Scientific Measurement Problem Solving SAVE PAPER AND INK
“Scientific Measurement & Problem Solving” SAVE PAPER AND INK!!! If you print out the notes on Power. Point, print "Handouts" instead of "Slides“ in the print setup. Also, turn off the backgrounds (Tools>Options>Print>UNche ck "Background Printing")!
Types of Observations and Measurements • We make QUALITATIVE observations of reactions — Describes using words Ex. Odor, color, texture, and physical state. • We also make QUANTITATIVE observations, using numbersmeasurements • Ex. 25. 3 m. L, 4. 239 g
Standards of Measurement When we measure, we use a measuring tool to compare some dimension of an object to a standard. For example, at one time the standard for length was the king’s foot. What are some problems with this standard?
Our measurements must be both Accurate & precise! Accuracy – how close a measurement comes to the true value of what is measured Precision – is concerned with the reproducibility of the measurement
Can you hit the bull's-eye? Three targets with three arrows each to shoot. How do they compare? Both accurate and precise Precise but not accurate Neither accurate nor precise
Stating a Measurement In every measurement there is a ¨Number followed by a ¨ Unit from a measuring device The number should also be as precise as the measurement!
SI Measurement • Le Systeme International d’Unites : SI Metrics • System of measurement agreed on all over the world in 1960 • Contains 7 base units • units are defined in terms of standards of measurement that are objects or natural occurrence that are of constant value or are easily reproducible • We still use some non-SI units!
Le Système international d'unités • SI units — based on the metric system • The only countries that have not officially adopted SI are Liberia (in western Africa) and Myanmar (a. k. a. Burma, in SE Asia), but now these are reportedly using metric regularly • Among countries with non-metric usage, the U. S. is the only country significantly holding out. The U. S. officially adopted SI in 1866. Information from U. S. Metric Association
The 7 Base Units of SI
S. I. (the ones you’re responsible for knowing!) Selected S. I. base (or standard) units Mass kg Length m Time sec Temperature K mole mol Amount of Substance
Derived Units made by combining Base Units! • Volume • Density length cubed m 3 cm 3 mass/volume g/m. L kg/L g/ cm 3 g/L • Speed length /time • Area length squared mi/hr m/s m 2 cm 2 km/hr
We use prefixes to expand on the base units!
S. I. prefixes you must memorize! Prefix Abbreviation Value kilo k 103 deci d 10 -1 centi c 10 -2 milli m 10 -3 micro m 10 -6 nano n 10 -9
Metric System • These prefixes are based on powers of 10. • From each prefix every “step” is either: • 10 times larger or • 10 times smaller • For example • Centimeters are 10 times larger than millimeters • 1 centimeter = 10 millimeters kilo hecto deca Base Units meter gram liter deci centi milli
Metric System • An easy way to move within the metric system is by moving the decimal point one place for each “step” desired Example: change meters to centimeters 1. 00 meter = 10. 0 decimeters = 100. centimeters kilo hecto deca meter liter gram deci centi milli
mass – measure of the quantity of matter SI unit of mass is the kilogram (kg) 1 kg = 1000 g = 1 x 103 g weight – mass + gravity force that gravity exerts on an object Mass does not vary from place to place! A 1 kg bar has a mass of 1 kg on earth and on the moon A 1 kg bar will weigh 1 kg on earth 0. 1 kg on moon
Volume – Amount of space occupied by matter SI derived unit for volume is cubic meter (m 3) m 3 = m x m We often use the Liter (L) when working with liquid volumes! 1 L = 1000 m. L = 1000 cm 3 = 1 dm 3 1 m. L = 1 cm 3 1 dm 3 = 1 L
Temperature Scales • Fahrenheit • Celsius • Kelvin Anders Celsius 1701 -1744 Lord Kelvin (William Thomson) 1824 -1907
TEMPERATURE SCALES In Chemistry, the terms heat and temperature are often used to describe specific properties of a sample. HEAT is the most common form of energy in nature and is directly related to the motion of particles of matter. The faster the motion of particles in a sample the greater its heat content.
TEMPERATURE is associated only with the intensity of heat and is not affected by the size of the sample. A forest fire and a lit match may both be at the same temperature, but there is a large difference in the amount of heat each possess. Heat always spontaneously flows from a hotter system (higher temp. ) to a colder system (lower temp. ).
Temperature Scales Boiling point of water Freezing point of water Fahrenheit Celsius Kelvin 212 ˚F 100 ˚C 373 K 180˚F 100˚C 32 ˚F 0 ˚C Notice that 1 Kelvin = 1 degree Celsius 100 K 273 K
Scientists do. Temperature not know of any limit on how high a temperature may be. The temperature at the center of the sun is about 15, 000 °C. However, nothing can have a temperature lower than – 273°C. This temperature is called absolute zero. It forms the basis of the Kelvin scale. Because the Kelvin scale begins at absolute zero, 0 K equals – 273°C, and 273 K equals 0 °C.
