Transcript Significant Figures DRB

Init fall 2009 by Daniel R. Barnes
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What’s the difference between these two pictures?
They’re both pictures of the same kitten, but the picture on
the left has a much higher resolution than the “lo-res”
picture on the right.
Perhaps the picture on the left was taken with a much
more fancy, expensive camera than what was used to take
the picture on the right.
Just for fun, squint your eyes really tight and look at both of
the pictures. Stare with your eyes squinted really tight for
about twenty seconds.
Now, open your eyes wide again.
Looks different, doesn’t it?
. . . explain why reporting numbers to the correct
number of significant digits is good practice.
When scientists report numerical data from an experiment, they
have to report the data using the correct number of digits so that
other scientists know just how exact their measurements and
calculations are.
When you report your numerical data using only the correct
number of significant digits, you are admitting that your measuring
devices are not perfect.
The practice of reporting the correct number of significant digits is
an exercise in humility.
Maybe I can illustrate this idea with an imaginary example . . .
Imagine that you are
driving from Los
Angeles to San
Francisco and you
want to calculate your
average speed when
you’re done with the
trip.
Your odometer reads
“73,294.8” when you
start your trip in Los
Angeles.
Your digital
wristwatch reads
“10:58:50 PM” when
you leave.
At the end of the trip, when you reach San Francisco, your
odometer reads “73,676.4”, and your digital wristwatch says
“5:16:52 AM”. (You drove all night in the dark and arrived just in
time to see the sun rise over the Golden Gate Bridge.)
Let’s do the math . . .
odoi = 73,294.8 miles
odof = 73,676.4 miles
distance = (73,676.4 – 73,294.8) miles
distance = 381.6 miles
ti = 10:58:50 PM
tf = 5:16:52 AM the next morning
Dt = 6 h, 18 min, 2 seconds = 6.300555556 hours
speed = distance / time
= 381.6 mi / 6.300555556 hours
speed = 381.6 mi / 6.300555556 hours
speed = 60.56608765 mi/h
That’s a very fancy-looking answer, but here’s the problem . . .
distance = 381.6 miles
time = 6.300555556 hours
D xxxxxx
speed = 60.56608765 mi/h
!
7 mi/h”
“60.56
Do we really have any business reporting our speed by giving a
number that has ten digits in it?
Our odometer only measures
down to a tenth of a mile,
so our distance calculation has
only four digits in it.
A chain is only as strong as its
weakest link, so our answer
really only deserves to have four
digits in it, too.
distance = 381.6 miles
time = 6.300555556 hours
xxxxxx
speed = 60.56608765 mi/h
“60.57 mi/h”
Do we really have any business reporting our speed by giving a
number that has ten digits in it?
Our odometer only measures
down to a tenth of a mile.
so our distance calculation has
only four digits in it.
A chain is only as strong as its
weakest link, so our answer
really only deserves to have four
digits in it, too.
Let’s look at another imaginary example.
Let’s say your
sister is on a diet
and she’s just
lost some
weight, so she
calls you on the
phone from
across town and
says she weighs
135.2 lbs.
Curious to
see how you
compare, you
weigh yourself
on your very
different
bathroom
scale and turn
out to be
114 lbs.
Your sister mentions how nice it is you’ve both lost so much
weight, but claims that if you both climbed onto your daddy’s
shoulders at the same time, your combined weight would still
be too much for him to carry.
You pull your calculator out of your purse and punch in the
numbers.
114 lbs + 135.2 lbs = 249.2 lbs . . . right?
Once again, there’s a bit of a problem. Your sister’s scale
measures down to a tenth of a pound.
However, your more old-fashioned scale isn’t that exact.
No matter how much you squint at that needle, you don’t feel
confident estimating your weight to the closest tenth of a
pound. Heck, when you’re standing on the scale, your eyes
are so far above that dial that you’re not even sure if you’re
closer to 114 lbs or 115 lbs!
Considering all this, when you report the combined weight of
your sister and you, do you really have any business claiming
that you know your combined mass down to a tenth of a
pound?
114 lbs + 135.2 lbs = 249.2 lbs,
but you better just report the total as 249 lbs.
Enough intro.
Let’s learn some skills.
. . . determine how many significant digits there
are in any number put before them.
Q: How many significant figures are there in the measurement. . .
