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# Chapter 2 – Analyzing Data

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1. Chapter 2 – Analyzing Data • 2.1 Units and Measurement [and standard problem solving technique illustrated using density] • 2.2 Scientific Notation and Dimensional Analysis • 2.3 Uncertainty in Data • 2.4 Representing Data

2. Section 2.1 Units and Measurements Chemists use an internationally recognized system of units to communicate their findings. Objectives • Know the first five of the seven base units in the SI system (time, length, mass, temperature and amount of substance) including their units, symbols, and the basic physical objects or phenomena on which they are defined. • Convert between the Kelvin and Celsius temperature scales. • Distinguish between and give examples of base units and derived units.

3. Section 2.1 Units and Measurements Objectives (cont) • Use the equation for density to solve for an unknown quantity using the problem solving process described on page 38 in example problem 2.1. • Understand both the concepts and experimental procedure involved in the Mini Lab on page 39 (Determine Density) and be able to answer the analysis questions listed at the end of the lab. • Know all the prefixes from giga to nano in table 2.2 including their symbols and associated powers of ten and be able to use them in a numerical problem.

4. Section 2.1 Units and Measurements Key Concepts • SI measurement units allow scientists to report data to other scientists. • Adding prefixes to SI units extends the range of possible measurements. • To convert to Kelvin temperature, add 273.15 to the Celsius temperature. K = °C + 273.15 • Volume and density have derived units. Density, which is a ratio of mass to volume, can be used to identify an unknown sample of matter.

5. Units of Measurement • SI - Systemè Internationale d’Unités • 7 base units • Base unit is based on an object or an eventin the physical world • Base unit is independent of other units

6. The 7 SI Base Units (see Table 2.1) Only base unit with prefix

7. Prefixes • To better describe range of possible measurements, base units (and other units) are modified by using prefixes • Correspond to a particular power of ten • See table 2.2 (following) • Memorize value and the symbol (including upper /lower case) of prefixes in 10-9 to 106 range

8. Capital letters Greek letter SI Prefixes – Page 33

9. SI Prefixes – Expanded

10. Base Unit - Time • The Second • In 1967, the 13th General Conference on Weights and Measures first defined the SI unit of time asthe duration of 9,192,631,770 cycles of microwave light absorbed or emitted by the hyperfine transition of cesium-133 atoms in their ground state undisturbed by external fields

11. Length The Meter Distance light travels through a vacuum in 1/(299 792 458) of a second

12. Mass • kilogram (kg) • Only base unit whose standard is a physical object • Defined by platinum-iridium metal cylinder kept in Sèvres, France

13. Temperature • kelvin (K) • The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water

14. Temperature • Kelvin scale • Unit kelvin (small k), abbreviation K (but no degree sign) • Absolute zero = zero K • Celsius scale • Unit °C (with degree sign) • T(C) = T(K) - 273.15 • Kelvin scale important for certain formulas we will use in later chapters

15. Boiling point water 373.15 K 100.00 C Freezing point water 273.15 K 0.00 C Absolute zero 0.00 K -273.15 C Temperatures – K vs C

16. The Mole and Avogadro’s Number • Abbreviation is mol • SI base unit for amount of substance • Defined as number of representative particles (carbon atoms) in exactly 12 g of pure carbon-12 • Mole of anything contains 6.022 X 1023 representative particles • 6.022 X 1023 = Avogadro’s number

17. Picky Details - Units • Abbreviations are avoided • Proper • s or second • cm3 or cubic centimeter • m/s or meter per second • Improper • sec • cc • mps

18. Picky Details - Units • Unit symbols are unaltered in the plural • Proper • l = 75 cm • Improper • l = 75 cms

19. Derived Units • Defined by a combination of base units • Velocity v = l/t m/s • Volume V = l  l  l m3 • liter (L) cubic decimeter dm3 • Density d = mass / V kg/m3

20. Derived Unit - Density • Density = mass volume • Most common unit is g/cm3 • Will return to using density in a word problem later in presentation

21. Chapter 2 – Analyzing Data • 2.1 Units of Measurement [and standard problem solving technique illustrated using density] • 2.2 Scientific Notation and Dimensional Analysis • 2.3 Uncertainty in Data • 2.4 Representing Data

22. Section 2.2 Scientific Notation and Dimensional Analysis Scientists often express numbers in scientific notation. • Express numbers correctly in standard scientific notation, convert them to and from numbers not expressed in scientific notation, and perform standard arithmetic operations using them.

23. Section 2.2 Scientific Notation and Dimensional Analysis KeyConcepts • A number expressed in scientific notation is written as a coefficient between 1 and 10 multiplied by 10 raised to a power.To add or subtract numbers in scientific notation, the numbers must have the same exponent. • To multiply or divide numbers in scientific notation, multiply or divide the coefficients and then add or subtract the exponents, respectively.

