Coursework

1. Chapter 2
Measurements and Calculations
Section 2.3
In this section of the chapter, you will learn about:
• Significant Figures in the numbers
• Increment and Uncertainty in Measurement
• How to record a measurement to appropriate
significant figures
• Accuracy and Precision 1

2. Reporting Measurements
• Chemistry involves many measurements and
many types of calculations.
• When we report a measurement, scientist use a
system called significant figures.
• There are several rules for determining
significant figures in a measurement or a
calculation.
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3. Rules for Counting Significant Figures
• Nonzero integers (non zero digits) are always
significant, irrespective of there is a decimal
point or not, in the number.
• Example-1: How many significant figures are in the
number 238. There are 3 sig figs.
• Example-2: How many significant figures are in the
number 348658942? There are 9 sig figs.
• Example-3: How many significant figures are in the
number 3.385? There are 4 sig figs.
• Example-4: How many significant figures are in the
number 359.45? There are 5 sig figs.
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4. Rules for Counting Significant Figures
• Zeros: There are three classes of zeros.
 Captive zeros (middle zeros) are those that fall
between nonzero digits. They are always significant.
Example: The number 1.304 has 4 sig figs.
 Leading zeros (front zeros) are those that precede all
of the nonzero integers. These zeros never count as
significant figures irrespective of whether there is
decimal point or not.
Example: The number 0.0012 has only 2 significant
figures coming from the non-zero digits. Because the
leading zeros are not significant
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5. Rules for Counting Significant Figures
 Trailing zeros are those that are at the right end of the
number. They are significant only if the number is
written with a decimal point.
Example-1: The number 2.480 has a decimal point in
it. Therefore, the trailing zero become significant.
Hence this number has total of 4 sig figs.
Example-2: The number 28.0000 has a decimal point
in it. Therefore, the trailing zeros become significant.
Hence this number has total of 6 sig figs.
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6. Rules for Counting Significant Figures
 How to represent the significant numbers that have
Trailing zeros but no decimal point in the number?
Example: The number 1400 Does it mean it has only 2 significant
figures just because it does not have a decimal point. Can it have
other significant figures possibilities too? May be! So it is
‘ambiguous”.
So, how to represent the 2 significant figures, but without losing
the value? Convert it to scientific notations.
So, 1400 will be 1.4 x 103; So, it has 2 significant figures, but did
not lose its original value.
Or it can also be shown as 1.40 x 103 with 3 sig figs, but same value
or 1.400 x 103; with 4 sig figs, but same value
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7. Rules for Counting Significant Figures
Rules for Exact numbers:
• Exact numbers are those that are determined by
counting something, or arise from definitions.
They can be assumed to have an unlimited
number of significant figures!
• For examples: 25 students, 50 books (obtained
by counting).
• 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
(obtained by definitions)
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8. Counting Significant Figures: Examples
Summary of zeros: The number 0.001 has one significant
figure. Because leading or starting zeros are not significant
whether there is decimal point or not.
The number 0.01008 has four significant figures. Because
the two leading zeros are not significant; but the two
middle zeroes and the non-zero integers are significant).
The number 0.00060050 has five significant figures.
Because the leading zeros are Never significant; The
middle zeros are always significant. The end or trailing
zeros are significant because of decimal point being
present.
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9. Counting Significant Figures: Examples
• The number 100 has only one significant figure.
• The number 100. (with a decimal point at the end) has
three significant figures.
• The number 6.0 x 104 has two significant figures (when
expressed in scientific notation).
• A box of pencils was counted to be 55 pencils. This
number has unlimited significant numbers because this
number is obtained by counting.
• There are unlimited significant figures in the definition,
one foot = 12 inches.
• Practice more examples from the book.
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10. Interactive websites
The following websites will provide you with
interactive practice problems
• http://science.widener.edu/svb/tutorial/sigfigurescs
n7.html
• http://www.sciencegeek.net/APchemistry/APtaters/
chap02counting.htm
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11. Uncertainty in Measurements
• A measurement always has some amount of uncertainty.
• To understand how reliable a measurement is, we need to
understand the limitations of the measurement.
• Uncertainty comes from limitations of the techniques
and instruments used for measuring.
• A measurement consists of “certain (sure) numbers” and
“uncertain” part. The numbers recorded in a
measurement (all the certain numbers and the first
uncertain number) are called significant figures.
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12. Uncertainty in Measurements
• The uncertainty in the last number (the estimated
number) is usually assumed to be ±1 (along with
appropriate decimal places ) unless otherwise indicated
