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1. Homework Assignment 02 __________________________________________ Homework Assignment 02...is from Chapter 3. Problems assigned are: 5,11,12,15,21,22This assignment is due at class time Friday, Sept 17 Prepare on regularly sized paper, one side only with multiple pages stapled.

2. Chapter 3 – Math Toolkit Logarithms __________________________________________ Consider the common logarithm of the number 4.265 x 10 2, that is log10 (4.265 x 10 2). This number has 4 significant figures, so its logarithm has 4 digits in its mantissa. The characteristic is related to the power of ten. The common log of 4.265 x 10 2. is 1.3701 which appears to have 5 significant figures, but the characteristic (1) is the power of ten. The preexponential part of the number (4.265) has 4 significant figures so the mantissa Should also have 4 digits.

3. Chapter 3 – Math Toolkit __________________________________________ For large numbers the characteristic of the logarithm is positive. Consider the common logarithm (base 10) of the number 78,436 or 7.8436 x 104. log107.8436 x 104= 4.89452. For numbers greater than 10, the characteristic is positive. (101 = 10). The characteristic is the same as if the number is written in scientific notation. There needs to be the same number of significant figures in the mantissa as present in the number itself.

4. Chapter 3 – Math Toolkit Types of Errors __________________________________________ • Systematic or determinate errors are repeatable. • Often systematic errors may be identified and then • corrected. • For example you might measure a length with a • faulty ruler, say one that had a cm cut off, or • measured the pH of a solution having standardized • with a buffer that you thought was 7.00, but whose • value was actually 8.00

5. Chapter 3 – Math Toolkit Types of Errors __________________________________________ The tolerance for a 50-mL class A buret is 0.05 mL. If you were pushing the limits of use of this device, you might calibrate within individual ranges as was discussed last time. Typical results of this calibration are shown above.

6. Chapter 3 – Math Toolkit Types of Errors __________________________________________ 2. Random or indeterminateerrors are those that arise from our limitations to make physical measurements. Random errors have equal chances to be positive or negative. They should occur at the level of the uncertainty of the measuring. Random errors cannot be eliminated, but may be reduced by better performed experiments.

7. Chapter 3 – Math Toolkit Standard Reference Materials __________________________________________ Standard Reference Materials are prepared and certified by the U.S. National Institute of Standards and Technology (NIST, formerly National Bureau of Standards, NBS). Standard reference materials are very important in establishing new methods of analysis and checking the results obtained by various laboratories and testing agencies.

8. Chapter 3 – Math Toolkit Standard Reference Materials __________________________________________ Errors in the analysis of anticonvulsant drugs before and after reference to standard materials.

9. Chapter 3 – Math Toolkit Precision vs. Accuracy __________________________________________ Precision is a measure of the reproducibility of the individual data within a set of results. Accuracy is a measure of how close the measured value is to the “true” value.

10. Chapter 3 – Math Toolkit Precision vs. Accuracy __________________________________________ Cheryl is neither accurate nor precise Cynthia is accurate but not precise Carmen is both accurate and precise Chastity is precise but not accurate

11. Chapter 3 – Math Toolkit Absolute vs. Relative Error __________________________________________ Absolute Error is the measurement of the uncertainty associated with the measurement. It is always given in terms of the same units as the measurement itself, such as for a balance, ± 0.1 mg. Relative Error is the measurement of the uncertainty expressed in terms of the magnitude of the measurement. Relative errors have no units, though they are often expressed as % or parts per thousand. relative error = absolute error / magnitude of measurement

12. Chapter 3 – Math Toolkit Absolute vs. Relative Error __________________________________________ What is the relative error whose absolute error is ± 0.1 mg if the measurement is 105 mg? relative error = 0.1 mg / 105 mg = 0.00095 or 0.095% or 0.95 ppt What is the relative error whose absolute error is ± 0.1 mg if the measurement is 40 mg? relative error = 0.1 mg / 40 mg = 0.25% or 2.5 ppt (Just like % means parts per 100 {per centum, Latin}, ppt means parts per thousand; or ppt = % X 10.)

13. Chapter 3 – Math Toolkit Propagation of Uncertainties in calculations __________________________________________ Whenever data is processed by doing the mathematical operations of addition, subtraction, multiplication or division, the uncertainties of each measurement is incorporated in the uncertainty we can associate with the final answer. Like the rules for significant figures, there are ways to account for the cumulative uncertainty 1) for addition or subtraction, and then 2) for multiplication or division.

14. Chapter 3 – Math Toolkit Propagation of Uncertainties – Addition/Subtraction __________________________________________ The uncertainties whenever data is added or subtracted is given by the expression _________________ en =  e12 + e22 + e32 + …..

15. Chapter 3 – Math Toolkit Propagation of Uncertainties – Addition/Subtraction __________________________________________ Consider the following measurements and their uncertainties. The desired result is X1 + X2 – X3. What is the value and the uncertainty in the final value? X1 1.76 (0.03) X2 1.89 (0.02) X3 0.59 (0.02) 1.76 + 1.89 – 0.59 = 3.06 _____________________ ______ e =  (0.03)2 + (0.02)2 + (0.02)2 =  0.0017 = 0.0412

16. Chapter 3 – Math Toolkit Propagation of Uncertainties – Addition/ subtraction __________________________________________ The resulting value could then be stated as 3.06  0.041 (The author’s {somewhat unique} use of subscripts in his way to indicate what the next digit is, although it is beyond what can be reported; in chain calculations where additional mathematical processing happens, carry one additional value, and then round off the final result.)

17. Chapter 3 – Math Toolkit Propagation of Uncertainties – Addition/subtraction __________________________________________ The error of  0.041 is in absolute measurements and in the same units as the 3.06 value. The error may also be expressed as relative uncertainty, commonly referred to as percent relative uncertainty, which is defined as % RU = e X 100 / value, or in this case % RU = (0.041)(100) / 3.06 = 1.34 % = 1.3% ; the same expressed in ppt is 13 ppt

18. Chapter 3 – Math Toolkit Propagation of Uncertainties __________________________________________ Multiplication and Division – First convert all errors to relative errors as described in the proceeding slide. _____________________ e =  (% e1)2 + (% e2)2 + (% e3)2 Note that e is the error in absolute measurement units. The relative error is e X 100 / value. (See example bottom of page 62)

19. Chapter 3 – Math Toolkit Propagation of Uncertainties __________________________________________ Mixed operations - whenever the calculation involves both addition or subtractionand multiplication or division, work through the addition/subtration first, then the multiplication/division. Note the example, middle of page 63.

20. Chapter 3 – Math Toolkit Summary Regarding Significant Numbers __________________________________________ The first uncertain figure of the answer is the last significant figure.

21. Chapter 3 – Math Toolkit Excel Spreadsheets __________________________________________ A spreadsheet consists of a collection of columns (vertical arrangement)and rows(horizontal arrangement). The intersection of a given column and row defines a cell. Within a cell you may enter text, numerical values, or formulas and functions. Click to Link to Spreadsheet