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# Mathematics in Chemistry

Download Presentation ## Mathematics in Chemistry

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1. Outline • Mathematics in Chemistry • Units • Rounding • Digits of Precision (Addition and Subtraction) • Significant Figures (Multiplication and Division) • Order of Operations • Mixed Orders • Scientific Notation • Logarithms and Antilogarithms • Algebraic Equations • Accuracy and Precision • Statistics • Serial Dilutions • Direct Dilutions • Graphing • Calibration Curves • MicroLAB™ • The Program • Reference Sheet • Pitfalls

2. Mathematics in Chemistry • Math is a very important tool, used in all of the sciences to model results and explain observations. • Chemistry in particular requires a lot of calculations before even trivial experiments can be performed. In this first exercise you will be introduced to some of the very basic calculations you will be required to perform in lab during the entire semester. • Remember, if you start memorizing rules and formulas now, you don’t have to do it the night before your exams!

3. Units • Units are very important! • Units give dimension to numbers. • They also allow us to use dimensional analysis in our calculations. • If a unit belongs next to a number, place it there!!! • Example: 6.23 mL The unit “mL” indicates to us that our measurement is a metric system volume and indicates to us the order of magnitude of that volume.

4. Rounding When you have to round to a certain number to obey significant figure rules, remember to do the following: For numbers 1 through 4, round down • For numbers 6 through 9, round up • For numbers with a terminal 5, round to the closest even number. 0.01255 rounded to three significant digits becomes 0.0126 0.01265 rounded to three significant digits becomes 0.0126 0.01275 rounded to three significant digits becomes 0.0128 0.012852 rounded to three significant digits becomes ? Why is this method statistically more correct?

5. Digits of Precision and Significant Figures • All measurements have some degree of uncertainty due to limitations of measuring devices. • Scientists have come up with a set of rules we can follow to easily specify the exact amount of significant figures, without sacrificing the accuracy of the measuring devices.

6. Digits of Precision: Addition and Subtraction Your answer must contain no more digits after the decimal point than the number with the least number of digits after the decimal point. 104.75 + 209.7852 + 1.1 = 315.6

7. Digits of Precision: Addition and Subtraction 205.12234 – 72.319 + 4.68 = 137.48334 137.48

8. Addition of Whole Numbers When you add or subtract whole numbers, your answer cannot be more accurate than any of your individual terms. 20 + 34 + 2400 – 100 = 2400 What about: 319 + 870 + 34,650 = ?

9. Addition of Whole Numbers When you add or subtract whole numbers, your answer cannot be more accurate than any of your individual terms. 20 + 34 + 2400 – 100 = 2400 What about: 319 + 870 + 34,650 = ? The answer is 35,840

10. Significant Figures Rule #1 Numbers with an infinite number of significant digits do not limit calculations. These numbers are found in definite relationships, otherwise known as conversion factors. 100 cm = 1 m 1000 mL = 1 L

11. Significant Figures Rule #2 All non-zero digits are significant. 1.23 has 3 significant figures 98,832 has 5 significant figures How many significant digits does 34.21 have?

12. Significant Figures Rule #2 All non-zero digits are significant. 1.23 has 3 significant figures 98,832 has 5 significant figures How many significant digits does 34.21 have? Correct! The answer is 4.

13. Significant Figures Rule #3 The number of significant figures is independent of the decimal point. 12.3, 1.23, 0.123 and 0.0123 have 3 significant figures 0.0004381 and 0.4381 have how many significant figures?

14. Significant Figures Rule #3 The number of significant figures is independent of the decimal point. 12.3, 1.23, 0.123 and 0.0123 have 3 significant figures 0.0004381 and 0.4381 have how many significant figures? Correct! The answer is 4.

15. Significant Figures Rule #4 Zeros between non-zero digits are significant. 1.01, 10.1, 0.00101 have 3 significant figures. How many significant digits are in 10,101?

16. Significant Figures Rule #4 Zeros between non-zero digits are significant. 1.01, 10.1, 0.00101 have 3 significant figures. How many significant digits are in 10,101? The answer is 5!

17. Significant Figures Rule #5 After the decimal point, zeros to the right of non-zero digits are significant. 0.00500 has 3 significant figures 0.030 has 2 significant figures. How many significant figures are in 34.1800?

18. Significant Figures Rule #5 After the decimal point, zeros to the right of non-zero digits are significant. 0.00500 has 3 significant figures 0.030 has 2 significant figures. How many significant figures are in 34.1800? This one has 6 significant digits.

19. Significant Figures Rule #6 If there is no decimal point present, zeros to the right of non-zero digits are not significant. 3000, 50000, 20 all have only 1 significant figure How many significant figures are in 32,000,000?

20. Significant Figures Rule #6 If there is no decimal point present, zeros to the right of non-zero digits are not significant. 3000, 50000, 20 all have only 1 significant figure How many significant figures are in 32,000,000? The answer is 2!

21. Significant Figures Rule #7 Zeros to the left of non-zero digits are never significant. 0.0001, 0.002, 0.3 all have only 1 significant figure How many significant figures are in 0.0231? How many significant figures are in 0.02310?

22. Significant Figures Rule #7 Zeros to the left of non-zero digits are never significant. 0.0001, 0.002, 0.3 all have only 1 significant figure How many significant figures are in 0.0231? This one has 3 significant digits. How many significant figures are in 0.02310? This one has 4 significant digits.