Calculations Using Temperature • Many chemistry equations require temp’s to be in Kelvin • K = ˚C + 273 • Body temp = 37 ˚C + 273 = 310 K ˚C = K - 273 • Liquid nitrogen = 273 -77 K = -196 ˚C
DENSITY – an important and useful physical property (Derived Unit) ratio of mass per unit of volume platinum mercury Mercury Platinum Aluminum 13. 6 g/cm 3 21. 5 g/cm 3 2. 7 g/cm 3
Get out those calculators! Problem A piece of copper has a mass of 57. 54 g. It is 9. 36 cm long, 7. 23 cm wide, and 0. 95 mm thick. Calculate density (g/cm 3). Pure copper metal Copper ore
SOLUTION 1. Get dimensions in common units. 2. Calculate volume in cubic centimeters. (9. 36 cm)(7. 23 cm)(0. 095 cm) = 6. 4 cm 3 3. Calculate the density.
Learning Check Which diagram represents the liquid layers in the cylinder? (K) Karo syrup (1. 4 g/m. L), (V) vegetable oil (0. 91 g/m. L, ) (W) water (1. 0 g/m. L) 1) 2) 3) V W K V K V W
Solution (K) Karo syrup (1. 4 g/m. L), (V) vegetable oil (0. 91 g/m. L, ) (W) water WATER!! (1. 0 g/m. L) 1) V W K Denser materials ‘sink’ in less dense materials! Most solids sink in their liquid form. Can you think of an exception to this? !
Volume of an Irregular Solid Water Displacement A solid displaces a matching volume of water when the solid is placed in water. Volume of solid is 8 m. L 33 m. L 25 m. L
Calculator Time! What is the density (g/cm 3) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 m. L to 33 m. L? a) 0. 2 g/ cm 3 b) 6. 0 g/m 3 c) 252 g/cm 3 33 m. L 25 m. L
Percent Error • Percent Error: • Measures the inaccuracy of experimental data • Can have + or – value • Accepted value : correct value based on reliable references • Experimental value: value you measured in the lab
Scientific Notation The number of atoms in 12 g of carbon: 602, 200, 000, 000 6. 022 x 1023 The mass of a single carbon atom in grams: 0. 00000000000199 1. 99 x 10 -23 N x 10 n N is a number between 1 and 10 n is a positive or negative integer
To change standard form to scientific notation… • Place the decimal point so that there is one non-zero digit to the left of the decimal point. • Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10. • If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive.
Examples • Given: 289, 800, 000 • Use: 2. 898 (moved 8 places) • Answer: 2. 898 x 108 • Given: 0. 000567 • Use: 5. 67 (moved 4 places) • Answer: 5. 67 x 10 -4
To change scientific notation to standard form… • Simply move the decimal point to the right for positive exponent 10. • Move the decimal point to the left for negative exponent 10. (Use zeros to fill in places. )
Example • Given: 5. 093 x 106 • Answer: 5, 093, 000 (moved 6 places to the right) • Given: 1. 976 x 10 -4 • Answer: 0. 0001976 (moved 4 places to the left)
Learning Check • Express these numbers in Scientific Notation: 1) 2) 3) 4) 5) 405789 0. 003872 300000 2 0. 478260
Scientific Notation 568. 762 0. 00000772 move decimal left move decimal right n>0 n<0 568. 762 = 5. 68762 x 102 0. 00000772 = 7. 72 x 10 -6 Addition or Subtraction 1. Write each quantity with the same exponent n 2. Combine N 1 and N 2 3. The exponent, n, remains the same 4. 31 x 104 + 3. 9 x 103 = 4. 31 x 104 + 0. 39 x 104 = 4. 70 x 104
Multiplication Scientific Notation Calculations (4. 0 x 10 -5) x (7. 0 x 103) = 1. Multiply N 1 and N 2 (4. 0 x 7. 0) x (10 -5+3) = 2. Add exponents n 1 and n 2 28 x 10 -2 = 3. Put in proper format, if necessary 2. 8 x 10 -1 Division 1. Divide N 1 and N 2 2. Subtract exponents n 1 and n 2 3. Put in proper format, if necessary 8. 5 x 104 ÷ 5. 0 x 109 = (8. 5 ÷ 5. 0) x 104 -9 = 1. 7 x 10 -5
Significant Figures n. The numbers reported in a measurement are limited by the measuring tool n. Significant Figures in a measurement include all certain digits plus one estimated digit
Significant Figures • All certain digits plus one estimated digit (used when recording measurements)
Known + Estimated Digits In 2. 85 cm… • Known digits 2 and 8 are 100% certain (there are lines on the ruler for these!) • The third digit, 5, is estimated (uncertain) • In the reported length, all three digits (2. 76 cm) are significant including the estimated one
Figure 5. 5: Measuring a pin. There are not really lines on the scale here – just estimate!