2009 y
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
3.14 ft
A: 3 significant figures
Q: How many significant figures are there in the measurement. . .
386,000,000.10 K
A: 11 significant figures
Q: How many significant figures are there in the measurement. . .
55.85 amu
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
x
x x x
0.00076 g
A: 2 significant figures
Q: How many significant figures are there in the measurement. . .
29.25 days
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
x
x x
0.003009 ms
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
1776 y
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
x
x x x
0.00071120 nm
A: 5 significant figures
Q: How many significant figures are there in the measurement. . .
x
x x x x x
0.000002870500 mm
A: 7 significant figures
Q: How many significant figures are there in the measurement. . .
14.01 amu
A: 4 significant figures
Q: How many significant figures are there in the measurement. . .
6.022 x
23
10
molecules/mole
A: 4 significant figures
Only the coefficient counts,
Not the power of ten.
Q: How many significant figures are there in the measurement. . .
12 inches/foot
12.0000000000000000000000000000000000000000000000 . . .
A: infinite significant figures
HUH? Well, because this isn’t a measured number, but is a
declared number, it is exact and 100% certain.
If you were to write “12” with an inifinite number of significant
digits, what would it look like?
Q: How many significant figures are there in the measurement. . .
9.10938188 ×
-28
10
grams
A: 9 significant figures
Only the coefficient counts,
Not the power of ten.
Q: How many significant figures are there in the measurement. . .
0.000000000000002 lb
A: 1 significant figure
Q: How many significant figures are there in the measurement. . .
75,106,200 mol
A: 6 significant figures
Well, it could be up to 8, but not likely.
Q: How many significant figures are there in the measurement. . .
x x x
186,000 mi/s
A: 3 significant figures
Well, it could be up to 6, but not likely.
Q: How many significant figures are there in the measurement. . .
1.86 x
5
10
mi/s
A: 3 significant figures
With no ambiguity : )
Q: How many significant figures are there in the measurement. . .
x x x
x x x
93,000,000 miles
A: 2 significant figures
Well, it could be up to 8, but not likely.
Q: How many significant figures are there in the measurement. . .
9.300 x
4
10
mi/s
A: 4 significant figures
With no ambiguity : )
Q: How many significant figures are there in the measurement. . .
9.3 x
6
10
mi/s
A: 2 significant figures
With no ambiguity : )
Q: How many significant figures are there in the measurement. . .
This is not a measured number. It is a declared number. The
guys who invented the metric system decided that a kilometer
would be exactly 1000 meters.
1000 m/km
1000.00000000000000000000000000000000000000000000 . . .
A: infinite significant figures
Remember, it is 100% certain with perfect exactitude that there
are EXACTLY one thousand meters in a kilometer.
If you were to write “1000” with an inifinite number of significant
digits, what would it look like?
. . . round the answers of calculations to the
correct number of significant digits.
23.43 g + 200 g = ?200
223.43
g g
Okay. That’s true, but how should we write the answer so that it
has the correct number of significant digits?
For addition and subtraction, it helps to arrange the numbers
vertically. DON’T ADD ANY PLACEHOLDER ZEROS!
23.43
+ 200
XX x x
I like to put an “x” in any position where a
number doesn’t have a digit, but other
numbers do have a digit.
223.43
x x xx
Any column that has even one “x” in it is insignifcant.
Also, any column is insignificant if it has even one insignificant
zero in it.
Two is less than five, so the one digit that remains, the “2” in front,
is not rounded up.
23.43 g + 200 g = ?200
223.43
g g
Yeah. I know that looks weird, but that’s the kind of results you
get sometimes when you follow the rules.
In real life, this kind of situation might arise
where you make two different mass
measurements with two different scales.
23.43
+ 200 x x
One of the scales, maybe, measures down
to the centigram, but the other scale is so
223.43
coarse in its level of detail that it only
x x xx
measures to the closest 100 g. (0.1 kg)
The “low-res” scale is probably a much larger scale used for
measuring much heavier objects (like people), whereas the scale
that measured the 23.43 g object is a much smaller, more
sensistive scale, used for measuring small piles of dust.
Or something.
34,836.982 – 40 = ?
= 34,796.982
= 34,800
Okay. That’s true, but how should we write the answer so that it
has the correct number of significant digits?
Just like with addition, it helps to arrange numbers vertically, in
columns, when subtracting.