23. Significant Figures: Multiplication and Division Your answer must contain no more digits total than the number with the least number of digits total. 5.10 x 6.213 x 5.425 = 172

24. Significant Figure Multiplication and Division = 76.016 76

25. Order of operations 1st: ( ), x2, square roots 2nd: x or / 3rd: + or –

26. Significant Figure Mixed Orders

27. Scientific Notation The three main items required for numbers to be represented in scientific notation are: • the correct number of significant figures • one non-zero digit before the decimal point, and the rest of the significant figures after the decimal point • this number must be multiplied by 10 raised to some exponential power 123 becomes 1.23 x 102 This number has three significant digits

28. Scientific Notation • Calculators could be a significant aid in performing calculations in scientific notation. • KNOW HOW TO USE YOUR CALCULATOR • Does your calculator retain or suppress zeros in its display? • In converting between scientific and decimal notation, the number of significant digits don’t change.

29. Scientific Notation Conversions • What is the scientific notation equivalent of 0.0432? 1043.50? • What is the standard decimal notation equivalent of 3.45 x 103? 6.500 x 10-2?

30. Scientific Notation • What is the scientific notation equivalent of 0.0432? The answer is 4.32 x 10-2 1043.50? The answer is 1.04350 x 103 • What is the standard decimal notation equivalent of 3.45 x 103? This is 3450 6.500 x 10-2? This is 0.06500

31. Scientific Notation Calculations • Addition: (4.22 x 105) + (3.97 x 106) = (4.22 x 105) + (39.7 x 105) = (4.22 + 39.7) x 105 = 43.9 x 105 = 4.39 x 106 Know how to perform these types of calculations on your calculator!

32. Scientific Notation Calculations • Subtraction: (4.22 x 105) - (3.97 x 106) = (4.22 x 105) - (39.7 x 105) = (4.22 – 39.7) x 105 = -35.5 x 105 = -3.55 x 106 Know how to perform these types of calculations on your calculator!

33. Scientific Notation Calculations • Multiplication: (4.22 x 105) x (3.97 x 106) = (4.22 x 3.97) x 10(5+6) = 16.8 x 1011 = 1.68 x 1012 Know how to perform these types of calculations on your calculator!

34. Scientific Notation Calculations • Division: (4.22 x 105) / (3.97 x 106) = (4.22 / 3.97) x 10(5-6) = 1.06 x 10-1 Know how to perform these types of calculations on your calculator!

35. Logarithms • Logarithms might seem strange, but they are nothing more than another way of representing exponents. • logbx = y is the same thing as x = by • Know how to use your calculator to perform these functions.

36. Logarithms We see logarithms frequently when working with pH chemistry. If you have a solution of pH 5.2, and you need to calculate the concentration of hydrogen ions, set the problem up as follows: pH = - log [H+] 5.2 = - log [H+] -5.2 = log [H+] 10-5.2 = 10log [H+] 10-5.2 = [H+] [H+] = 6.3 x 10-6

37. Logs and Antilogs To enter log 100 on your calculator: • Press: log  1  0  0  Enter or • Press: 1  0  0  log for reverse entry To enter the antilog 2 on your calculator: • Press: 2nd  log  2  Enter or • Press: 2  2nd  log for reverse entry Did you notice anything?

38. Significant Figure Rules • Logarithms log (4.21 x 1010) = 10.6242821  10.624 • Antilogarithms antilog (- 7.52) = 10-7.52 = 3.01995 x 10-8 3.0 x 10-8

39. Significant Figures of EquipmentElectronics • Always report all the digits electronic equipment gives you. • When calibrating a probe, the digits of precision of your calibration values determine the digits of precision of the output of the data.

40. Algebraic Equations • It is important to understand how to manipulate algebraic equations to determine unknowns and to interpolate and extrapolate data. Don’t forget about significant figures. For y = 1.0783 x + 0.0009 If x = 0.021, find y (answer = 0.024) If y = 4.3, find x (answer = 4.0)

41. Accuracy • The accuracy of a measurement represents a comparison of the measured value (experimental value) to the “true” value. • A measure of accuracy is indicated by: Percent Error = • Tolerances of glassware affect the accuracy of volume measurements.

42. Precision • Precision of a measurement reflects reproducibility of an experimental procedure. • Refer to the bull’s eye experiment on page 60. • Graduations on glassware affect the precision of the glassware in question.

43. Statistics • We use statistics in the laboratory in order to validate our results. • We evaluate the central tendency of our work by calculating the mean (related to accuracy) of our data. • We evaluate the variability in our work by calculating the standard deviation (s) (related to precision) of our data. • The relative standard deviation gives us a more meaningful number than the standard deviation.

44. Calculation of the Mean xi = individual values N = number of measurements For significant figures, always keep as many digits after the decimal point as the original values. Remember units!

45. Calculation of the standard deviation of a set of numbers s = xi = individual values = the average of the individual values N = number of measurements For significant digits, report the same digits of precision as the xi values. The units are the same as the units for the x values.

46. Calculation of Relative Standard Deviation RSD% = s = standard deviation of a set of data = average of the individual measurements The calculation itself dictates the number of significant digits. What would the units be?

47. Dilutions Using a solution of known concentration for the preparation of a solution with a lower concentration is commonly called dilution.

48. Solution Preparation from Solids • Determine the mass of the solid needed. You will need the following: • Molar mass of the solid • Total volume desired • Final concentration desired • Calculation: m = M x MM x V g = mol/L x g/mol x L • Remember the precision of your glassware!

49. Solution Preparation from Solids Make the solution: • Weigh out the appropriate mass of solid. • Place a small volume of distilled water in the volumetric flask. • Add the solid to the volumetric flask. • Add some more distilled water to the flask, stopper, and invert several times. • Add distilled water to the calibration line (fill to volume) using a medicine dropper, stopper, and invert several times.