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Chemistry 15, Fall 2015 Measurement & Significant Figures • Precision must be tailored for the situation – Result cannot be more precise than input data • Data has certain + uncertain aspects – Certain digits are known for sure – Final (missing) digit is the uncertain one – 2/3 cups of flour (intent is not 0.66666666667) • Fraction is exact, but unlimited precision not intended • Context says the most certain part is 0.6 • Uncertain part is probably the 2nd digit • Recipe probably works with 0.6 or 0.7 cups • How to get rid of ambiguity? Significant Figures • “Sig Figs” = establish values of realistic influence – 1cup sugar to 3 flour does not require exact ratio of 0.3333333 – Unintended accuracy termed “superfluous precision” – Need to define actual measurement precision intended – “Cup of flour” in recipe could be +/- 10% or 0.9 to 1.1 cup • Can’t be more Sig-Figs than least accurate measure – Final “Sig Fig” is “Uncertainty Digit” … least accurately known – adding .000001 gram sugar to 1.1 gram flour = 1.1 gram mixture How to Interpret Sig-Figs (mostly common sense) • All nonzero digits are significant – 1.234 g has 4 significant figures, – 1.2 g has 2 significant figures. • “0” between nonzero digits significant: – 3.07 Liters has 3 significant figures. – 1002 kilograms has 4 significant figures Handling zeros in Sig-Figs • Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: – 0.001 oC has only 1 significant figure – 0.012 g has 2 significant figures – 1.51 nanometers (0.00000000151 meter), 3 sig figs • Trailing zeroes that are to the right of a decimal point with numerical values are always significant: – 0.0230 mL has 3 significant figures – 0.20 g has 2 significant figures – 1.510 nanometers (0.000000001510 meters), 4 sig figs More examples with zeros • Leading zeros don’t count – Often just a scale factor (0.000001 = microgram) • Middle zeros between numbers always count – 1.001 measurement has 4 decades of accuracy • Trailing zeros MIGHT count – YES if part of measured or defined value: 5280 feet/mile – YES if placed intentionally, 7000 grains ≡ 1 pound – NO if zeros to right of non-decimal point • 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs – NO if only to demonstrate scale or orders of magnitude • Carl Sagan’s “BILLIONS and BILLIONS of stars” – Does NOT mean “BILLIONS” + 1 = 1,000,000,001 Sig-Fig Class Quiz … How many sig figs below? • Zeros between – 60.8 has __ significant figures – 39008 has __ sig-figs • Zeros in front – 0.093827 has __ sig-figs – 0.0008 has __ sig-fig – 0.012 has __ sig-figs • Zeros at end – 35.00 has __ sig-figs – 8,000.000 has __ sig-figs – 1,000 has ___ sig figs Sig Fig quiz answers • Zeros between – 60.8 has 3 significant figures – 39008 has 5 sig-figs • Zeros in front – 0.093827 has 5 sig-figs – 0.0008 has 1 sig-fig – 0.012 has 2 sig-figs • Zeros at end – 35.00 has 4 sig-figs – 8,000.000 has 7 sig-figs – 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4 Exact Values • Some numbers are exact because they are known with complete certainty. • Most exact numbers are simple integers: – 12 inches per foot, 12 eggs per dozen, 3 legs to a tripod • Exact numbers are considered to have an infinite number of significant figures. • When using an exact number in a calculation, the idea of significant figures for that item is ignored when determining the number of significant figures in the result of a calculation – 2.54 cm per inch (exact, NOT 3 sig figs) – 5/9 Centigrade/Fahrenheit degree (exact) – 5280 feet per mile (exact, based on definitions) – The challenge is to remember which numbers are exact Sig-Figs with Exponents • A number ending with zeroes NOT to right of decimal point are not necessarily significant: – 190 miles could be 2 or 3 significant figures – 50,600 calories could be 3, 4, or 5 sig-figs • Ambiguity is avoided using exponential notation to exactly define significant figures of 3, 4, or 5 by writing 50,600 calories as: – 5.06 × 10E4 calories (3 significant figures) or – 5.060 × 10E4 calories (4 significant figures), or – 5.0600 × 10E4 calories (5 significant figures). – Remember values right of decimal ARE significant Sig-Fig Addition & Subtraction Least Significant Figure determines outcome • Solve this problem: 1.023E3 + 1.0E-4 – 15.22 • First get the decimals (blue #) to align – Take 1.0234E3 same as 1,023.4 – Then add 1.0E-4 same as – Then subtract 15.22 same as + 0.0001 - 15.22 – Do the math 1,008.1803 – Round to least decimal sig fig 1,008.2 – 1.0E-4 vanishes …“spitting in the ocean” analogy … if you measure ocean volume by cubic meters or miles, adding a teaspoon is undetectable ! Avoid ambiguity! • 2+3*4 = ? – Is it : (2+3)*4 = 5*4 = 20 – Or : 2+(3*4) = 2+12 = 14 Avoid ambiguity! • 2+3*4 = ? – Is it : (2+3)*4 = 5*4 = 20 NO ! – Or : 2+(3*4) = 2+12 = 14 YES • Always do multiplications first, computers work the same way • Do what’s inside parentheses first • Add parentheses for clarification Sig-Fig Multiplication & Division Least Significant Figure determines outcome 1.01 x 1.0000001 = 1.01 1.01 / 1.0000001 = 1.01 • Write equation, do calculation, set sig fig – 1,023.4 x 15.0 = 15,351 15,400 = 1.54E4 3 sig figs due to 15.0 value – 1,023.4 / 15.0 = 68.22666 68.2 = 6.82E1 3 sig figs due to 15.0 value Mixed additon & multiplication (0.0048965 – 0.00347) x (3.248E4 – 4.58983E3) • • Solve what’s inside parenthesis FIRST – Initial value 1st parenthesis 0.0048965 4.8965 E-3 – Subtract 2nd value 0.00347 3.47 – Result after subtraction 0.0014265 1.4265 E-3 – Round to least accurate 0.00143 1.43 32,480 32.48 E-3 Second Parenthesis Calculation – 3.248E4 same as – Subtract 4.58983E3 same as • E-3 E3 4,589.83 - 4.58983 E3 – Result after subtraction 27,890.17 27.89017 E3 – Round to low of 4 sig fig 27,890 27.89 E3 Multiply results from parenthesis calculations – 0.00143 * 27,890 = 39.88270 39.9 – Multiplication accuracy limited to least sig figs = 3 in this case Conversions should be comparable in size • 1.2 miles ? Feet – 1.000 Mile ≡ 5280 feet (by definition) – 1.2 mile * 5280 ft/mi = 6336 feet calculated – Do we round to 6300 feet ?? (2 sig fig) • Maybe not, mile dimension >> foot dimension • Rounding off 36 feet may be excessive (look to context) – What about tolerances? • 3rd sig fig on 1.2(?) mile = +/- .05 mile = +/- 264 feet • 3rd sig fig on 1.2(?) foot = +/- .05 foot = +/- 0.6 inch • Very different practical result for different size units – Engineering practice, metric versus english • cm with 2 sig fig inch with 3 sig fig • 2.5 cm 1.00 inch