Experiment-2: Scientific Notation and Measurements

Significant figures

Rounding off

Measurements of Length, Mass, Volume, and Temperature.

Unit Conversions.

Learn to use: balances, metric rulers, graduated cylinders, and thermometers.
Measurements in science can be divided into two broad categories: exact and inexact.
Some numbers, by their nature, are exact. When we say that we have 3 pencils or 2
sisters or that there are 5 beakers in the drawer, we indicate the exact number of those
items. Similarly, in the conversion factors:
1 ft = 12 inch
1 liter = 1000 ml
1 dozen = 12 units
numbers 12, 1000, and 12 are exact. By definition, there are exactly 12 inches in 1
foot, precisely 1000 milliliters in 1 liter, and exact 12 eggs per dozen.
Inexact Numbers: All other numbers are inexact. A "100 ml beaker" may hold
anywhere from 90 ml to 110 ml. Physical measurements are always inexact. The
number obtained in the experiment will depend upon the skill of the experimenter and
the sensitivity of the equipment used.
ACCURACY and PRECISION
Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.
Accurate
(the average is accurate)
not precise
Precise
not accurate
Accurate
and
precise
It is important to be scientific when reporting a measurement, so that it does not appear
to be more accurate than the equipment used to make the measurement allows. You can
achieve this by controlling the number of digits, or significant figures, used to report
the measurement. The number of significant figures in a measurement, such as 2.541 g,
is equal to the number of digits that are known with some degree of confidence (2, 5,
and 3) plus the last digit (1), which is an estimate or approximation. As we improve the
sensitivity of the equipment used to make a measurement, the number of significant
figures increases.
Postage Scale
3g
1 significant figure
 1 g precision
Open-pan balance
2.54 g
3 significant figure
 0.01 g precision
Analytical balance
2.541 g
4 significant figure
 0.001 g precision
In any measurement, the number of significant figures is critical. It includes one
estimated digit. For example: measuring volume of water in the laboratory using a:
o
beaker with volumes marked on the side,
o
graduated cylinder, or
o
buret.
Which glassware would give you the most precise volume measurement?
Beaker
The smallest division is 10 mL, so one can read the
volume to 1/10 of 10 mL or 1 mL. The volume
one reads from the beaker has a reading error of 1
mL. The volume in this beaker is 47 1 mL. You
might have read 46 mL; your friend might read the
volume as 48 mL. All the answers are correct within
the reading error of 1 mL. So, the volume of 47 1
mL has 2 significant figures. The "4" you know for
sure plus the "7" you had to estimate.
Graduated
First, note that the surface of the liquid is curved. It
Cylinder
is called the meniscus. You read the volume at the
BOTTOM of the meniscus. The smallest division of
this graduated cylinder is 1 mL. Therefore, the
reading error will be 0.1 mL or 1/10 of the smallest
division. An appropriate reading of the volume is
36.5 0.1 mL. An equally precise value would be
36.6 mL or 36.4 mL. Your measurement has 3
significant figures. The "3" and the "6" you know
for sure and the "5" you had to estimate.
Buret
The smallest division in this buret is 0.1 mL.
Therefore, your reading error is 0.01 mL. A good
volume reading is 20.38 0.01 mL. An equally
precise acceptable answer would be 20.39 mL or
20.37 mL. The answer has 4 significat figures. The
"2", "0", and "3" you definitely know and the "8"
you had to estimate.
The higher the number of significant figures in a measurement, the better the precision.
It is obviously important, then, to know how many significant figures a measured
number has. Rules for deciding the number of significant figures in a measured
quantity are:
(1) All nonzero digits are significant:
1.234 g has 4 significant figures;
1.2 g has 2 significant figures.
(2) Zeroes between nonzero digits are significant:
1002 kg has 4 significant figures;
3.07 mL has 3 significant figures.
(3) Zeroes to the left of the first nonzero digits are not significant; such zeroes merely
indicate the position of the decimal point:
0.001 oC has only 1 significant figure;
0.012 g has 2 significant figures.
(4) Zeroes to the right of a decimal point in a number are significant:
0.023 mL has 2 significant figures;
0.200 g has 3 significant figures.
(5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes
are not necessarily significant:
190 miles may be 2 or 3 significant figures;
50,600 calories may be 3, 4, or 5 significant figures.