34836.982
40 x xx
As with addition, I like to put an “x” in any
position where a digit is missing.
34796.982
x xxx
This answer’s last three digits, are, therefore, insignificant.
However, the zero in 40 is also insignificant, so its whole column
is insignificant as well.
Because the insignificant “6” in the answer is five or greater, it
rounds the previous digit up as its last, dying act.
78.1 x 32,510,000 = ?
= 2,539,031,000
= 2,540,000,000
The multiplication may be correct, but we’re going to need to trim
off some insignificant digits.
Luckily, with multiplication and division, you don’t have to re-write
the numbers with their digits all lined up in the correct columns.
With multiplication and division, all you have to do is count the
number of significant digits in each number.
The answer gets to have as many significant digits as the
ingredient number with the smallest number of significant digits.
As with addition and subtraction, the chain is, once again, only as
strong as its weakest link. This time, however, it’s easier to deal
with.
12 3
1 2 34
78.1 x 32,510,000 = ?
1 23
= 2,539,031,000
= 2,540,000,000
How many significant digits are in the answer?
Why are there only three significant digits in the answer?
Notice that although “78.1” goes all the way to the tenths digit, this
doesn’t mean squat in multiplication or division.
In multiplication and division, it doesn’t matter what column the
digits are in, it just matters how many digits there are.
Notice, also, that the leftmost insignificant digit in “2,539,031,000”
is a “9”.
Remember that if the leftmost insignificant digit in the answer is
five or greater, it causes the rightmost significant digit to round up.
32,000 / 486.2 = ?
= 65.8165364
= 66
Of course, as always, we need to trim some digits off of this
answer.
How do we handle significant figures during division?
Our most primitive ingredient number, “32,000”, has only two
significant digits, so it’s the weakest link.
Therefore, our answer only gets to keep two significant digits.
Will there be any rounding as we trim off insignificant digits?
Eight is greater than or equal to five, so, yes, we do.
We don’t even have a decimal point anymore. That’s how low-res
this answer has become.
Q: The calculator says that 37.2 x 58,327.67 =
2,169,789.324
How should this number be reported?
A: There are three sig figs in the first #, and seven sig figs in
the 2nd #, so the answer only deserves to have three. As the
nine disappears, it rounds the “6” up to a “7”, yielding . . .
“2,170,000” as the properly-reported answer.
Q: The calculator says that 78 + 347.23 + 3.14 = 428.37
How should this answer be reported?
A: The first number has no hundredths or tenths digit, so
those two columns are insignificant. However, all three
numbers have a ones digit, so the ones column and every
column to the left of it (tens, hundreds) are significant.
As the “.37” disappears, no rounding occurs because three is
less than 5. The properly-reported answer, therefore, is . . .
“428”.
Q: The calculator says that 0.0003407 / 36.24 =
0.00000940121412803532
How should this number be reported?
A: There are four sig figs in both the divisor and the
dividend, so the quotient gets to have four also. As the
leftmost insignificant digit, “2”, disappears, it causes no
rounding because it’s less than 5, yielding . . .
“0.000009401” as the properly-reported answer.
Q: The calculator says that 26,394.85 – 1,200 = 25,194.85
How should this answer be reported?
A: The first number has a digit in every position all the way
to the hundredths, but the second number only goes to the
hundreds. It’s got a tens digit and a ones digit, but they’re
both just placeholder zeros whose only purpose is to let us
know where the decimal point goes. Therefore, the hundreds
column is the farthest-right significant column. The tens
column in the answer has a “9”, so as it disappears, it rounds
the “1” up to a “2”, yielding:
“25,200” as the properly-reported answer.
The next practice problem is kind of
tricky, so make sure you’re good at all
the easy problems first before trying it.
Consider the following molar mass calculation for stearic acid:
C18H36O2:
C: 18 x 12.01 = 216.18
H: 36 x 1.01 = 36.36
O: 2 x 16.00 = 32
284.54 g/mol
Get together with your seating group and discuss how this
calculation should be done if the rules for significant figures are
properly taken into account. Be ready to present your group’s
conclusions in two minutes.
