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# Lecture_overheads_-_Ch03 (1) - CHEM 321 Quantitative...

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CHEM 321 – Quantitative Analysis Ch. 3 – Experimental Error 3-1 Significant Figures 3-2 Significant Figures in Arithmetic (standard rules for SF) 3-3 Types of Error 3-4 Propagation of Uncertainty ( real rule for SF) Skim article on “Significant Figures - Is 8°C = 50°F?” on iLearn Read Lab Manual Appendix B, especially example on how standard rules for SF can fail Review Text Appendix C for calculus-based error propagation

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3-1 Significant figures (SF) Definition Minimum # digits needed to express a number in scientific notation without loss of precision Working definition # of digits in a number up to and including 1 st uncertain digit 1. Last digit is 1 st uncertain digit and has an uncertainty of ± 1 unless otherwise specified 2. Errors are usually expressed to 1 SF
Examples 9.25 × 10 4 3 SF 9.250 × 10 4 4 SF 9.2500 × 10 4 5 SF 31.20 4 SF count zeros at end – significant otherwise wouldn’t write down 30.02 4 SF count embedded zeros 0.0021 2 SF don’t count leading zeros To avoid ambiguity, use scientific notation 82,000 # SF is ambiguous (could 2, 3, 4, or 5) 8.2 x 10 4 2 SF

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Measured quantity versus an exact quantity Measured quantities always have error Examples Weights measured on the balance (tolerance = ±0.0002 g) Volumes from volumetric glassware (tolerances in lab manual appendix A) Readings from an instrument (replicate measurements will give std deviation) Exact quantities generally do not have error, you’ll recognize an exact quantity when you see one Examples Exact mathematical quantities (2, 1/2, π ) Self-designated chemical concentrations (“target” concentrations) Physical constants (use error if given)
3-2 Significant figures in arithmetic (standard gen chem rules for SF) Rule Always round at the end of a calculation to avoid accumulation round-off errors Addition and Subtraction Rule # SF in an answer to addition or subtraction is determined by the # with the smaller # of decimal places or # with 1 st uncertainty in a decimal place Example 3.4 + 0.020 + 7.31 = 10.730 = 10.7 (why?) line up numbers at the decimal place 3.4 ← result limited by decimal places by this number 0.020 + 7.31 10.730 = 10.7 ← Note: first uncertainty in the decimal place

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More examples 5.345 + 6.728 = 12.073 Result may have more SF 7.26 - 6.69 = 0.57 Result may have fewer SF Rounding conventions Rule Round up if subsequent digit > 5 Rule Round down if subsequent digit < 5 Rule Round to nearest even number if subsequent digits = 50000... (note this is a rare case) Examples Round the following calibrated volumes to 4 SF 50.05499 → 50.05 50.05501 → 50.06 50.05500 → 50.06 50.06500 → 50.06
Multiplication and Division Rule # SF in an answer to multiplication or division is equal to the # digits in the # with the fewest SF Examples ( 29 ( 29 3 3 4 10 1.14 10 1.14124 2.77 10 4.12 - - - × = × = × × (3 SF) ( 29 ( 29 6 6 1 6 10 5.2 10 5.1980 10 8.5 10 6.1153 × = × = × × × - (2 SF) ( 29 ( 29 0 0 10 7.518 10 7.51761 8.9563 67.33 × = × = ÷ (4 SF)

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Significant Figures and Logs ( y = log x ) Rule for y = log x # digits in mantissa of y = # SF in x Example log x = y log 339 = 2.530 = y Note : 2 = "exponent of y " 530 = "mantissa of y " Example log 0.001237 = -2.9076 Example log 1237 = 3.0924
Significant Figures and antilogs (antilog y = 10 y ) Rule For antilog y = x # of digits in mantissa of y = # SF in x Example

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Lecture_overheads_-_Ch03 (1) - CHEM 321 Quantitative...

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