The potential ambiguity in the last rule can be avoided by the use of standard
exponential, or "scientific," notation. For example, depending on whether 3, 4, or 5
significant figures is correct, we could write 50,6000 calories as:
5.06 × 104 calories (3 significant figures)
5.060 × 104 calories (4 significant figures), or
5.0600 × 104 calories (5 significant figures).
Rules for mathematical operations:
In carrying out calculations, the general rule is that the accuracy of a calculated result is
limited by the least accurate measurement involved in the calculation.
Addition and subtraction: when the quantities are added or subtracted, the number of
decimal places in the answer is equal to the number of decimal places in the quantity
with the smallest number of decimal places. For example, 1.234 + 567.89
1.234 has three digits right of the decimal point; 567.89 has two. The result will have
the smaller of these - two digits right of the decimal point.
This is easier to see if you line up the figures in a column:
1.234
+ 567.89
Your calculator produces 569.124 as a result; round it to two significant digits
right of the decimal point and report 569.12.
Multiplication and division: the result should be rounded off so as to have the
same number of significant figures as in the component with the least number of
significant figures. For example,
3.0 (2 significant figures) × 12.60 (4 significant figures) = 37.8000, which should
be rounded off to 38 (2 significant figures).
Rules for rounding off numbers
(1) If the digit to be dropped is greater than 5, the last retained digit is increased by one.
For example, 12.6 is rounded to 13.
(2) If the digit to be dropped is less than 5, the last remaining digit is left as it is. For
example, 12.4 is rounded to 12.
(3) If the digit to be dropped is 5, and if any digit following it is not zero, the last
remaining digit is increased by one. For example, 12.51 is rounded to 13.
(4) If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit
is increased by one if it is odd, but left as it is if even. For example, 11.5 is rounded to
12, 12.5 is rounded to 12.
Addition
Even though your calculator gives you the answer
8.0372, you must round off to 8.04. Your answer
must only contain 1 doubtful number. Note that the
doubtful digits are underlined.
Subtraction
Subtraction is interesting when concerned with
significant figures. Even though both numbers
involved in the subtraction have 5 significant figures,
the answer only has 3 significant figures when
rounded correctly. Remember, the answer must only
have 1 doubtful digit.
Multiplication
The answer must be rounded off to 2 significant
figures, since 1.6 only has 2 significant figures.
Division
The answer must be rounded off to 3 significant
figures, since 45.2 has only 3 significant figures.
More examples:
1. 37.76 + 3.907 + 226.4 = 268.1
7.
0.0032 × 273 = 0.87
2. 319.15 - 32.614 = 286.54
8.
(5.5)3 = 1.7 x 102
3. 104.630 + 27.08362 + 0.61 = 132.32
9.
0.556 × (40 - 32.5) = 4
4. 125 - 0.23 + 4.109 = 129
10. 45 × 3.00 = 1.4 x 102
5. 2.02 × 2.5 = 5.0
6. 600.0 / 5.2302 = 114.7
11. Average of 0.1707, 0.1713, 0.1720,
0.1704, and 0.1715 is = 0.1712
Summary of Rules of Significant Figures:
Multiplication and Division: Your answer should have the same number of significant
figures as the number with the least number of significant figures.
Addition and Subtraction: Your answer should be reported to the decimal position of
least significance in the number involved in the calculation
Scientific notation
Scientific notation also referred to as exponential notation. The notation is based on
powers of base number 10. The general format looks something like this:
N X 10x where N= number greater than 1 but less than 10 and x = exponent of 10.
Placing numbers in exponential notation has several advantages. For very large numbers
and extremely small ones, these numbers can be placed in scientific notation in order to
express them in a more concise form. In addition, numbers placed in this notation can be
used in a computation with far greater ease. This last advantage was more practical
before the advent of calculators and their abundance.
Numbers Greater Than 10
Locate the decimal and move it either right or left so that there are only one non-zero
digit to its left. The resulting placement of the decimal will produce the N part of the
standard scientific notational expression. Convert 23419 into standard scientific notation?
Position the decimal so that there is only one non-zero digit to its left. In this case we end
up with 2.3419
Count the number of positions we had to move the decimal to the left and that will be x.
So we have: 2.3419 X 10 4
Numbers less than one
We generally follow the same steps except in order to position the decimal with only one
non-zero decimal to its left, we will have to move it to the RIGHT. The number of
positions that we had to move it to the right will be equal to -x. In other words we will
end up with a negative exponent.
