IB Chemistry on Uncertainty, significant figures and scientific notation

1. Significant figures
Used in measurements
Degree of precision
Show digits believed to be
correct/certain + 1 estimated/uncertain
All reads 80
80
80.0
80.00
80.000
least precise
more precise
Certain
23.00
Uncertain
5
Zeros bet
(significant)
4.109 = 4sf
902 = 3sf
5002.05 = 6sf
Zeros after
decimal point
(significant)
4.580 = 4 sf
9.30 = 3sf
86.90000 = 7sf
3.040 = 4sf
67.030 = 5sf
measurement
15.831g
23.005g
(15.831 ± 0.001)g
(5 sig figures)
Rules for significant figures
All non zero digit
(significant)
31.24 = 4 sf
563 = 3 sf
23 = 2sf
Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain
Zero right of decimal point and
following a non zero digit
(significant)
0.00500 = 3sf
0.02450 = 4sf
0.04050 = 4sf
0.50 = 2sf
Deals with precision NOT accuracy!!!!!!!!
Precise measurement doesnt mean, it’s accurate
( instrument may not be accurate)
Zeros to left of digit
(NOT significant)
0.0023 = 2sf
0.000342 = 3sf
0.00003 = 1sf
Zero without decimal
(ambiguous)
80 = may have 1 or 2 sf
500 = may have 1 or 3 sf
Click here and here for notes on sig figures

2. Significant figures
1
22
Smallest division = 0.1
22
Max = 21.63
2
Certain
21.6
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 21.6
4
Uncertain = 21.62 ±0.01
5
(21.62 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 21.61
Answer = 21.62 (4 sf)
21.6
(certain)
1
Smallest division = 1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 1
= 1/10 x 1 = ±0.1
2
(uncertain)
Certain
36
3
Certain = 36
4
Measurement = Certain digits + 1 uncertain digit
(36.5 ±0.1)
Uncertain = 36.5 ±0.1
5
Max = 36.6
Min = 36.4
Answer = 36.5 (3 sf)
36.
5
(certain) (uncertain)

3. Significant figures
1
Smallest division = 10
Max = 47
2
Certain
40
Uncertainty = 1/10 of smallest division.
= 1/10 of 10
= 1/10 x 10 = ±1
3
Certain = 40
4
Uncertain = 46 ±1
5
(46 ±1)
Measurement = Certain digits + 1 uncertain digit
Min = 45
Answer = 46 (2 sf)
4
(certain)
1
Certain
3.4
Smallest division = 0.1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 3.4
4
Uncertain = 3.41±0.01
5
Measurement = Certain digits + 1 uncertain digit
6
(uncertain)
Max = 3.42
(3.41 ±0.01)
Min = 3.40
Answer = 3.41 (3sf)
3.4
(certain)
1
(uncertain)

4. Significant figures
1
Smallest division = 0.05
Max = 0.48
0.1
2
0.2
0.3
Certain
0.45
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.05
= 1/10 x 0.05 = ±0.005 (±0.01)
Certain = 0.45
Uncertain = 0.47 ± 0.01
5
0.5
3
4
0.4
(0.47 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 0.46
Answer = 0.47 (2 sf)
0.4
(certain)
Measurement
1
Smallest division = 0.1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 5.7
4
Uncertain = 5.72 ± 0.01
(5.72 ±0.01)
Answer = 5.72 (3sf)
5.7
(certain)
2
(uncertain)
1
Smallest division = 1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 1
= 1/10 x 1 = ±0.1
3
Certain = 3
4
Uncertain = 3.0 ± 0.1
(3.0 ±0.1)
Answer =3.0 (2 sf)
3
0
(certain) (uncertain)
7
(uncertain)

