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 ) Fraction is exact, but unlimited precision not intended Context says the most certain part is 0.6 Uncertain part is probably the 2 nd 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 –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 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 o C has only 1 significant figure –0.012 g has 2 significant figures – 1.51 nanometers ( meter), 3 sig figs Trailing zeroes that are to the right of a decimal point with numerical values are always significant: – mL has 3 significant figures –0.20 g has 2 significant figures –1.510 nanometers ( meters), 4 sig figs

More examples with zeros Leading zeros don’t count –Often just a scale factor ( = 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 – has __ sig-figs – has __ sig-fig –0.012 has __ sig-figs Zeros at end –35.00 has __ sig-figs –8, 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 – has 5 sig-figs – has 1 sig-fig –0.012 has 2 sig-figs Zeros at end –35.00 has 4 sig-figs –8, 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 – × 10E4 calories (5 significant figures). –Remember values right of decimal ARE significant

Sig-Fig Addition & Subtraction Least Significant Figure determines outcome Solve this problem: 1.023E E-4 – First get the decimals (blue #) to align –Take E3 same as 1,023.4 –Then add 1.0E-4same as –Then subtract 15.22same as –Do the math 1, –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 = / = 1.01 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 =  68.2 = 6.82E1 3 sig figs due to 15.0 value

Mixed additon & multiplication ( – ) x (3.248E4 – E3) Solve what’s inside parenthesis FIRST –Initial value 1 st parenthesis E-3 –Subtract 2 nd value E-3 –Result after subtraction E-3 –Round to least accurate E-3 Second Parenthesis Calculation –3.248E4 same as 32, E3 –Subtract E3 same as 4, E3 –Result after subtraction 27, E3 –Round to low of 4 sig fig 27, E3 Multiply results from parenthesis calculations – * 27,890 =  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