1. Chapter 2 Measurement and Problem Solving

2. Homework Exercises (optional) Exercises (optional) 1 through 27 (odd) 1 through 27 (odd) Problems Problems 29-65 (odd) 29-65 (odd) 67-91 (odd) 67-91 (odd) 93-99 (odd) 93-99 (odd) Cumulative Problems Cumulative Problems 101-117 (odd) 101-117 (odd) Highlight Problems (optional) Highlight Problems (optional) 119, 121 119, 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 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 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 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 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 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) 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 Used to write very large or very small numbers Based on powers of 10 Based on powers of 10

5. 2.2 Scientific Notation: Writing Large and Small Numbers Scientific notation uses exponents (i.e. powers of numbers) which are numbers that are written as superscripts (following another number) which indicate how many times the number is multiplied by itself Scientific notation uses exponents (i.e. powers of numbers) which are numbers that are written as superscripts (following another number) which indicate how many times the number is multiplied by itself (e.g., 6 2 = 6 × 6 = 36, 3 5 = 3 × 3 × 3 × 3 × 3 = 243 (e.g., 6 2 = 6 × 6 = 36, 3 5 = 3 × 3 × 3 × 3 × 3 = 243 Scientific notation exclusively uses powers of 10 Scientific notation exclusively uses powers of 10 When ten is raised to a power, its decimal equivalent is the number 1 followed by as many zeros as the power itself When ten is raised to a power, its decimal equivalent is the number 1 followed by as many zeros as the power itself For example: For example: 10 2 = 100 (two zeros and power of 2) 10 2 = 100 (two zeros and power of 2) 10 4 = 10,000 (four zeros and power of 4) 10 4 = 10,000 (four zeros and power of 4) 10 6 = 1,000,000 (six zeros and power of 6) 10 6 = 1,000,000 (six zeros and power of 6)

6. 2.2 Scientific Notation: Writing Large and Small Numbers A negative sign in front of an exponent indicates that the number and the power to which it is raised are in the denominator of a fraction in which 1 is in the numerator. A negative sign in front of an exponent indicates that the number and the power to which it is raised are in the denominator of a fraction in which 1 is in the numerator. The number of zeros between the decimal point and the one is always one less than the absolute power of the exponent The number of zeros between the decimal point and the one is always one less than the absolute power of the exponent For example: For example: 10 -1 = 1/10 1 = 1/10 = 0.1 10 -1 = 1/10 1 = 1/10 = 0.1 10 -2 = 1/10 2 = 1/10 × 10 = 1/100 = 0.01 10 -2 = 1/10 2 = 1/10 × 10 = 1/100 = 0.01 10 -3 = 1/10 3 = 1/10 × 10 × 10 = 1/1000 = 0.001 10 -3 = 1/10 3 = 1/10 × 10 × 10 = 1/1000 = 0.001

7. 2.2 Scientific Notation: Writing Large and Small Numbers Numbers written in scientific notation consist of a number (coefficient) followed by a power of 10 (x 10 n ) Numbers written in scientific notation consist of a number (coefficient) followed by a power of 10 (x 10 n ) Negative exponent: number is less than 1 Negative exponent: number is less than 1 Positive exponent: number is greater than 1 Positive exponent: number is greater than 1 coefficient or decimal part exponential term or part exponent

8. 2.2 Scientific Notation: Writing Large and Small Numbers In an ordinary cup of water there are: In an ordinary cup of water there are: Each molecule has a mass of: Each molecule has a mass of: 0.0000000000000000000000299 gram In scientific notation: 7.91 х 10 24 molecules 2.99 х 10 -23 gram 7,910,000,000,000,000,000,000,000 molecules

9. To Express a Number in Scientific Notation: For small numbers (<1): 1)Locate the decimal point 2)Move the decimal point to the right to give a number (coefficient) between 1 and 10 3)Write the new number multiplied by 10 raised to the “n th 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. 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 ) If the decimal point is moved to the right, from its initial position, then the exponent is a negative number (× 10 -n )

10. To Express a Number in Scientific Notation: For large numbers (>1): 1) Locate the decimal point 2) Move the decimal point to the left to give a number (coefficient) between 1 and 10 3) Write the new number multiplied by 10 raised to the “n th 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. 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 (× 10 n ) If the decimal point is moved to the left, from its initial position, then the exponent is a positive number (× 10 n )