24. Scientific Notation • Expresses a number as a multiple of two factors: • 1  number  10 3.1 -7.9 • 10 raised to a power 103 10-7 • 100 = 1 • 10n > 1 n positive integer • 0 < 10-n < 1 n positive integer

25. Scientific Notation • 3.1  104 = 31000 • 3.1  10-4 = 0.00031 • Have moved decimal point 4 places in both cases • Some exponents match one of the standard SI prefixes • 4.27  10-6 s = 4.27 s (microseconds)

26. Scientific Notation • Addition/Subtraction • Must be same power of ten • Multiplication/Division • Multiply first factors • Add exponents if multiplying • Subtract exponents if dividing • Put result back into standard form • z.yyy  10n z = any non-zero digit • One digit in front of decimal point

27. Practice – Scientific Notation General Problems 11(a-h), 12(a-d) page 41 Problems 76(a-d), 77(a-d) page 62 Addition & Subtraction Problems 13(a-d), 14(a-d) page 42 Problems 78(a-j) page 62 Multiplication & Division Problems 15(a-d), 16(a-d) page 43 Problems 79(a-f) page 62

28. Chapter 2 – Analyzing Data • 2.1 Units of Measurement [and standard problem solving technique illustrated using density] • 2.2 Scientific Notation and Dimensional Analysis • 2.3 Uncertainty in Data • 2.4 Representing Data

29. Section 2.3 Uncertainty in Data Measurements contain uncertainties that affect how a result is presented. • Defineand compare accuracy and precision and correctly identify which term or terms apply to a given value based on a description of how the value was determined. • Calculatethe percent error associated with a given measurement. • Determine the number of significant figures associated with a given number.

30. Section 2.3 Uncertainty in Data Objectives (cont) • Determine the appropriate number of significant figures to record when using an analog measuring device such as a ruler. • Determine the number of significant figures associated with a result obtained from simple arithmetic operations (addition, subtraction, multiplication, division) on numbers. • Round a number to a specified number of significant digits.

31. Section 2.3 Uncertainty in Data Objectives (cont) • Show all work for a problem following the Problem-Solving Process described on page 38 in example problem 2.1 while utilizing dimensional analysis, significant figures, rounding and standard algebra skills.

32. Section 2.3 Uncertainty in Data Key Concepts • An accurate measurement is close to the accepted value. A set of precise measurements shows little variation. • The measurement device determines the degree of precision possible. • Error is the difference between the measured value and the accepted value. Percent error gives the percent deviation from the accepted value. • error = experimental value – accepted value

33. Section 2.3 Uncertainty in Data (cont.) Key Concepts • The number of significant figures reflects the precision of reported data. • Calculations should be rounded to the correct number of significant figures. • The rules for determining the number of significant figures in a number produced in a mathematical operation are different for multiplication/division and addition/subtraction.

34. Accuracy and Precision • Accuracy refers to agreement of particular value with true value (sometimes true value is difficult to determine; may require appropriate calibration) • Precision refers to degree of agreement among several measurements of same quantity

35. Calibration • [Formal definition – don’t need to know] • The set of operations which establish, under specified conditions, the relationship between values indicated by a measuring instrument or measuring system, and the corresponding standard or known values derived from the standard. • Purpose is to determine and/or improve the accuracy of the device being calibrated

36. Calibration - Example • To calibrate an electronic balance, might use following approach: • Assign a value of zero to the reading of the output of the balance when nothing is on the balance pan • Assign a value equal to the value of a calibration mass (e.g., 1000.00 g) to the reading of the output of the balance when the calibration mass is placed on the balance pan • Assume linear behavior of the reading of the balance between these two calibration points

37. Calibration - Example • For calibration to be possible, need a calibration mass (e.g., a 100.00 g mass) • The actual (true) value of the calibration mass has to be determined by the organization/company supplying the mass – this ultimately requires tracing back to the primary standard of mass (making comparisons that are linked to the kilogram standard mass) • It is only through calibration that the accuracy of a measurement device (i.e., a balance) can be determined and improved

38. Accuracy and Precision • Refer to figure 2.10

39. Accuracy and Precision • Refer to figure 2.10

40. precise and accurate precise but not accurate Precision and Accuracy

41. Precision and Accuracy random error Not precise, individual measurement not generally accurate but average is systematic error May be precise (only marginally for this example); average not accurate

42. Percent Error • % Error = 100 (actual – measured) actual • (actual – measured) = error • Can ignore plus or minus signs for now – need only absolute value of error • Actual = “true value” • In some cases, true value unknown • (Often compare a predicted value from some model with measured value)

43. Practice Accuracy and Precision Problems 46, 48, page 54 Problem 87 page 63 Percent Error Problems 32-34 page 49 Problems 49, 51 page 54 Problems 93(a-d), 94(a-d) page 63

44. Significant Figures • 2 different types of numbers: • Exact • Measured • Exact numbers have infinite precision • Measured - obtained from measuring device, have error and limited precision • Measured number written to reflect both its numerical value and the precision to which it was measured

45. Significant Figures and Measurement Precision • For lab data, sig figs determined by precision of measurement device • For digital device, last number to right in display is limit to precision • For analog device, precision limited by the estimated digit obtained from “eyeballing” reading between markings • Examples to follow

46. Balances and Precision Student  0.1 g (digital) Electronic analytical  0.0001 g (digital) Triple beam  0.01 g (analog)

47. Significant Figures • Mass of object measured on student balance (precision ± 0.1g) is 23.6 g • This quantity contains 3 significant figures, i.e., three experimentally meaningful digits • If same measurement made with analytical balance (precision ± 0.0001g) , mass might be 23.5820 g (6 sig. fig.)

48. Significant Figures and Precision 2 measurements of mass of same object Same quantity described at two different levels of precision or certainty

49. Significant Figures Determined by Measuring Device 32.3 C 32.33 C

50. Significant Figures Determined by Measuring Device 0.1 mL graduations Can estimate to 0.01 mL Reading 1.70 mL 3 significant figures