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Lecture #3
Measurement in Scientific Study
and
Uncertainty in Measurement
Chemistry 142 A
James B. Callis, Instructor
Winter Quarter, 2006
Precision and Accuracy
Errors in Scientific Measurements
Precision - Refers to reproducibility or how close the
measurements are to each other.
Accuracy - Refers to how close a measurement is to the
‘true’ value.
Systematic Error - produces values that are either all higher
or all lower than the actual value.
Random Error - in the absence of systematic error, produces
some values that are higher and some that
are lower than the actual value.
Rules for Determining Which Digits Are
Significant
1. Make sure that the measured quantity has a decimal point.
2. Start at the left of the number and move right until you
reach the first nonzero digit.
3. Count that digit and every digit to its right as significant.
4. Zeros that end a number and lie either after or before the
decimal point are significant; thus 1.030 mL has four
significant figures, and 5300. L has four significant figures
also.
5. Situation unclear if there is no decimal point.
We will adopt convention that numbers such as 5300 L
have 2 sig. figs.
A terminal decimal point is often used to
clarify the situation, 5300. L has 4 sig. figs.
Scientific notation is clearer, 5.30 x 103 L has 3 sig figs.
6. Exact numbers have an infinite number of significant
figures. The alphabet has 26 letters. There are 12 eggs in
a dozen eggs.
Examples of Significant Digits in Numbers
Problem - Sig digits
P 3-1a
0.0050 L
P 3-1b
18.00 g
P 3-1c
1.089 x 10–6 L
P 3-1d
P 3-1e
83.0001 L
0.006002 g
P 3-1f
875,000 oz
P 3-1g
30,000 kg
P 3-1h
5.0000 m3
P 3-1i
23001.00 lbs
P 3-1j
0.000108 g
P 3-1k
1,470,000 L
Rules for Significant Figures in answers
1. For multiplication and division. The number with the
least certainty limits the certainty of the result. therefore, the
answer contains the same number of significant figures as
there are in the measurement with the fewest significant
figures. Multiply the following numbers:
9.2 cm x 6.8 cm x 0.3744 cm =
2. For addition and subtraction. The answer has the same
number of decimal places as there are in the measurement
with the fewest decimal places. Example, adding two volumes
(a) 83.5 mL + 23.28 mL =
Example subtracting two volumes:
(b) 865.9 mL - 2.8121393 mL =
Rules for Rounding Off Numbers
(1) In a series of calculations*, carry the extra digits through to the final result,
then round off. **
(2) If the digit to be removed
a.
b.
is less than 5, the preceding digit stays the same. For example, 1.33
rounds to 1.3.
is equal to or greater than 5, the preceding digit is increased by one. For
example, 1.36 rounds to 1.4.
(3) When rounding, use only the first number to the right of the last significant
figure. Do not round off sequentially. For example, the number 4.348 when
rounded to two significant figures is 4.3, not 4.4.
Notes:
* Your TI-93 calculator has the round function which you can use to get the
correct result. Find round by pressing the math key and moving to NUM. Its
use is round(num, no of decimal places desired), e.g. round(2.746,1) =2.7.
** Your book will show intermediate results rounded off. Don’t use these rounded
results to get the final answer.
Rounding Off Numbers – Problems
(3-4a) Round 5.379 to three significant figures
Ans:
(3-4b) Round 5.379 to two significant figures
Ans:
We used the rule: If the digit removed is greater than or equal to 5, the preceding
number increases by 1.
(3-4a) Round 0.2413 to three significant figures
Ans:
(3-4b) Round 0.2413 to two significant figures
Ans:
We used the rule: If the digit removed is less than 5, the preceding number is
unchanged
P 3-5: A small rectangular slab of lithium has the
dimensions 20.9 mm by 11.1 mm by 11.9 mm. Its
mass is 1.49 x 103 mg. What is the density of lithium
in g/cm3?
Step 1: task:
Find the density of lithium in g/cm3.
Step 2: given information:
The mass is 1.49 x 103 mg and the dimensions are
20.9 by 11.1 by 11.9 mm.
Step 3: strategy:
Use density = mass/volume, so the volume needs to
be calculated. For a rectangular slab, use volume =
length x width x height. Need to convert units.
Step 4: set up the problem:
density = mass volume
d=
Are we ready to calculate?
density = mass volume
d=
=
Step 5: right units?
d=
Step 6: calculate, sig figs.
d=
Step 7: check result
Problem 3-6: Volume by Displacement
Problem: Calculate the density of an irregularly shaped metal
object that has a mass of 567.85 g if, when it is placed into a
2.00 liter graduated cylinder containing 900.00 mL of water,
the final volume of the water in the cylinder is 1277.56 mL ?
Plan: Calculate the volume from the different volume
readings, and calculate the density using the mass that
was given.
Solution:
Volume =
Density =
mass
volume
Definitions - Mass & Weight
Mass - The quantity of matter an object contains
kilogram - ( kg ) - the SI base unit of mass, is a
platinum - iridium cylinder kept in
Paris as a standard!
Weight - depends upon an object’s mass and the strength
of the gravitational field pulling on it, i.e.
w = f = ma.
Problem 3-7: Computer Chips
Future computers might use memory bits which require an
area of a square with 0.250 mm sides. (a) How many bits
could be put on a 1.00 in x 1.00 in computer chip? (b) If
each bit required
that 25.0 % of its area to be coated with a gold film 10.0 nm
thick,what mass of gold would be needed to make one chip?
Approach:
(a)use Achip =
(b) use r = m/V
Solution to Chip Problem (3-7)
a 
A chip
N  bit 
A

b  m 
ρA chipf goldt 
Temperature Scales and Interconversions
Kelvin ( K ) - The “Absolute temperature scale” begins at
absolute zero and only has positive values.
Celsius ( oC ) - The temperature scale used by science,
formally called centigrade and most
commonly used scale around the world.
Water freezes at 0oC, and boils at 100oC.
Fahrenheit ( oF ) - Commonly used scale in America for
our weather reports. Water freezes at 32oF,
and boils at 212oF.
T (in K) = T (in oC) + 273.15 T (in oF) = 9/5 T (in oC) + 32
T (in oC) = T (in K) - 273.15 T (in oC) = [ T (in oF) - 32 ] 5/9
Problem 3-8:Temperature Conversions
(a) The boiling point of Liquid Nitrogen is -195.8 oC, what is
the temperature in Kelvin and degrees Fahrenheit?
T (in K) = T (in oC) + 273.15
T (in K) =
T (in oF) = 9/5 T (in oC) + 32
T (in oF) =
(b)The normal body temperature is 98.6oF, what is it in Kelvin
and degrees Celsius?
T (in oC) = [ T (in oF) - 32] 5/9
T (in oC) =
T (in K) = T (in oC) + 273.15
T (in K) =
Answers to Problems in Lecture #3
1. (a)2; (b) 4; (c) 4; (d) 6; (e) 4; (f) 3; (g) 1; (h) 5; (i) 7
(j) 3; (k) 3
2. 23 cm3
3(a) 106.8 mL 3(b) 863.1 mL
4. (a) 5.38; (b) 5.5; (c) 0.241; (d) 0.24
5. 0.536 g/cm3
6. 1.5040 g / mL
7. 31 mg gold
8. (a) 77.4 K; -320.4 oF; (b) 37.0 oC; 310.2 K