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Chapter 2

Chapter 2. Measurement and Problem Solving. Homework. Exercises (optional) 1 through 27 (odd) Problems 29-65 (odd) 67-91 (odd) 93-99 (odd) Cumulative Problems 101-117 (odd) Highlight Problems (optional) 119, 121. 2.2 Scientific Notation: Writing Large and Small Numbers.

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Chapter 2

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  1. Chapter 2 Measurement and Problem Solving

  2. Homework • Exercises (optional) • 1 through 27 (odd) • Problems • 29-65 (odd) • 67-91 (odd) • 93-99 (odd) • Cumulative Problems • 101-117 (odd) • Highlight Problems (optional) • 119, 121

  3. 2.2 Scientific Notation: Writing Large and Small Numbers • In scientific (chemistry) work, it is not unusual to come across very large and very small numbers • Using large and small numbers in measurements and calculations is time consuming and difficult • Recording these numbers is also very prone to errors due to the addition or omission of zeros • A method exists for the expression of awkward, multi-digit numbers in a compact form: scientific notation

  4. 2.2 Scientific Notation: Writing Large and Small Numbers • Scientific Notation • A system in which an ordinary decimal number (m) is expressed as a product of a number between 1 and 10, multiplied by 10 raised to a power (n) • Used to write very large or very small numbers • Based on powers of 10

  5. 2.2 Scientific Notation: Writing Large and Small Numbers • Numbers written in sci. notn. consist of a number (coefficient) followed by a power of 10 (x 10n) • Negative exponent: number is less than 1 • Positive exponent: number is greater than 1 exponent coefficient or decimal part exponential term or part

  6. 2.2 Scientific Notation:Writing Large and Small Numbers 7,910,000,000,000,000,000,000,000 molecules • In an ordinary cup of water there are: • Each molecule has a massof: 0.0000000000000000000000299 gram In scientific notation: 7.91 х1024 molecules 2.99 х10-23 gram

  7. To Express a Number in Scientific Notation: For small numbers (<1): • Locate the decimal point • Move the decimal point to the right to give a number (coefficient) between 1 and 10 • Write the new number multiplied by 10raised to the “nth power” • where“n” is the number of places you moved the decimal point so there is one non-zero digit to the left of the decimal. • If the decimal point is moved to the right, from its initial position, then the exponent is a negative number (× 10-n)

  8. To Express a Number in Scientific Notation: For large numbers (>1): • Locate the decimal point • Move the decimal point to the left to give a number (coefficient) between 1 and 10 • Write the new number multiplied by 10 raised to the “nth power” • where“n” is the number of places you moved the decimal point so there is one non-zero digit to the left of the decimal. • If the decimal point is moved to the left, from its initial position, then the exponent is a positive number (× 10n)

  9. Examples • Write each of the following in scientific notation • 12,500 • 0.0202 • 37,400,000 • 0.0000104

  10. Examples • 12,500 • Decimal place is at the far right • Move the decimal place to a position between the 1 and 2 (one non-zero digit to the left of the decimal) • Coefficient (1.25) • The decimal place was moved 4 places to the left (large number) so exponent is positive • 1.25x104

  11. Examples • 0.0202 • Move the decimal place to a position between the 2 and 0 (one non-zero digit to the left of the decimal) • Coefficient (2.02) • The decimal place was moved 2 places to the right (small number) so exponent is negative • 2.02x10-2

  12. Examples • 37,400,000 • Decimal place is at the far right • Move the decimal place to a position between the 3 and 7 • Coefficient (3.74) • The decimal place was moved 7 places to the left (large number) so exponent is positive • 3.74x107

  13. Examples • 0.0000104 • Move the decimal place to a position between the 1 and 0 • Coefficient (1.04) • The decimal place was moved 5 places to the right (small number) so exponent is negative • 1.04x10-5

  14. Using Scientific Notation on a Calculator • Enter the coefficient (number) • Push the key: Then enter only the power of 10 • If the exponent is negative, use the key: • DO NOT use the multiplication key: to express a number in sci. notation EE EXP or (+/-) X

  15. Converting Back to Standard Notation • Determine the sign of the exponent, n • If n is + the decimal point will move to the right (this gives a number greater than one) • If n is – the decimal point will move to the left (this gives a number less than one) • Determine the value of the exponent of 10 • The “power of ten” determines the number of places to move the decimal point • Zeros may have to be added to the number as the decimal point is moved

  16. Using Scientific Notation • To compare numbers written in scientific notation, with the same coefficient, compare the exponents of each number • The number with the larger power of ten (the exponent) is the larger number • If the powers of ten (exponents) are the same, then compare coefficients directly • Which number is larger? 3.4 х 104 < 3.4 х 107 21.8 х 103 or 2.05 х 104 2.18 х 104 > 2.05 х 104

  17. 2.3 Significant Figures:Writing Numbers to Reflect Precision • Two kinds of numbers exist: • Numbers that are counted (exact) • Numbers that are measured • It is possible to know the exact value of a counted number • The exact value of a measured number is never known • Counting objects does not entail reading the scale of a measuring device

