemistry students measured the mass of a chemistry
textbook. They each weighed the book 4 times. Knowing that the true
mass is 2.31 kg, which student weighed the book:
a. accurately and precisely
b. inaccurately but precisely
c. accurately but imprecisely
d. inaccurately and imprecisely
Weighing
1
2
3
4
Average:
student #1
2.38 kg
2.23 kg
2.07 kg
2.55 kg
2.31±0.16 kg
student #2
2.06 kg
1.94 kg
2.09 kg
2.40 kg
2.12±0.14 kg
student #3
2.32 kg
2.30 kg
2.31 kg
2.32 kg
2.31±0.01 kg
student #4
2.71 kg
2.63 kg
2.66 kg
2.68 kg
2.67±0.03kg
Solution:
You have to ask yourself two questions about each data set:
1. Is the average close to the accepted (true) value? If it is,
then the result is accurate.
2. Is the average deviation small relative to the actual
value? If it is, then the result is precise.
Student #1
2.38 kg
2.23 kg
2.07 kg
2.55 kg
2.31±0.16 kg
Student #2
2.06 kg
1.94 kg
2.09 kg
2.40 kg
2.12±0.14 kg
Student #3
2.32 kg
2.30 kg
2.31 kg
2.32 kg
2.31±0.01 kg
Student #1 – accurate but imprecise
Student #2 – inaccurate and imprecise
Student #3 – accurate and precise
Student #4 – inaccurate but precise
Student #4
2.71 kg
2.63 kg
2.66 kg
2.68 kg
2.67±0.03 kg
•When taking measurements all certain digits plus the
uncertain digit are significant.
•Example: Your bathroom scale weighs in 10 Newton
increments and when you step onto it, the pointer stops
between 550 and 560. You look at the scale and determine
your weight to 557 N. You are certain of the first two
places, 55, but not the last place 7. The last place is a guess
and if it is your best guess, it also is significant.
•When given measurements, the numbers that are significant are the
digits 1 – 9, and the 0 when it is not merely a place holder.
1. When 0’s are between significant figures, 0’s are always significant.
Example: 101 has 3 sig. fig. and 34055 has 5 sig. fig.
2. When the measurement is a whole number ending with 0’s, the 0’s
are never significant.
Example: 210 has 2 sig. fig. and 71,000,000 also has 2 sig. fig.
Removal of the 0’s DO change the value (size) of the
measurement, but the 0’s are place holders and are thus not
significant.
3. When the measurement is less than a whole number, the 0’s
between the decimal and other significant figures are never
significant (they are place holders).
Example: 0.0021 has 2 sig. fig. and 0.0000332 has 3 sig. fig.
Removal of the 0’s DO change the value (size) of the measurement, the 0’s are
place holders and are thus not significant.
4. When the measurement is less than a whole number and the 0’s
fall after the other significant numbers, the 0’s are always
significant.
Example: 0.310 has 3 sig. fig. and 0.3400 has 4 sig. fig.
The 0’s have no effect on the value (size) of the measurement. Therefore, these
0’s must have been included for another reason and that reason is to show
precision of the measurement. Since these 0’s show precision they must be
significant.
5. When the measurement is less than a whole number, and there is
a 0 to the left of the decimal, the 0 is not significant.
Example: 0.02 has only 1 sig. fig. and 0.110 has 3 sig. fig.
The 0 to the left of the decimal is only for clarity, it is neither a
place holder nor adds to the accuracy of the measurement.
6. When the measurement is a whole number but ends with 0’s to
the right of the decimal, the 0’s are significant.
Example: 20.0 has 3 sig. fig. and 18876.000 has 8 sig. fig.
The 0’s have no effect on the value (size) of the measurement.
Therefore, these 0’s must have been included for another reason
and that reason is to show precision of the measurement. Since
these 0’s show precision they must therefore be significant.
1. For addition or subtraction, the result has the same number of
decimal places as the least precise measurement (the
measurement with the least number of decimal places) used in the
calculation.
Example:
12.011
Limiting term has one decimal place
18.0
1.013
________
Corrected 31.1
31.123
One decimal place
2. When adding or subtracting numbers written with the ± notation,
always add the ± uncertainties and then round off the ± value to
the largest significant digit. Round off the answer to match.
Example: (22.4 ± 0.5) + (14.76 ± 0.25) = 37.16 ± 0.75 = 37.2 ± 0.8
The uncertainty begins in the tenths place… it is the last significant
digit.
3. For multiplication or division, the number of significant figures in
the result is the same as the number in the least precise
measurement (the measurement with the least number of
significant figures) used in the calculation.
Example:
4.56 x 1.4 = 6.38
Corrected
Limiting term has
two significant figures
6.4
Two significant
figures
Example: How should the result of the following calculation be
expressed?
322.45 x 12.75 x 3.92 = 16116.051
= 16100
(3 sig. fig.)
1. In a series of calculations, carry the extra digits through to the final
result, then round.
2. If the digit to be removed
a. is less than 5, the preceding digit stays the same. For example,
1.33 rounds to 1.3.
b. is equal to or greater than 5, the preceding digit is increased by
1. For example, 1.36 rounds to 1.4.
•You will often have to solve problems where there is a
combination of mathematical operations. To get reasonable
answers, you need to recall your order of operations, given
the following table:
Order of Operations
1. parentheses
2. exponents and logs
3. multiplication and division
4. addition and subtraction
•Review Appendix A1.1 and handout.
•Dimensional analysis (or unit factor method) – used to
convert from one system of units to another.
•In dimensional analysis we treat a numerical problem as
one involving a conversion of units from one kind to
another.
•To do this we need one or more conversion factors (unit
factors) to change the units of the given quantity to the
units of the answer.
•A conversion factor is a fraction formed from a valid
relationship or equality between units that is used to switch
from one system of measurement and units to another.
 given   conversion   desired 