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
C: 18 x 12.01 = 216.18
H: 36 x 1.01 = 36.36
O: 2 x 16.00 = 32
284.54 g/mol
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
The numbers of atoms of each
element per molecule are
C: 18 x 12.01 = 216.18
known with 100% certainty, so
H: 36 x 1.01 = 36.36
these numbers do not affect the
O: 2 x 16.00 = 32
number of significant digits in
the answer.
284.54 g/mol
Every stearic acid moleulce has exactly 18 carbon atoms, so the
number of carbon atoms is:
18.00000000000000000000000000000000000000000000000 . . .
The number of significant digits in a certain number is infinity.
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
“18” may have an infinite
number of significant digits, but
C: 18 x 12.01 = 216.18
“12.01” has only four, so it’s the
H: 36 x 1.01 = 36.36
weak link in the chain.
O: 2 x 16.00 = 32
Since this is multiplication, the
284.54 g/mol answer only gets to have four
digits, also.
HOWEVER, since “216.18” is what we call an “intermediate result”
(a result of a calculation, but not our final answer), we’re not going
to do any rounding right now. We will, however, mark the fifth digit
as insignificant . . . for later . . .
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
“36” may have an infinite
number of significant digits, but
C: 18 x 12.01 = 216.18
“1.01” has only three, so it’s the
H: 36 x 1.01 = 36.36
weak link in the chain.
O: 2 x 16.00 = 32
Since this is multiplication, the
284.54 g/mol answer only gets to have three
digits, also.
HOWEVER, since “36.36” is an “intermediate result”, we’re not
going to do any rounding right now. We will, however, mark the
fourth digit as insignificant . . . for later . . .
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
“2” has an inifinite number of
significant digits, and “16.00”
C: 18 x 12.01 = 216.18
has four.
H: 36 x 1.01 = 36.36
Therefore, the answer,
O: 2 x 16.00 = 32 .00
according to the calculator, “32”,
284.54 g/mol really deserves to have two
more digits.
Usually, rounding to the correct number of significant digits is a
matter of getting rid of digits, not adding, them, but in this case,
we do add a couple of zeros.
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
When you’re adding or
subtracting, it’s always best to
C: 18 x 12.01 = 216.18
line up your columns. This
H: 36 x 1.01 = 36.36
problem, as it appears, has bad
O: 2 x 16.00 = 32 .00
column alignment, so let’s slide
the intermediate results and the
284.54 g/mol final result around until they all
line up right . . .
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
There. That’s better.
C: 18 x 12.01 = 216.18
H: 36 x 1.01 = 36.36
O: 2 x 16.00 =
32 .00
Now, all the tenths digits are
in the same column, all the
hundredths digits are in the
same column, et cetera.
284.54 g/mol
Now that all the columns are lined up, we can start disqualifying
columns, starting from the right and moving left.
Is the hundredths column significant? Nope. The top two
numbers don’t have a significant digit in this column. That
disqualifies the whole column.
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
Is the tenths column any
good?
C: 18 x 12.01 = 216.18
H: 36 x 1.01 =
36.36
O: 2 x 16.00 =
32 .00
284.54 g/mol
All three numbers have a
significant digit in this
column, so the tenths digit
is significant. We get to
keep it.
Since we’ve gotten to the first column that isn’t disqualified, our
job disqualifying columns is done. We know now that the only
digit in the final result that we have to get rid of is the last one.
TIME’S UP! Your instructor will now raise the projector screen
and choose a random person to come to the whiteboard and show
the rest of the class what his/her group decided.
C18H36O2:
As the “4” disappears, does
it round up the “5” next to
C: 18 x 12.01 = 216.18
it?
H: 36 x 1.01 = 36.36
Nope. If it were a “5” or
O: 2 x 16.00 = 32 .00
higher, we’d round up, but it
284.54 g/mol isn’t, so we don’t.
284.5 g/mol
Our final answer, expressed to the correct number of significant
digits, is . . .
Read pp 66 – 71, including practice
problems 1, 2, 3, 4, 5, 6, 7, and 8. Answers
are on page R83.
3.1 Section Assessment, pg 72:
questions 11, 12, 14, and 15. Answers not
in book! Click the red button to go to
Barnes’ ch 3 section assessment answers
power point.
Push me for answers
Chapter 3 Assessment, pg 96: questions 58, 59, 60, and 61.
Answers to odd #’d questions are on page R84. For all
answers, click the yellow button to get a helpful pdf.
Push me for answers