  18. 2.3 Exact Numbers • Exact numbers occur in definitions or in counting • These numbers have no uncertainty • Unlimited number of significant figures (never limit the no. of sig. figures in a calculation) • They are either • Counting numbers • 7 pennies, 6 apples, 4 chairs • Defined numbers (one exact value) • 12 in = 1 ft • 1 gal = 4 quarts • 1 minute = 60 seconds

  19. Measured Numbers • Unlike counted (or defined) numbers, measured numbers always contain a degree of uncertainty (or error) • A measurement: • involves reading a measuring device • always has some amount of uncertainty • uncertainty comes from the tool used for comparison • A measuring device with a smaller unit will give a more precise measurement, e.g., some rulers show smaller divisions than others

  20. Measured Numbers • Whenever a measurement is made, an estimate is required, i.e., the value between the two smallest divisions on a measuring device • Every person will estimate it slightly differently, so there is some uncertainty present as to the true value 2.8 cm 2.9 cm 2.8 to 2.9 cm

  21. 2.3 Significant Figures: Writing Numbers to Reflect Precision • Scientific numbers are reported so that all digits are certain except the last digit which is estimated • To indicate the uncertainty of a single measurement, scientists use a system called significant figures • Significant Figures: All digits known with certainty plus one digit that is uncertain

  22. 2.3Counting Significant Figures • The last digit written in a measurement is the number that is considered to be uncertain (estimated) • Unless stated otherwise, the uncertainty in the last significant digit is ±1 (plus or minus one unit) • The precision of a measured quantity is determined by number of sig. figures • A set of guidelines is used to interpret the significance of: • a reported measurement • values calculated from measurements

  23. 2.3Counting Significant Figures • Four rules (the guidelines): • Nonzero integers are always significant • Zeros (may or may not be significant) • significant zeros • place-holding zeros (not significant) • It is determined by its position in a sequence of digits in a measurement • Leading zeros never count as significant figures • Captive (interior) zeros are always significant • Trailing zeros are significant if the number has a decimal point

  24. 2.4 Significant Figures in Calculations • Calculations cannot improve the precision of experimental measurements • The number of significant figures in any mathematical calculation is limited by the least precise measurement used in the calculation • Two operational rules to ensure no increase in measurement precision: • addition and subtraction • multiplication and division

  25. 2.4 Significant Figures in Calculations: Multiplication and Division • Product or quotient has the same number of significant figures as the factor with the fewestsignificant figures • Count the number of significant figures in each number. The least precise factor (number) has the fewest significant figures • Rounding • Round the result so it has the same number of significant figures as the number with the fewest significant figures

  26. 2.4 Significant Figures in Calculations: Rounding • To round the result to the correct number of significant figures • If the last (leftmost) digit to be removed: • is less than 5, the preceding digit stays the same (rounding down) • is equal to or greater than 5, the preceding digit is rounded up • In multiple step calculations, carry the extra digits to the final result and then round off

  27. 2.4 Multiplication/Division Example: 5 SF 3 SF 4 SF • The number with the fewest significant figures is 1.1 so the answer has 2 significant figures 2.1 2 SF 2 SF

  28. 2.4 Multiplication/Division Example: • The number with the fewest significant figures is 273 (the limiting term) so the answer has 3 significant figures 3 SF 3 SF 4 SF 5 SF 0.1021 × 0.082103 × 273 = 2.288481 2.29

  29. 2.4 Significant Figures in Calculations: Addition and Subtraction • Sum or difference is limited by the quantity with the smallest number of decimal places • Find quantity with the fewest decimal places • Round answer to the same decimal place

  30. 2.4 Addition/Subtraction Example: 2 d.p. 1 d.p. • The number with the fewest decimal places is 171.5 so the answer should have 1 decimal place 3 d.p. 236.2 1 d.p.

  31. 2.5 The Basic Units of Measurement • The most used tool of the chemist • Most of the basic concepts of chemistry were obtained through data compiled by taking measurements • How much…? • How long…? • How many...? • These questions cannot be answered without taking measurements • The concepts of chemistry were discovered as data was collected and subjected to the scientific method

  32. 2.5 The Basic Units of Measurement • The estimation of the magnitude of an object relative to a unit of measurement • Involves a measuring device: • meter stick, scale, thermometer • The device is calibrated to compare the object to some standard (inch/centimeter, pound/kilogram) • Quantitative observation with two parts: A number and a unit • Number tells the total of the quantity measured • Unit tells the scale (dimensions)

  33. 2.5 The Basic Units of Measurement • A unit is a standard (accepted) quantity • Describes what is being added up • Units are essential to a measurement • For example, you need “six of sugar” • teaspoons? • ounces? • cups? • pounds?