Reading a Meterstick . l 2. . . . I 3. . . . I 4. . First digit (known) = 2 Second digit (known) cm 2. ? ? cm = 0. 8 2. 8? cm Third digit (estimated) between 0. 03 - 0. 05 Length reported = 2. 83 cm or 2. 84 cm or 2. 85 cm
Learning Check. l 8. . . . I 9. . . . I 10. . cm What is the length of the line? 1) 9. 3 cm 2) 9. 40 cm 3) 9. 30 cm How does your answer compare with your neighbor’s answer?
Rules for Counting Significant Figures RULE 1. All non-zero digits in a measured number are significant. Number of Significant Figures? 38. 15 cm 5. 6 m. L 65. 6 kg 122. 55 m 4 2 3 5
Sandwiched Zeros RULE 2. Zeros between nonzero numbers are significant. Number of Significant Figures? 50. 8 mm 2001 min 3 4 . 702 mg 3 400005 m 6
Leading Zeros (in front) RULE 3. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures? 0. 008 mm 1 0. 0156 g 3 0. 0042 cm 2 0. 0002602 m. L 4
Trailing Zeros (at end) RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders. Number of Significant Figures? 25, 000 m 2 200 L 1 48, 600 mg 3 5 25, 000 kg
Trailing Zeros, cont. RULE 5. Trailing zeros in numbers with decimals ARE significant. Number of Significant Figures? 35, 000. 0 m 6 700. s 3 48. 600 L 5 25, 005. 000 g 8
How many significant figures are in each of the following measurements? 24 m. L 2 significant figures 3001 g 4 significant figures 0. 0320 m 3 3 significant figures 6. 4 x 104 molecules 2 significant figures 560 kg 2 significant figures
Significant Numbers in Calculations n. A calculated answer cannot be more precise than the measuring tool. n. A calculated answer must match the least precise measurement. n. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing
Rounding • Need to use rounding to write a calculation involving measurements correctly. • Calculator gives you lots of insignificant numbers so you must round to the correct decimal place • When rounding, look at the digit after the one you can keep • Greater than or equal to 5, round up • Less than 5, keep the same
Examples Round each of the following measurements so they have 3 sig figs: Ø Ø Ø 761. 50 14. 334 10. 44 10789 8024. 50 203. 514 762 14. 3 10. 4 10800 8020 204
Series of operations: keep all non-significant digits during the intermediate calculations, and round to the correct number of SF only when reporting an answer. Ex: (4. 5 + 3. 50001) x 2. 00 = (8. 00001) x 2. 00 = 16. 0002 → 16
Using Sig Figs in Calculations • Adding/Subtracting: • end with the least number of decimal places
Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25. 2 one decimal place (to right of decimal pt. ) + 1. 34 two decimal places (to right of decimal pt. ) 26. 54 Answer: 26. 5 (one decimal place)
Using Sig Figs in Calculations • Adding/Subtracting: • end with the least number of decimal places
Significant Figures Addition or Subtraction (con’t, ) 89. 332 +1. 1 90. 432 3. 70 -2. 9133 0. 7867 one significant figure after decimal point round off to 90. 4 two significant figures after decimal point round off to 0. 79
Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235. 05 + 19. 6 + 2. 1 = 1) 256. 75 2) 256. 8 3) 257 B. 58. 925 1) 40. 725 18. 2 = 2) 40. 73 3) 40. 7
Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures. (Sometimes you’ll need to put the answer into Sci. Notation to get correct # of sig figs!)