11. 2.2 Scientific Notation: Writing Large and Small Numbers Write each of the following in scientific notation Write each of the following in scientific notation 12,500 12,500 0.0202 0.0202 37,400,000 37,400,000 0.0000104 0.0000104

12. Examples 12,500 12,500 Decimal place is at the far right 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) 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): only significant digits become part of the coefficient Coefficient (1.25): only significant digits become part of the coefficient The decimal place was moved 4 places to the left (large number) so exponent is positive The decimal place was moved 4 places to the left (large number) so exponent is positive 1.25x10 4 1.25x10 4

13. Examples 0.0202 0.0202 Move the decimal place to a position between the 2 and 0 (one non-zero digit to the left of the decimal) 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): only significant digits become part of the coefficient Coefficient (2.02): only significant digits become part of the coefficient The decimal place was moved 2 places to the right (small number) so exponent is negative The decimal place was moved 2 places to the right (small number) so exponent is negative 2.02x10 -2 2.02x10 -2

14. Examples 37,400,000 37,400,000 Decimal place is at the far right Decimal place is at the far right Move the decimal place to a position between the 3 and 7 Move the decimal place to a position between the 3 and 7 Coefficient (3.74): only significant digits become part of the coefficient Coefficient (3.74): only significant digits become part of the coefficient The decimal place was moved 7 places to the left (large number) so exponent is positive The decimal place was moved 7 places to the left (large number) so exponent is positive 3.74x10 7 3.74x10 7

15. Examples 0.0000104 0.0000104 Move the decimal place to a position between the 1 and 0 Move the decimal place to a position between the 1 and 0 Coefficient (1.04): only significant digits become part of the coefficient Coefficient (1.04): only significant digits become part of the coefficient The decimal place was moved 5 places to the right (small number) so exponent is negative The decimal place was moved 5 places to the right (small number) so exponent is negative 1.04x10 -5 1.04x10 -5

16. 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 (+/-) X EXPEE or

17. Converting Back to Decimal Notation 1)Determine the sign of the exponent, n If n is positive (×10 n ), the decimal point will move to the right (this gives a number greater than one) If n is positive (×10 n ), the decimal point will move to the right (this gives a number greater than one) If n is negative(×10 -n ), the decimal point will move to the left (this gives a number less than one) If n is negative(×10 -n ), the decimal point will move to the left (this gives a number less than one) 2)Determine the value of the exponent of 10 The “power of ten” determines the number of places to move the decimal point 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 Zeros may have to be added to the number as the decimal point is moved

18. Using Scientific Notation To compare numbers written in scientific notation, with the same coefficient, compare the exponents of each number 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 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 If the powers of ten (exponents) are the same, then compare coefficients directly Which number is larger? Which number is larger? 21.8 х 10 3 or 2.05 х 10 4 2.18 х 10 4 > 2.05 х 10 4 3.4 х 10 4 < 3.4 х 10 7

19. 2.3 Significant Figures: Writing Numbers to Reflect Precision Two kinds of numbers exist: Two kinds of numbers exist: Numbers that are exact (defined) Numbers that are exact (defined) Numbers that are measured Numbers that are measured It is possible to know the exact value of a counted number It is possible to know the exact value of a counted number The exact value of a measured number is never known The exact value of a measured number is never known Counting objects does not entail reading the scale of a measuring device Counting objects does not entail reading the scale of a measuring device

20. 2.3 Exact Numbers Exact numbers occur in definitions or in counting Exact numbers occur in definitions or in counting These numbers have no uncertainty These numbers have no uncertainty Counting numbers Counting numbers You can count the number of peaches in a bushel of peaches with absolute certainty You can count the number of peaches in a bushel of peaches with absolute certainty You can count the number of chairs in a room with absolute certainty You can count the number of chairs in a room with absolute certainty Defined numbers (one exact value) Defined numbers (one exact value) There are exactly twelve inches in one foot (1 ft = 12 in) There are exactly twelve inches in one foot (1 ft = 12 in) There are exactly four quarts in one gallon (1 gal = 4 quarts) There are exactly four quarts in one gallon (1 gal = 4 quarts) There are exactly sixty seconds in one minute (1 min = 60 sec) There are exactly sixty seconds in one minute (1 min = 60 sec)