Negative exponents can be rewritten as values with positive exponents by taking the
inversion of the number. For example: 10-5 can be rewritten as 1/ 105
Express 0.000436 in scientific notation:
First, we will have to move the decimal to the right in order to satisfy the condition of
having one non-zero digit to the left of the decimal. That will give us: 4.36
Then we count the number of positions that we had to move it, which was 4. That will
equal -X or x = -4. And the expression will be 4.36 X 10-4
Numbers between 1 and 10
In those numbers we do not need to move the decimal so the exponent will be zero. For
example: 7.92 can be rewritten in notational form as: 7.92 X 100
UNITS
By the 18th century, dozens of different units of measurement were commonly used
throughout the world. Length, for example, could be measured in feet, inches, miles,
spans, cubits, and more. The lack of common standards led to a lot of confusion and
significant inefficiencies in trade between countries. At the end of the century, the
French government sought to alleviate this problem by devising a system of
measurement that could be used throughout the world. In 1790, the French National
Assembly commissioned the Academy of Science to design a simple, decimal-based
system of units; the system they devised is known as the metric system. In 1960, the
metric system was officially named the Système International d'Unités (or SI for short)
and is now used in nearly every country in the world except for the United States. The
metric system is almost always used in scientific measurement.
The simplicity of the metric system stems from the fact that there is only one unit of
measurement (or base unit) for each type of quantity measured (length, mass, etc.). The
three most common base units in the metric system are the meter, gram and liter. The
meter is a unit of length equal to 3.28 feet; the gram is a unit of mass equal to
approximately 0.0022 pounds (about the mass of a paper clip); and the liter is a unit of
volume equal to 1.05 quarts. So length, for example, is always measured in meters in
the metric system, regardless of whether you are measuring the length of your finger or
the length of the Nile River - you use the meter. To simplify things, very large and very
small objects are expressed as multiples of 10 of the base unit. For example, rather than
saying that the Nile river is 6,650,000 meters long, we can say that it is 6,650 thousandmeters long. This would be done by adding the prefix 'kilo' (meaning 1,000) to the base
unit 'meter' to give us 6,650 kilometers for the length of the Nile River. This is much
simpler than the American system of measurement in which we have to remember
inches, feet, miles, and many more units of measurement. Metric prefixes can be used
with any base unit. For example, while a kilometer is 1,000 meters, a kilogram is 1,000
grams and a kiloliter is 1,000 liters. Some important relationships are:
Length
Mass
Volume
10 mm = 1 cm
1g
= 1000 mg
1L
= 1000 mL
10 cm = 1 dm
1 kg
= 1000g
1 dL
= 100 mL
100 cm = 1 m
1 metric ton = 1000 kg
1 mL = 1 cm3 = 1cc
1000 m = 1 km
1 lb.
= 453.59 g
1 gallon = 3.7854 L
1 inch = 2.54 cm exact
1 lb
= 16 oz
1 gallon = 4 quarts= 8 pints
12 in. = 1 foot
1 oz
= 28.35 g
1 pint = 16 fl oz
3ft.
1 amu = 1.66 x 10 –24 g 1 fl oz. = 29.57 mL
= 1 yard
1 mile = 5280 feet = 1.609 km 1  g = 1 x 10 –6 g
1 quart = 946.3 mL
Prefix Symbol Value
Description
pico
p
10-12 1 picoliter, (pL) = 0.000000000001 L
nano
n
10-9 1 nanogram, (ng) = 0.0000000001 g
micro µ or u
10-6 1 micrometer (µm) = 0.000001 m
milli
m
10-3 1 milliliter (mL) = 0.001 L
centi
c
10-2 1 centimeter (cm) = 0.01 m
deci
d
10-1 1 decigram (dg) = 0.1 g
kilo
k
103 1 kilogram (kg) = 1000 g
mega
M
106 1 megagram (Mg) = 1,000,000 g
giga
G
109 1 gigameter (Gm) = 1,000,000,000 m
tera
T
1012 1 teraliter (TL) = 1,000,000,000,000 L
Metric System Conversions
Kilometer = km, meter = m, decimeter = dm, centimeter = cm, millimeter = mm
Three temperature scales are in common use in science and industry.
