Download chapter one: scientific notation, significant figures, units, density and

CHAPTER ONE: SCIENTIFIC NOTATION, SIGNIFICANT FIGURES,
UNITS, DENSITY AND % COMPOSITION
Note: These notes and examples are intended for self-study and will be examined. Some examples will
be discussed in the lectures.
REFERENCE
Brown, LeMay, Burnsten, Murphy, Langford and Sagatys, 2nd Edition, Chapter 1.4 - 1.5
SCIENTIFIC NOTATION
A number in scientific notation may be converted into any value of n by moving the decimal point (d.p.).
Example 1
5.68762 × 102  56.8762 × 101  0.00568762 × 105  56876.2 × 10-2
Calculations with numbers expressed in scientific notation
Multiplication and Division
Multiplication:
(N1 × 10n1) × (N2 × 10n2) = (N1 × N2) x 10n1+n2
Division:
(N1 × 10n1) / (N2 × 10n2) = (N1 / N2) x 10n1-n2
Addition and Subtraction
First convert all numbers to the same value of n (ie. the same power of 10):
Addition:
(N1 × 10n1) + (N2 × 10n1) + (N3 × 10n1) = (N1 + N2 + N3) × 10n1
Subtraction:
(N1 × 10n1) - (N2 × 10n1) = (N1 - N2) × 10n1
Note:
The value of n to which the numbers are converted does not affect the final answer:
Example 2
1043.36 - 12.13 = 1.04336 × 103 - 1.213 × 101
Choose n = 1
Choose n = 3
1043.36 - 12.13 = 104.336 × 101 - 1.213 x 101
1043.36 - 12.13 = 1.04336 × 103 - 0.01213 × 103
= (104.336 - 1.213) × 101
= (1.04336 - 0.01213) × 103
= 103.123 × 101
= 1.03123 × 103
= 1031.23
= 1031.23
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SIGNIFICANT FIGURES
Determining the number of significant figures in a given measurement
1.
Any figure that is not zero is significant:
845 mL
Note:
2.
____ s.f.
1243.29 mg
____ s.f.
changing to scientific notation:
1.24329 x 103 mg
____ s.f.
changing the units:
1.24329 g
____ s.f.
The number of s.f. is not affected by:
Zero figures between non-zero figures are significant:
1906 mL
3.
____ s.f.
____ s.f.
Zero figures to the left of the first non-zero figure are not significant:
0.008 kg
Note:
40501.09 J
____ s.f.
0.003798 L
____ s.f.
changing to scientific notation:
3.798 × 10-3 L
____ s.f.
changing the units:
3.798 mL
____ s.f.
The number of s.f. is not affected by:
The zeroes are used only to show the position of the decimal point.
4.
Exact numbers (from counting numbers of objects) have an  number of s.f. by definition.
5.
Physical constants/quantities defined to be exact numbers have their properties:
1 atm  101.325 kPa  760 mmHg
6.
0 oC  32 oF  273.15 K
all
_____ s.f.
Zero figures to the right of the last non-zero figure are significant only if the number has a decimal
point:
300.0 mm
7.
____ s.f.
0.0300 mm
____ s.f.
In measurements without a decimal point, the number of s.f. is ambiguous:
1200 mg is unclear; could be
1200 ± 100 mg
2 s.f.
1200 ± 10 mg
3 s.f.
1200 ± 1 mg
4 s.f.
This occurs when measurements are supplied from a source not using the s.f. system (eg. an older text);
obtain a second opinion (demonstrator, lecturer) if an estimation cannot be made from the context of the
reference. All questions supplied in tuts, tests & exams should (hopefully!) avoid this.
Writing such ambiguous measurements may be avoided in two ways:
1.
2.
using scientific notation
1200 ± 100 mg 2
s.f.
1.2 × 103 mg
1200 ± 10 mg
3 s.f.
1.20 × 103 mg
1200 ± 1 mg
4 s.f.
1.200 × 103 mg
using a decimal point without any following figures:
1200 ± 1 mg
4 s.f.
1200. mg
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Calculations using measurements with different numbers of significant figures
Calculators do NOT count significant figures; this must be done by you.
Multiplication/Division
The number of s.f. in the final answer is equal to the least of the numbers of s.f. in each of the
original measurements.
Note:
As the number of s.f. in a measurement remains unchanged by conversion to scientific notation or
vice-versa, s.f. counting in multiplication and/or division calculations can be done directly
irrespective of notation:
Example 3
0.4323 × 232.36
multiply them on calculator:
100.44923
look at s.f. in each measurement 0.4323 _____ s.f.
232.36
lesser of these is ____  answer must have ____ s.f.
_____ s.f.
 _______
Note: rounding off
In scientific notation:
(4.323 × 10-1) × (2.3236 × 102) on calculator:
s.f.
____, ____  _____
10.044923 × 101
10.044923 × 101
 __________
 __________
mixed:
s.f.
0.4323 × (2.3236 × 102)
on calculator:
100.44923
____, ____  _____
100.44923
 __________
on calculator:
3(!!)
Example 4
27.315/9.105
s.f.
Note:
____, ____  _____
 _________
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Figures are not always rounded off; in certain cases they must be added on.
Example 5
1.279/250.0
-3
1.279/250
1.279/250.
5.116 × 10
5.116 × 10-3
___, ___  ___
___, ___  ___
___, ___  ___
____________
____________
____________
calculator:
5.116 × 10 (0.005116)
s.f.
answer
-3
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Example 6
What is the total mass of 7 identical test tubes each of mass 15.912g?
calculator:
111.384 g
s.f.