21. Measured Numbers Counting objects does not involve a measuring device and it is not subject to uncertainties Counting objects does not involve a measuring device and it is not subject to uncertainties Unlike counted (or defined) numbers, measured numbers always contain a degree of uncertainty (or error) Unlike counted (or defined) numbers, measured numbers always contain a degree of uncertainty (or error) A measurement: A measurement: involves reading the scale of a measuring device involves reading the scale of a measuring device always has some amount of uncertainty which comes from the tool used for comparison always has some amount of uncertainty which comes from the tool used for comparison A measuring device with a smaller unit will give a more precise measurement, but some degree of uncertainty will always be present A measuring device with a smaller unit will give a more precise measurement, but some degree of uncertainty will always be present

22. Measured Numbers This ruler has divisions every one millimeter This ruler has divisions every one millimeter Whenever a measurement is made, an estimate is required, i.e., the value between the two smallest divisions on a measuring device 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 Every person will estimate it slightly differently, so there is some uncertainty present as to the true value 2.8 to 2.9 cm 2.8 cm2.9 cm Mentally divide the space into 10 equal spaces to estimate the last digit 2.85 cm

23. Measured Numbers This balance has divisions every one gram This balance has divisions every one gram Whenever a measurement is made, an estimate is required, i.e., the value between the two smallest divisions on a measuring device Whenever a measurement is made, an estimate is required, i.e., the value between the two smallest divisions on a measuring device The estimate will be in the tenths place The estimate will be in the tenths place Mentally divide the space into 10 equal spaces to estimate the last digit Mentally divide the space into 10 equal spaces to estimate the last digit 1 g 2 g 3 g 1 g 2 g 1.2 g

24. 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 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 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 Significant Figures: All digits known with certainty plus one digit that is uncertain

25. 2.3 Counting Significant Figures The last digit written in a measurement is the number that is considered to be uncertain (estimated) 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) 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 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 set of guidelines is used to interpret the significance of a reported measurement a reported measurement values calculated from measurements values calculated from measurements

26. 2.3 Counting Significant Figures Four rules (the guidelines): Four rules (the guidelines): 1. Nonzero integers are always significant Zeros (may or may not be significant) Zeros (may or may not be significant) significant zeros significant zeros place-holding zeros (not significant) place-holding zeros (not significant) It is determined by its position in a sequence of digits in a measurement It is determined by its position in a sequence of digits in a measurement 2. Leading zeros never count as significant figures 3. Captive (interior) zeros are always significant 4. Trailing zeros are significant if the number has a decimal point

27. 2.4 Significant Figures in Calculations Calculations cannot improve the precision of experimental measurements 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 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: Two operational rules to ensure no increase in measurement precision: addition and subtraction addition and subtraction multiplication and division multiplication and division

28. 2.4 Significant Figures in Calculations: Multiplication and Division Product or quotient has the same number of significant figures as the factor with the fewest significant figures Product or quotient has the same number of significant figures as the factor with the fewest significant figures Count the number of significant figures in each number. The least precise factor (number) has the fewest significant figures Count the number of significant figures in each number. The least precise factor (number) has the fewest significant figures Rounding Rounding Round the result so it has the same number of significant figures as the number with the fewest significant figures Round the result so it has the same number of significant figures as the number with the fewest significant figures

29. 2.4 Significant Figures in Calculations: Rounding To round the result to the correct number of significant figures To round the result to the correct number of significant figures If the last (leftmost) digit to be removed: 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

30. 2.4 Multiplication/Division Example: The number with the fewest significant figures is 273 (the limiting term) so the answer has 3 significant figures The number with the fewest significant figures is 273 (the limiting term) so the answer has 3 significant figures 0.1021 × 0.082103 × 273 = 2.288481 2.29 3 SF 5 SF4 SF

31. 2.4 Multiplication/Division Example: The number with the fewest significant figures is 1.1 so the answer has 2 significant figures The number with the fewest significant figures is 1.1 so the answer has 2 significant figures 2 SF 5 SF3 SF 2.1 4 SF 2 SF

32. 2.4 Significant Figures in Calculations: Addition and Subtraction Sum or difference is limited by the quantity with the smallest number of decimal places Sum or difference is limited by the quantity with the smallest number of decimal places Find quantity with the fewest decimal places Find quantity with the fewest decimal places Round answer to the same decimal place Round answer to the same decimal place

33. 2.4 Addition/Subtraction Example: The number with the fewest decimal places is 171.5 so the answer should have 1 decimal place The number with the fewest decimal places is 171.5 so the answer should have 1 decimal place 1 d.p. 3 d.p. 2 d.p. 236.2 1 d.p.