5. Scientific notation
How many significant figures
Written as
a = 1 to 9
Number too big/small
b = integer
3 sf
Scientific  notation  a 10b
6,720,000,000
 6.72109
Size sand
4 sf
0.0000000001254
 1.2541010
3 sf
Speed of light
 3.00108
300000000
Scientific notation
80
3 ways to write 80
80
How many significant figures
4.66 x
4660000
10 6
3 sf
4.660 x 10 6
5 sf
80. – 8.0 x 101 – (2sf)
Digit 8 is certain
It can be 79 to 81
80.0
80.0 – 8.00 x 101 – (3sf)
Digit 80 is certain
It can be 79.9 or 80.1
4 sf
4.6600 x 10 6
80 – 8 x 101 – (1sf)
Digit 8 uncertain
It can be 70 to 90
80.
90 or 9 x 101
80 or 8 x 101
70 or 7 x 101
81 or 8.1 x 101
80 or 8.0 x 101
79 or 7.9 x 101
80.1 or 8.01 x 101
80.0 or 8.00 x 101
79.9 or 7.99 x 101
More precise
Click here practice scientific notation
Click here practice scientific notation
✔

6. Rules for sig figures addition /subtraction:
• Last digit retained is set by the first doubtful digit.
• Number decimal places be the same as least number of decimal places in any numbers being added/subtracted
23.112233
1.3324
+ 0.25
24.694633
uncertain
least number
decimal places
round down
4.7832
1.234
+ 2.02
8.0372
uncertain
least number
decimal places
round down
1247
134.5
450
+ 78
1909.5
uncertain
least number
decimal places
1.0236
- 0.97268
0.05092
8.04
4.2
2.32
+ 0.6157
7.1357
least number
decimal places
uncertain
round down
0.03
3 x 10-2 (1sf)
1.910 x 103 (4sf)
uncertain
least number
decimal places
5.09 x 10-2 (3sf)
least number
decimal places
68.7
- 68.42
0.28
uncertain
7.987
- 0.54
7.447
uncertain
least number
decimal places
round up
round down
0.3
16.96
7.1 x 100 (2sf)
round up
12.587
4.25
+ 0.12
16.957
0.0509
round up
7.1
1.367
- 1.34
0.027
1910
8.04 x 100 (3sf)
2.469 x 101 (4sf)
uncertain
round down
round up
24.69
least number
decimal places
1.696 x 101 (4sf)
uncertain
least number
decimal places
2.300 x 103
+ 4.59 x 103
6.890 x 103
7.45
3 x 10-1 (1sf)
least number
decimal places
7.45 x 100 (3sf)
Convert to same exponent
47.68 x 104
+ 23.2 x 103
476.8 x 103
+ 23.2 x 103
500.0 x 103
round up
6.89 x 103
6.89 x 103 (3sf)
500.0 x 103
5.000 x 105
least number
decimal places

7. Rules for sig figures - multiplication/division
• Answer contains no more significant figures than the least accurately known number.
12.34
3.22
x 1.8
71.52264
23.123123
x
1.3344
30.855495
least sf (2sf)
round up
21.45
x 0.023
0.49335
least sf (5sf)
round down
round down
30.855
72
7.2 x 101 (2sf)
3.0855 x 101 (5sf)
16.235
0.217
x
5
17.614975
least sf (1sf)
round up
4.52
÷ 6.3578
7.1093775
0.00435
x
4.6
0.02001
923
÷ 20312
0.045441
1300
x 57240
round down
74412000
least sf (2sf)
0.020
7.11 x 100 (3sf)
least sf (3sf)
4.6 x 100 (2sf)
round down
7.11
2 x 101 (1sf)
4.6
4.9 x 10-1 (2sf)
least sf (3sf)
2.0 x 10-2 (2sf)
least sf (2sf)
Scientific notation
least sf (2sf)
round up
0.49
round up
20
2.8723
x
I.6
4.59568
least sf (2sf)
6305
÷ 0.010
630500
least sf (2sf)
round down
630000
6.3 x 105 (2sf)
I.3*103
x 5.724*104
7.4412 x 107
round down
0.0454
74000000
7.4 x 107
4.54 x 10-2 (3sf)
Click here for practice notes on sig figures