Using Sig Figs in Calculations • Multiplying/Dividing: • end with the least number of sig figs (Counting sig figs from left)
Using Sig Figs in Calculations • Multiplying/Dividing: • end with the least number of sig figs
Significant Figures Multiplication or Division The number of significant figures in the result is set by the original number that has the smallest number of significant figures 4. 51 x 3. 6666 = 16. 536366 = 16. 5 3 sig figs round to 3 sig figs 6. 8 ÷ 112. 04 = 0. 0606926 = 0. 061 2 sig figs round to 2 sig figs
Learning Check A. 2. 19 X 4. 2 = 1) 9 2) 9. 2 3) 9. 198 B. 4. 311 ÷ 0. 07 = 1) 61. 58 2) 62 3) 60 C. 2. 54 X 0. 0028 = 0. 0105 X 0. 060 1) 11. 3 2) 11 3) 0. 041
For exact numbers (e. g. 4 beakers) and those used in conversion factors (e. g. 1 inch = 2. 54 cm), there is no uncertainty in their measurement. Therefore, IGNORE exact numbers when finalizing your answer with the correct number of significant figures. (Numbers from definitions or numbers of objects are considered to have an infinite number of significant figures) The average of three measured lengths, 6. 64, 6. 68 and 6. 70 is: 6. 64 + 6. 68 + 6. 70 3 = 6. 67333 = 6. 67 Because 3 is an exact number =7
Chemistry In Action On 9/23/99, $125, 000 Mars Climate Orbiter entered Mar’s atmosphere 100 km lower than planned and was destroyed by heat. 1 lb = 1 N 1 lb = 4. 45 N “This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time. ”
Conversion Factors • Ratio that comes from a statement of equality between 2 different units • every conversion factor is equal to 1 Example: statement of equality conversion factor = 4 quarters 1 dollar
Conversion Factors (con’t. ) Fractions in which the numerator and denominator are EQUAL quantities expressed in different units Example: 1 in. = 2. 54 cm Factors: 1 in. 2. 54 cm and 2. 54 cm 1 in.
Learning Check Write conversion factors that relate each of the following pairs of units: 1. Liters and m. L 1 L. and 1000 m. L 2. Hours and minutes 1 hr. and 60 mins. 1000 m. L 1 L 60 mins. 1 hr 3. Meters and kilometers 1000 m 1 km and 1 km__ 1000 m .
Conversion Factors • can be multiplied by other numbers without changing the value of the number (since you are just multiplying by 1)
Dimensional Analysis Method of Solving Problems 1. Start with the given 2. Determine what unit label is needed on the answer 3. Add conversion factor(s) & cancel units until you are left with the desired unit label! How many m. L are in 1. 63 L? 1 L = 1000 m. L 1. 63 L x = 1630 m. L 1 L 2 1 L L 1. 63 L x = 0. 001630 1000 m. L 1. 9
Example Convert 5. 2 cm to mm • Known: 100 cm = 1 m 1000 mm = 1 m • MUST use m as an intermediate
Example Convert 0. 020 kg to mg • Known: 1 kg = 1000 g 1000 mg = 1 g • Must use g as an intermediate
Sample Problem • You have $7. 25 in your pocket in quarters. How many quarters do you have? 7. 25 dollars X 4 quarters 1 dollar = 29 quarters
Learning Check How many seconds are in 1. 4 days? Unit plan: 1. 4 days x days hr min seconds
Solution Unit plan: days hr min seconds 1. 4 day x 24 hr x 60 min x 60 sec 1 day 1 hr 1 min = 1. 2 x 105 sec
Advanced Conversions • A more difficult type of conversion deals units that are fractions themselves • Be sure convert one unit at a time; don’t try to do both at once • Work on the unit on top first; then work on the unit on the bottom • Setup your work the exact same way
When unit labels are fractions (or ratios), them! 11. 3 g/m. L can be written as 11. 3 g 1 m. L Convert 11. 3 g/m. L to g/L 11. 3 g 1000 m. L 1 L = 1. 13 x 104 g/L unzip
PROBLEM: Mercury (Hg) has a density of 13. 6 g/cm 3. What is the mass of 95 m. L of Hg in grams? In pounds? Remember 1 cm 3 = 1 m. L Solve the problem using DIMENSIONAL ANALYSIS.
The speed of sound in air is about 343 m/s. What is this speed in miles per hour? meters to miles seconds to hours 1 mi = 1609 m 343 1 min = 60 s m 1 mi 60 s s x 1609 m x 1 min 1 hour = 60 min mi x = 767 hour 1. 9
Advanced Conversions • Another difficult type of conversion deals with squared or cubed units • Be sure to square or cube the conversion factor you are using to cancel all the units • If you tend to forget to square or cube the number in the conversion factor, try rewriting the conversion factor instead of just using the exponent
Square and Cubic units • Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also! • Best way: Square or cube the ENTIRE conversion factor • Example: Convert 4. 3 cm 3 to mm 3 ( 4. 3 cm 3 10 mm 1 cm ) 3 = 4. 3 cm 3 103 mm 3 13 cm 3 = 4300 mm 3
Example • Convert: 2000 cm 3 to m 3 • No intermediate needed OR Known: 100 cm = 1 m cm 3 = cm x cm m 3 = m x m
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