The degree Celsius (°C) scale was devised by dividing the range of temperature
between the freezing and boiling temperatures of pure water at standard atmospheric
conditions (sea level pressure) into 100 equal parts. Temperatures on this scale were at
one time known as degrees centigrade, however it is no longer correct to use that
terminology. In 1948 the official name was changed from "centigrade degree" to
"Celsius degree" by the 9th General Conference on Weights and Measures.
The kelvin (K) temperature scale is an extension of the degree Celsius scale down to
absolute zero, a hypothetical temperature characterized by a complete absence of heat
energy. Temperatures on this scale are called kelvins, NOT degrees kelvin, kelvin is
not capitalized, and the symbol (capital K) stands alone with no degree symbol. In
1967 the new official name and symbol for "kelvin" were set by the 13th General
Conference on Weights and Measures.
The degree Fahrenheit (°F) non-metric temperature scale was devised and evolved
over time so that the freezing and boiling temperatures of water are whole numbers,
but not round numbers as in the Celsius temperature scale.
Baseline Temperatures
temperature
kelvins degrees Celsius degrees Fahrenheit
symbol
K
°C
°F
Boiling point of water
373.15
100.
212.
Melting point of ice
273.15
0.
32.
Absolute zero
0.
-273.15
-459.67
Temperature conversions between the three temperature scales
kelvin / degree Celsius conversions degree Fahrenheit / degree Celsius conversions
kelvins = degrees Celsius + 273.15
degrees F = (degrees C x 1.8) + 32
degrees Celsius = kelvins - 273.15
degrees C = (degrees F - 32.) / 1.8
Procedure:
Part-A, Mass Measurement using a digital balance
Digital balances are very sensitive. First, press the “zero or tare” button once to read
0.000 g. Then, put the object to be weighed. Record all the digits from the readout
window. Never weigh any hot object. Never put chemicals directly on the pan, always
use a container. Keep the balance clean. Use the same scale throughout the experiment.
1. Measure the mass of a 100-mL beaker using the electronic scale.
2.
Measure the mass of a 10-mL graduated cylinder using the same electronic scale.
3. Measure the mass of a 125-mL Erlenmeyer flask using the same electronic scale.
Part-B, Length Measurement
4. Measure the diameter of a watchglass in cm.
5. Measure the length and inside-mouth diameter of a test-tube in cm.
Part-C, Volume Measurement
2
d
6. Calculate the volume from step-5 measurements. (V=      L) cc
2
7. Fill-up the same test-tube with water. Transfer the water into a graduated cylinder
to find the volume. Remember to read bottom of the meniscus. Compare this
experimental volume with the calculated volume from step-6. Which one do you
think is more reliable?
Part-D, Temperature Measurement
Use a Celsius thermometer. Do not to shake! Do not touch the bottom of a hot container.
Hold it vertical and read it while immersed in the sample.
8. A “Hot-bath” has been made for you. Record its temperature in 0C.
9. An “Ice-bath” has been made for you. Record its temperature 0C.
10. Half-fill a 250-mL beakers with distilled water. Immerse the thermometer into the
beaker for about 5 minutes. Record the temperature. This is known as “the room
temperature”.
Laboratory Report#2
Scientific Notation and Measurements
Last Name_____________________________, first name________________
Date of Experiment___________ Instructor’s Initials_______
Part-A, Mass Measurement using a digital balance
1. Mass of a 100-mL beaker
= _________________________ g
2. Mass of a 10-mL graduated cylinder
= _________________________ g
3. Mass of a 125-mL Erlenmeyer flask
= _________________________ g
Part-B, Length Measurement
4. Diameter of a watchglass =__________ cm.
5. Length a test-tube =______ cm; Inside-mouth diameter of the test-tube = ______ cm
Part-C, Volume Measurement
6. Volume of the test-tube
=___________ cc (calculated)
7. Volume of the test-tube
=___________ mL (experimental)
Part-D, Temperature Measurement
8. “Hot-bath” temperature
= ___________ 0C
9. “Ice-bath” temperature
= ___________ 0C
10. “Room” temperature
= ___________ 0C
11. Convert#1 into kg (#1/1000)
= __________ kg
12. Convert #4 into mm (#4 x 10)
= __________ mm
13. Convert #7 into L (#7 /1000)
= __________ L
14. Convert #8 into k
= __________ k
15. Convert #9 into 0F
= __________ 0F
16. Convert #7 into gallons (#13 / 3.785)
= __________ gl