34. 2.5 The Basic Units of Measurement The most used tool of the chemist The most used tool of the chemist Most of the basic concepts of chemistry were obtained through data compiled by taking measurements Most of the basic concepts of chemistry were obtained through data compiled by taking measurements How much…? How much…? How long…? How long…? How many...? How many...? These questions cannot be answered without taking measurements These questions cannot be answered without taking measurements The concepts of chemistry were discovered as data was collected and subjected to the scientific method The concepts of chemistry were discovered as data was collected and subjected to the scientific method

35. 2.5 The Basic Units of Measurement A measurement is the process or the result of determining the magnitude of a quantity (e.g., length or mass) relative to a unit of measurement A measurement is the process or the result of determining the magnitude of a quantity (e.g., length or mass) relative to a unit of measurement Involves a measuring device: Involves a measuring device: meter stick, scale, thermometer meter stick, scale, thermometer The device is calibrated to compare the object to some standard (inch/centimeter, pound/kilogram) 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 Quantitative observation with two parts: A number and a unit Number tells the total of the quantity measured Number tells the total of the quantity measured Unit tells the scale (dimensions) Unit tells the scale (dimensions)

36. 2.5 The Basic Units of Measurement A unit is a standard (accepted) quantity A unit is a standard (accepted) quantity Describes what is being added up Describes what is being added up Units are essential to a measurement Units are essential to a measurement For example, you need “six of sugar” For example, you need “six of sugar” teaspoons? teaspoons? ounces? ounces? cups? cups? pounds? pounds?

37. 2.5 The Standard Units (of Measurement) The unit tells the magnitude of the standard The unit tells the magnitude of the standard Two most commonly used systems of units of measurement Two most commonly used systems of units of measurement U.S. (English) system: Used in everyday commerce (USA and Britain*) U.S. (English) system: Used in everyday commerce (USA and Britain*) Metric system: Used in everyday commerce and science (The rest of the world) 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) 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)

38. 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 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 It has a single base unit per physical quantity All other units are multiples of 10 of the base unit All other units are multiples of 10 of the base unit The power (multiple) of 10 is indicated by a prefix The power (multiple) of 10 is indicated by a prefix

39. 2.5 The Standard Units: The Metric System In the metric system there is one base unit for each type of measurement In the metric system there is one base unit for each type of measurement length length volume volume mass mass (also, time, temperature) (also, time, temperature) The base units multiplied by the appropriate power of 10 form smaller or larger units 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 The prefixes are always the same, regardless of the base unit milligrams and milliliters both mean 1/1000 (10 -3 ) of the base unit milligrams and milliliters both mean 1/1000 (10 -3 ) of the base unit

40. 2.5 The Standard Units: Length Meter Meter Base unit of length in metric and SI system Base unit of length in metric and SI system About 3 ½ inches longer than a yard About 3 ½ inches longer than a yard 1 m = 1.094 yd 1 m = 1.094 yd

41. 2.5 The Standard Units: Length Other units of length are derived from the meter Other units of length are derived from the meter Commonly use centimeters (cm) Commonly use centimeters (cm) 1 m = 100 cm 1 m = 100 cm 1 inch = 2.54 cm (exactly) 1 inch = 2.54 cm (exactly)

42. 2.5 The Standard Units: Volume Volume: Measure of the amount of three-dimensional space occupied by a object Volume: Measure of the amount of three-dimensional space occupied by a object Derived from length Derived from length Since it is a three-dimensional measure, its units have been cubed Since it is a three-dimensional measure, its units have been cubed SI base unit = cubic meter (m 3 ) SI base unit = cubic meter (m 3 ) Metric base unit = liter (L) or 10 cm 3 Metric base unit = liter (L) or 10 cm 3 Commonly measure smaller volumes in cubic centimeters (cm 3 ) Commonly measure smaller volumes in cubic centimeters (cm 3 ) Volume = side × side × side volume = side × side × side