___, ___  ____
111.384 g
 _________
Example 7
[0.0021281 × 104 × 63.2596(0.00045/100.062)] / [(2.345 × 10-6 (21.561/17.2)]
calculator:
2059.5793
s.f.
____, ____, ____, ____, ____, ____, ____ ____
2059.5793
 ___________
(0.000450, 0.0004500)
Note: Intermediate rounding off in complex calculations is unnecessary provided that all calculations are x
and /; do complete calculation and round off only at end.
Addition/Subtraction
The number of decimal places in the final answer is equal to the least of the numbers of decimal places
(d.p.) in each of the original measurements.
Note: As conversion of a measurement to scientific notation or vice-versa changes its number of d.p.,
addition and/or subtraction of quantities in different notations will give incorrect results. Quantities
must be converted to the same notation before d.p. can be counted:
Example 8
(1.123 × 103) + 25.72 + 0.0651
normal:
on calculator:
1148.7851
1148.7851
 __________
1123 + 25.72 + 0.0651
d.p.
scientific:
___, ___, ___  ____
(1.123 × 103) + (2.572 × 101) + (6.51 x 10-2)
choose n = 1
(112.3 + 2.572 + 0.00651) × 101
d.p.
___, ___, ___  ___
on calculator:
114.87851 × 101
114.8765 × 101  _________ _________
Example 9
d.p.
scientific:
d.p.
0.00177 - 0.0013
on calculator:* 4.7 × 10-4
___, ___  ____
0.00047
(1.77 - 1.3) × 10-3
on calculator:** 4.7 × 10-4
___, ___  ____
0.47 × 10-3
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 _________
 _________  _________
Example 10
1130. + 4.9
1130.0 + 4.9
1130 + 4.9
calculator:
1134.9
1134.9
1134.9
d.p.
___, ___  ____
___, ___  ____
see below
answer
___________
___________
___________
In real terms 1130 (3 s.f.) means a number that is accurate only to the nearest 10; 1130 + 4.9 = 1134.9 which
remains 1130 (3 s.f.) when rounded off to the nearest 10. However, 1130 (4 s.f.) is accurate to the nearest 1;
1130. + 4.9 = 1134.9 which becomes 1135(4 s.f.) when rounded off to the nearest 1.
To summarise:
Multiplication/Division:
Count significant figures
Addition/Subtraction:
Count decimal places
Addition/subtraction in scientific notation:
(i)
Convert all quantities to same notation &/or value of n before counting d.p.
(ii)
Convert answer to that same value of n before rounding off d.p.
Combined calculations and when to round off
S.f. & d.p. counting and rounding off must be done every time the operation changes from
addition/subtraction to multiplication/division and vice versa.
UNITS
Determining Conversion Factors For Prefixes
Imagine a ruler, with the various powers as the graduations, count the number of “jumps” that have to be
made to go from the prefix given the prefix desired. The example uses length (m), but this applies to all
units:
Power 10 3 10 2
Unit
km
hm
101
100 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9
dam
m
dm
cm
mm
μm
nm
Converting km  m: looking at the powers of both, km is 103 while m is 100, a difference (jump) of 3 ie.
1000, .: multiply km × 1000 = m, while from m  km: divide m by 1000 = km.
Converting km  nm: looking at the powers of both, km is 103 while m is 10-9, involves a difference
(jump) of 12 ie. 1000 000 000 000 (1012), .: multiply km × 1012 = nm, while from nm  km: divide m by
1012 = km.
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Question 1: How many significant figures are in each of the following measurements?
(i) 5269 pm, (ii) 4.07 m, (iii) 42.300 cm, (iv) 0.0065 m, (v)
320 nm and (iv) 4.8 × 10-9 m
Question 2: Convert all of the numbers in question 1 into micrometres (μm).
Determining Conversion Factors For Temperatures and Pressures
Temperature
The main conversion will be converting °C to K, and vice versa. For:
°C  K: °C + 273.15 K
K  °C:
K – 273.15 K = °C
Pressure
1 atm = 760 mmHg = 101325 Pa
Therefore the conversion factors are:
760 mmHg / 1 atm = 101325 Pa / 1 atm = 101325 Pa / 760 mmHg
Select the relevant one, e.g., convert 1.045 atm into mmHg, and use dimensional analysis:
x mmHg = 1.045 atm × (760 mmHg / 1 atm) = 794.2 mmHg
Now convert this answer in mmHg into Pa:
x Pa = 794.2 mmHg × (101325 Pa / 760 mmHg) = 1.059 × 105 Pa
Finally, convert this into kPa.
Question 3: Express the following temperatures:
(i) 300.89 K in °C, (ii) 25.06 °C in K and (iii) 200.25 K in °C
Question 4: Express:
(i) 710.9 mmHg in Pa, (ii) 104328 Pa in atm and (iii) 102.5 kPa in mmHg
DENSITY & PERCENT COMPOSITION
Question 5: The density of diamond is 3.51 g cm-3. The international (but non-SI) unit for reporting the
masses of diamonds is the “carat”, with 1 carat = 200.0 mg. What is the volume of a diamond of mass 0.300
carat?
Question 6: A supersonic transport (SST) airplane consumes about 18000 L of kerosene per hour of flight.
Kerosene has a density of 0.965 kg L-1. What mass of kerosene is consumed on a flight lasting 3.0 hours?
Question 7: Silver iodide (AgI) is relatively insoluble in water. At 25 °C, only 214 μg (micrograms) will
dissolve in 1.00 litre of water. How many litres of water are needed to dissolve 5.00 g of silver iodide?
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