  34. 2.5 The Standard Units (of Measurement) • The unit tells the magnitude of the standard • Two most commonly used systems of units of measurement • U.S. (English) system: Used in everyday commerce (USA and Britain*) • Metric system: Used in everyday commerce and science (The rest of the world) • SI Units (1960): A modern, revised form of the metric system set up to create uniformity of units used worldwide (world’s most widely used)

  35. 2.5 The Standard Units (of Measurement):The Metric/SI System • The metric system is a decimal system of measurement based on the meter and the gram • It has a single base unit per physical quantity • All other units are multiples of 10 of the base unit • The power (multiple) of 10 is indicated by a prefix

  36. 2.5 The Standard Units: The Metric System • In the metric system there is one base unit for each type of measurement • length • volume • mass • The base units multiplied by the appropriate power of 10 form smaller or larger units • The prefixes are always the same, regardless of the base unit • milligrams and milliliters both mean 1/1000 of the base unit

  37. 2.5 The Standard Units: Length • Meter • Base unit of length in metric and SI system • About 3 ½ inches longer than a yard • 1 m = 1.094 yd

  38. 2.5 The Standard Units: Length • Other units of length are derived from the meter • Commonly use centimeters (cm) • 1 m = 100 cm • 1 inch = 2.54 cm (exactly)

  39. 2.5 The Standard Units: Volume Volume = side × side × side • Measure of the amount of three-dimensional space occupied by a object • Derived from length • SI unit = cubic meter (m3) • Metric unit= liter (L) or 10 cm3 • Commonly measure smaller volumes in cubic centimeters (cm3) volume = side × side × side

  40. 2.5 The Standard Units: Volume • Since it is a three-dimensional measure, its units have been cubed • SI base unit = cubic meter (m3) • This unit is too large for practical use in chemistry • Take a volume 1000 times smaller than the cubic meter, 1dm3

  41. 2.5 The Standard Units: Volume • Metric base unit=1dm3 = liter (L) • 1L = 1.057 qt • Commonly measure smaller volumes in cubic centimeters (cm3) • Take a volume 1000 times smaller than the cubic decimeter, 1cm3

  42. 2.5 The Standard Units: Volume • Metric base unit=1dm3 = liter (L) • 1L = 1.057 qt • Commonly measure smaller volumes in cubic centimeters (cm3) • Take a volume 1000 times smaller than the cubic decimeter, 1cm3

  43. 2.5 The Standard Units: Volume • The most commonly used unit of volume in the laboratory: milliliter (mL) • 1 mL = 1 cm3 • 1 L= 1 dm3 = 1000 mL • 1 m3 = 1000 dm3 = 1,000,000 cm3 • Use a graduated cylinder or a pipette to measure liquids in the lab

  44. 2.5 The Standard Units: Mass • Measure of the total quantity of matter present in an object • SI unit (base) = kilogram (kg) • Metric unit (base) = gram (g) • Commonly measure mass in grams (g) or milligrams (mg) • 1 kg = 1000 g • 1 g = 1000 mg • 1 kg = 2.205 pounds • 1 lb = 453.6 g

  45. 2.5 Prefixes Multipliers • One base unit for each type of measurement • Length (meter), volume (liter), and mass (gram*) • The base units are then multiplied by the appropriate power of 10 to form larger or smaller units base unit = meter, liter, or gram

  46. 2.5 Prefixes Multipliers (memorize) × base unit • Mega (M) 1,000,000 106 • Kilo (k) 1,000 103 • Base 1 100 • Deci (d) 0.1 10-1 • Centi (c) 0.01 10-2 • Milli (m) 0.001 10-3 • Micro (µ) 0.000001 10-6 • Nano (n) 0.000000001 10-9 liter meter gram

  47. 2.5 Prefix Multipliers • For a particular measurement: • Choose the prefix which is similar in size to the quantity being measured • Keep in mind which unit is larger • A kilogram is larger than a gram, so there must be a certain number of grams in one kilogram • Choose the prefix most convenient for a particular measurement n < µ < m < c < base < k < M

  48. 2.6 Converting from One Unit to Another: Dimensional Analysis • Many problems in chemistry involve converting the units of a quantity or measurement to different units • The new units may be in the same measurement system or a different system, i.e., U.S. System to Metric and the converse • Dimensional Analysis is the method of problem solving used to achieve this unit conversion • Unit conversion is accomplished by multiplication of a given quantity (or measurement) by one or more conversion factors to obtain the desired quantity or measurement

  49. 2.6 Converting from One Unit to Another: Equalities • An equality is a fixed relationship between two quantities • It shows the relationship between two units that measure the same quantity • The relationships are exact, not measured • 1 min = 60 s • 12 inches = 1 ft • 1 dozen = 12 items (units) • 1L = 1000 mL • 16 oz = 1 lb • 4 quarts = 1 gallon

  50. 2.6 Converting from One Unit to Another: Dimensional Analysis • Conversion factor: An equality expressed as a fraction • It is used as a multiplier to convert a quantity in one unit to its equivalent in another unit • May be exact or measured • Both parts of the conversion factor should have the same number of significant figures

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