43. 2.5 The Standard Units: Volume SI base unit = 1m 3 SI base unit = 1m 3 The volume equal to that occupied by a perfect cube that is one meter on each side The volume equal to that occupied by a perfect cube that is one meter on each side This unit is too large for practical use in chemistry This unit is too large for practical use in chemistry Take a volume 1000 times smaller than the cubic meter, 1dm 3 Take a volume 1000 times smaller than the cubic meter, 1dm 3

44. 2.5 The Standard Units: Volume Metric base unit = 1dm 3 (one liter, L) Metric base unit = 1dm 3 (one liter, L) The volume equal to that occupied by a perfect cube that is ten centimeters on each side The volume equal to that occupied by a perfect cube that is ten centimeters on each side 1L = 1.057 qt 1L = 1.057 qt Commonly measure smaller volumes in cubic centimeters (cm 3 ) Commonly measure smaller volumes in cubic centimeters (cm 3 ) Take a volume 1000 times smaller than the cubic decimeter, 1cm 3 Take a volume 1000 times smaller than the cubic decimeter, 1cm 3 V =10 cm × 10 cm × 10 cm 10 cm = 1 dm 3

45. 2.5 The Standard Units: Volume The most commonly used unit of volume in the laboratory: milliliter (mL) The most commonly used unit of volume in the laboratory: milliliter (mL) The volume equal to that occupied by a perfect cube that is one centimeter on each side The volume equal to that occupied by a perfect cube that is one centimeter on each side 1 mL = 1 cm 3 1 mL = 1 cm 3 1 L= 1 dm 3 = 1000 mL 1 L= 1 dm 3 = 1000 mL 1 m 3 = 1000 dm 3 = 1,000,000 cm 3 1 m 3 = 1000 dm 3 = 1,000,000 cm 3 Use a graduated cylinder or a pipette to measure liquids in the lab Use a graduated cylinder or a pipette to measure liquids in the lab

46. 2.5 The Standard Units: Mass Measure of the total quantity of matter present in an object Measure of the total quantity of matter present in an object SI unit (base) = kilogram (kg) SI unit (base) = kilogram (kg) Metric unit (base) = gram (g) Metric unit (base) = gram (g) Since the gram is such a relatively small unit, the kilogram is a very commonly used unit Since the gram is such a relatively small unit, the kilogram is a very commonly used unit 1 kg = 1000 g 1 kg = 1000 g 1 g = 1000 mg 1 g = 1000 mg 1 kg = 2.205 pounds 1 kg = 2.205 pounds 1 lb = 453.6 g 1 lb = 453.6 g

47. 2.5 Prefixes Multipliers One base unit for each type of measurement One base unit for each type of measurement Length (meter), volume (liter), and mass (gram*) 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 The base units are then multiplied by the appropriate power of 10 to form larger or smaller units base unit = meter, liter, or gram

48. 2.5 Prefixes Multipliers (memorize) mega (M) 1,000,000 10 6 mega (M) 1,000,000 10 6 kilo (k) 1,000 10 3 kilo (k) 1,000 10 3 base 1 10 0 base 1 10 0 deci(d) 0.1 10 -1 deci(d) 0.1 10 -1 centi(c) 0.01 10 -2 centi(c) 0.01 10 -2 milli(m) 0.001 10 -3 milli(m) 0.001 10 -3 micro(µ) 0.000001 10 -6 micro(µ) 0.000001 10 -6 nano (n) 0.000000001 10 -9 nano (n) 0.000000001 10 -9 × base unit meter liter gram

49. 2.5 Prefix Multipliers For a particular measurement: For a particular measurement: Choose the prefix which is similar in size to the quantity being measured Choose the prefix which is similar in size to the quantity being measured Keep in mind which unit is larger 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 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 Choose the prefix most convenient for a particular measurement n < µ < m < c < base < k < M

50. 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 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 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 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 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