by the manufacturer.
• For example, if a weighing balance has uncertainty in
the second digit after the decimal point, and if an object
weighs 1.86 grams, it can be interpreted as 1.86 ±
0.01 grams.
• Because, it is the second digit after the decimal point,
that is uncertain.
• The ± sign Means, it could be either 1.85g or 1.87g
interval.
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13. Uncertainty in Measurements
• In a metric ruler, if the length measurement of an object
is 3.57cm, only the first two are certain, while the third
digit is uncertain.
• Therefore, it should be recorded as 3.57cm ± 0.01cm,
because the uncertainty is in the second digit after the
decimal point.
• You should always record the measurement with all
the significant figures, along with its uncertainty and
its units.
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14. Example: On the ruler on the left, observe the
difference in any two main divisions.
• It is 1 cm (because 6cm-5 cm= 1 cm)
• Next, this 1 cm difference is divided by
another 10 small subdivisions.
• Each small subdivision, is called as
Increment.
• How to calculate Increment?
Increment = Difference in main divisions
# of subdivisions
In our example:
Increment = 1 cm = 0.1cm
10
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15. Uncertainty: Any uncertainty Comes after the
increment in the next digit, because there are no
further markings within the subdivisions. And it
is another 10th of the increment
Hence Uncertainty = Increment
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In this case, Uncertainty = 0.1cm = 0.01cm
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Therefore, our reading for the mark shown in the
figure should be: 5.95cm  0.01 cm
Because the mark is in between 5.9 and 6.0; but
we are guessing that it could be 5.95. Someone
else might look at it and say it might be 5.94, or
5.97cm.
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16. • This is a volume
measuring device.
• Called graduated
cylinder.
• The units of its
measurement is mL
(milli liter) as shown
on the device
• The liquid level should
be read to lower
meniscus.
• Before you record the
correct value, first
calculate the
increment and
uncertainty.
t
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17. Increment = Difference in Main Divisions
# of subdivisions within the main division
= 5mL = 1mL
5 subdivisions
Therefore, each small division = increment = 1mL
t
Uncertainty=Increment=1mL=0.1mL
10 10
The lower meniscus of liquid is
therefore between 21 mL and 22 mL
• So, you could estimate as may be as
21.6 mL may be, because that last
number you are not sure whether it
is .5 or .6 or .7
• Hence it should be reported as:
21.6  0.1 mL
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18. Do you agree with the readings shown at
the top? How do you report these
measurements showing uncertainty?
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19. More Examples
• In the same way as we did before, calculate Increment and
uncertainty and check if the values shown above are correct, for
the blue colored line that is being measured in the 2 rulers.
• What is the correct way of recording the above readings?
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20. More Examples
• What is the reading in the circular meter shown above, for the
line with arrow? What is the correct way of writing it?
• What is the length of the blue object shown below the ruler?
What is the correct way of writing it?
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21. Lab experiments
• In this week’s lab experiment, it is about using
various measuring devices
• You should be able to calculate increments and
uncertainty
• And show the measured values to:
 correct significant figures
 Show the uncertainty
 Show the correct units
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22. Accuracy and Precision
Accuracy: It is how close is your measured value to the
correct or true value.
Or how good a measuring device is, with least amount of
errors, so it can give the true value of the measurement.
For example, if an actual true value of the mass of an
object is 3.57grams, does your weighing balance give you
correct 3.57 grams when you use it for weighing the
object? If not to what extent does it deviate?
Suppose the weighing balance showed the mass as 4.80
grams, instead of the true 3.57 grams, then, it is not
accurate. 22

23. Accuracy and Precision
Precision: If you take the same measurement several
times for a particular object, by using the same device, if
all the times the measurements match closely to each other,
then it is called precision, or the device is precise.
The values even if not accurate, are they precise?
For example, suppose an object gave the following values,
each time you measured it’s mass, using the same
weighing scale: 3.56, 3.55, 3.55, 3.57, 3.56grams.
They are fairly in agreement with each other, compared to
say another weighing scale, which might give the values
are 3.56, 3.94, 4.05, 3.24, 3.00 grams. This is not precise.
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24. Accuracy and Precision
Measurements, depending on the device, can be:
• Accurate, or
• Precise, or
• can neither be accurate nor precise,
• or can be both accurate and precise.
It is this last aspect (both precise and accurate) scientists
strive to achieve in designing the devices and experiments.
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25. Accuracy and Precision
In the above archery targets, or dartboard targets:
which one is precise but not accurate?
Which one is both precise and accurate?
which one is neither precise nor accurate?
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