Transcript Chapter 1.09 sig figs_21sep15

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