51. 2.6 Converting from One Unit to Another: Equalities An equality is a fixed relationship between two quantities An equality is a fixed relationship between two quantities It shows the relationship between two units that measure the same quantity It shows the relationship between two units that measure the same quantity These relationships are exact, not measured These relationships are exact, not measured 1 min = 60 s 1 min = 60 s 12 inches = 1 ft 12 inches = 1 ft 1 dozen = 12 items (units) 1 dozen = 12 items (units) 1L = 1000 mL 1L = 1000 mL 16 oz = 1 lb 16 oz = 1 lb 4 quarts = 1 gallon 4 quarts = 1 gallon

52. 2.6 Converting from One Unit to Another: Dimensional Analysis Conversion factor: An equality expressed as a fraction 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 It is used as a multiplier to convert a quantity in one unit to its equivalent in another unit May be exact or measured May be exact or measured Both parts of the conversion factor should have the same number of significant figures Both parts of the conversion factor should have the same number of significant figures

53. 2.7 Solving Multistep Conversion Problems: Dimensional Analysis Example (Conversion Factors Stated within a Problem) The average person in the U.S. consumes one-half pound of sugar per day. How many pounds of sugar would be consumed in one year? The average person in the U.S. consumes one-half pound of sugar per day. How many pounds of sugar would be consumed in one year?  State the initial quantity given (and the unit): One year State the final quantity to find (and the unit): Pounds  Write a sequence of units (map) which begins with the initial unit and ends with the desired unit: year day pounds

54. 2.7 Solving Multistep Conversion Problems: Dimensional Analysis Example  For each unit change, State the equalities: Every equality will have two conversion factors Every equality will have two conversion factors year day pounds 0.5 lb sugar =1day 365 days = 1 year

55. 2.7 Solving Multistep Conversion Problems: Dimensional Analysis Example State the conversion factors: State the conversion factors:  Set Up the problem:

56. Guide to Problem Solving when Working Dimensional Analysis Problems Identify the known or given quantity and the units of the new quantity to be determined Identify the known or given quantity and the units of the new quantity to be determined Write out a sequence of units which starts with your initial units and ends with the desired units (“solution map”) Write out a sequence of units which starts with your initial units and ends with the desired units (“solution map”) Write out the necessary equalities and conversion factors Write out the necessary equalities and conversion factors Perform the mathematical operations that connect the units Perform the mathematical operations that connect the units Check that the units cancel properly to obtain the desired unit Check that the units cancel properly to obtain the desired unit Does the answer make sense? Does the answer make sense?

57. 2.9 Density The ratio of the mass of an object to the volume occupied by that object The ratio of the mass of an object to the volume occupied by that object Density tells how tightly the matter within an object is packed together Density tells how tightly the matter within an object is packed together Units for solids and liquids = Units for solids and liquids = 1 cm 3 = 1 mL so can also use 1 cm 3 = 1 mL so can also use Unit for gases = g/L Unit for gases = g/L Density of three states of matter: solids > liquids >>> gases Density of three states of matter: solids > liquids >>> gases g/mL g/cm 3

58. 2.9 Density Can use density as a conversion factor between mass and volume Can use density as a conversion factor between mass and volume Density of some common substances given in Table 2.4, page 33 Density of some common substances given in Table 2.4, page 33 You will be given any densities on tests EXCEPT water You will be given any densities on tests EXCEPT water Density of water is 1.0 g/cm 3 at room temperature Density of water is 1.0 g/cm 3 at room temperature 1.0 mL of water weighs how much? 1.0 mL of water weighs how much? How many mL of water weigh 15 g? How many mL of water weigh 15 g?

59. 2.9 Density To determine the density of an object To determine the density of an object Use a balance to determine the mass Use a balance to determine the mass Determine the volume of the object Determine the volume of the object Calculate it if possible (cube shaped) Calculate it if possible (cube shaped) Can also calculate volume by determining what volume of water is displaced by an object Can also calculate volume by determining what volume of water is displaced by an object Volume of Water Displaced = Volume of Object

60. Density Problem Iron has a density of 7.87 g/cm 3. If 52.4 g of iron is added to 75.0 mL of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise? Iron has a density of 7.87 g/cm 3. If 52.4 g of iron is added to 75.0 mL of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise?

61. Density Problem Solve for volume of iron

62. End