8. Rules for sig figures – Multiplication/Division/Addition/Subtraction
Answer contains no more significant figures than the least accurately known number.
Avoid rounding off error.
•
•
0.0000673 x 291 ÷ 0.125 = ?
least sf (3sf)
0.0000673
x
291
0.0195843
0.0000673
x
291
0.0195843
Intermediate step
– leave more sf
DO NOT ROUND UP/DOWN
intermediate steps
Leave extra sf to avoid
rounding off error
0.0195843
÷
0.125
0.15632
ROUND UP/DOWN
intermediate step
0.0196
÷ 0.125
0.1568
round down
✔
0.156
1.56 x
10-1
round up
Rounding off error
(3sf)
✗
0.157
1.57 x
10-1
(3sf)
(21.5 + 21.53 + 22.548 ) x 8.45 = ?
least number
decimal places
21.5
21.53
+ 22.548
65.578
Intermediate step
– leave more sf
65.578
x 8.45
554.1341
round down
554
least number
decimal places
✔
5.54 x 102 (3sf)
DO NOT ROUND UP/DOWN
Leave extra sf to avoid
rounding off error
least sf (3sf)
ROUND UP/DOWN
intermediate step
least sf (3sf)
21.5
21.53
22.548
65.578
65.5
x 8.45
553.47
round down
553 ✗
5.53 x 102 (3sf)

9. Significant figures in measurement
Recording measurement
using significant figures
Radius, r = 2.15 cm
Volume, V = 4/3πr3
V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60
4/3 – constant
π – constant
sf is not taken
(not a measurement)
least sf (3sf)
round down
41.6
Recording measurement
using significant figures
Radius, r = 3.0 cm
Circumference, C = 2πr
2 and π – constant
sf is not taken
(not a measurement)
C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495
least sf (2sf)
round up
19
Recording measurement
using significant figures
Time, t = 2.25 s
Displacement, s = ½ gt2
s = 1/2 x 9.8 x (2.25)2
= 24.80625
g and ½ – constant
sf is not taken
(not a measurement)
least sf (3sf)
round down
24.8
Recording measurement
using significant figures
G = (20 )
H = (16 )
Z = (106)
Speed, s = (G + H)
Z
20 + 16 = 36
÷ 106
0.339
least sf (2sf)
round down
0.34

10. Significant figures in measurement
Recording measurement
using significant figures
Length, I = 1.25 m
Period, T = 2π √L
√g
least sf (3sf)
T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
round down
2.24
Recording measurement
using significant figures
Length, I = 4.52 cm
Height, h = 2.0 cm
Area, A = I x h
4.52
x 2.0
9.04
least sf (2sf)
round down
9.0
Recording measurement
using significant figures
Conc, c
= (2.00) M
Volume, v = (2.0 )dm3
Moles, n = Conc x Vol
2.00
x 2.0
4.00
least sf (2sf)
round down
4.0
2, π and g – constant
sf is not taken
(not a measurement)

11. Significant figures in measurement
Recording measurement
using significant figures
Density = Mass
Volume
Mass, m = 482.63g
Volume, v = 258 cm3
482.63
÷
258
1.870658
least sf (3sf)
round down
1.87
Recording measurement
using significant figures
Enthalpy, H = m x c x ΔT
Mass water, m = 2.00 g
ΔTemp, ΔT
= 2.0 C
x
2.00
4.18
2.0
16.72
c (4.18) - constant
sf is not taken
(not a measurement)
least sf (2sf)
round up
Recording measurement
using significant figures
Volt, v
= 2.0 V
Current, I = 3.0A
Time, t = 4.52s
17
tI2
Energy  1/ 2
v
4.52
3.0
x 3.0
40.68
÷ 1.414
28.769
round up
29
least sf (2sf)