x


 quantity  factor
  quantity
•For example, suppose we want to express a person’s
height of 72.0 in. in centimeters.
•We need relationship or equality between inches and
centimeters.
2.54 cm = 1 in. (exactly)
•If we divide both sides by 1 in., we obtain a conversion factor.
2.54 cm 1 in.

1
1 in.
1 in.
•Notice we have canceled the units from both the numerator and the
denominator of the center fraction.
•Units behave just as numbers do in mathematical operations, which
is a key part of dimensional analysis.
•Now let’s multiply 72.0 in. by this fraction:
3 sig. fig.
2.54 cm
72.0 in. x
 183 cm
1 in.
3 sig. fig.
 given   conversion   desired 

x


 quantity   factor
  quantity 
•Because we have multiplied 72.0 in. by something that is
equal to 1, we know we haven’t changed the magnitude of
the person’s height.
•We have changed the units (notice that inches cancel
leaving centimeters which is what we wanted).
•Notice that our given quantity and desired quantity have
the same number of significant figures (conversion factors
are considered exact numbers and do not dictate significant
figures).
•One of the benefits of dimensional analysis is that it lets
you know when you have done the wrong arithmetic.
•From the relationship
2.54 cm = 1 in.
we can construct two conversion factors.
2.54 cm
1 in.
and
1 in.
2.54 cm
•In the previous problem what if we used the incorrect
conversion factor? Would it make a difference in our
answer?
1 in.
2
72.0 in. x
2.54 cm
 28.3 in. / cm
•None of our units cancel.
•In this case we get the wrong units!!!!
•Can we use more than one conversion factor in a single problem?
•Example:
In 1975, the world record for the long jump was 29.21 ft. What is this
distance in meters?
Solution: We can state this problem as
29.21 ft = ? m
One of several sets of relationships we can use is
1 ft = 12 in.
1 in. = 2.54 cm
100 cm = 1 m
Notice they take us from inches to centimeters to meters.
12 in. 2.54 cm
1m
29.21 ft x
x
x
 8.903 m
1 ft
1 in.
100 cm
•Notice if we had stopped after the first conversion factor, the units of
the answer would be inches; if we stop after the second, the units
would be centimeters, and after the third we get meters – the units
we want.
•Note: this is not the only way we could have solved this problem.
Other conversion factors could have been chosen.
•Important: you should be able to reason your way through a
problem and find some set of relationships that can take you from
the given information to the answer.
Three systems:
•Note the size of the temperature
unit (the degree) is the same for
the Kelvin and Celsius scales.
•The difference is in their zero
points.
•To convert between Kelvin
and Celsius scales add or
subtract 273.15.
Temperature (Kelvin) = temperature (Celsius) + 273.15
Temperature (Celsius) = temperature (Kelvin) – 273.15
Unit for Celsius = oC
Unit for Kelvin = K
No degree
symbol
•To convert between the Celsius
and Fahrenheit scales two
adjustments must be made:
(1) degree size
(2) zero point
Degree size
180o F = 100oC
Conversion factor =
o
o
180 F
9 F
or o
o
100 C
5 C
or the reciprocal
Zero Point
32o F = 0oC
Subtract or add 32 when converting
5o C
TC  (TF  32 F) o
9 F
9o F
TF  TC x o  32o F
5 C
o
•Property of matter used by chemists as an “identification
tag”.
•Density = mass of substance per unit volume of the
substance
mass
Density 
volume
•Each pure substance has a characteristic density.
•Density can also be used to convert between mass and
volume.
•Be able to manipulate the formula for density.
mass
Density 
volume
•If given the density and mass of a substance how could you
mass
determine volume?
Volume 
density
•If given the density and volume of a substance how could
Mass  density x volume
you determine mass?
•Remember!! A material will float on the surface of a liquid if
the material has a density less than that of the liquid.