Experiment-2: Scientific Notation and Measurements
ü Significant figures
ü Rounding off
ü Measurements of Length, Mass, Volume, and Temperature.
ü Unit Conversions.
ü 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.
(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 / 3 g / 1 significant figure / ± 1 g precisionOpen-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
Cylinder / First, note that the surface of the liquid is curved. It 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
2. 319.15 - 32.614=286.54
3. 104.630 + 27.08362 + 0.61=132.32
4. 125 - 0.23 + 4.109=129
5. 2.02 × 2.5=5.0
6. 600.0 / 5.2302=114.7
7. 0.0032 × 273=0.87
8. (5.5)3=1.7 x 102
9. 0.556 × (40 - 32.5)=4
10. 45 × 3.00=1.4 x 102
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 thousand-meters 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
10 mm = 1 cm10 cm = 1 dm
100 cm = 1 m
1000 m = 1 km
1 inch = 2.54 cm exact
12 in. = 1 foot
3ft. = 1 yard
1 mile = 5280 feet = 1.609 km /
Mass
1g = 1000 mg1 kg = 1000g
1 metric ton = 1000 kg
1 lb. = 453.59 g
1 lb = 16 oz
1 oz = 28.35 g
1 amu = 1.66 x 10 –24 g
1 m g = 1 x 10 –6 g /
Volume
1 L = 1000 mL1 dL = 100 mL
1 mL = 1 cm3 = 1cc
1 gallon = 3.7854 L
1 gallon = 4 quarts= 8 pints
1 pint = 16 fl oz
1 fl oz. = 29.57 mL
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