IB Chemistry on Uncertainty calculation and significant figures

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
Certain
23.00
least precise
Uncertain
5
Zeros bet
(significant)
4.109 = 4sf
902 = 3sf
5002.05 = 6sf
measurement
15.831g
23.005g
more precise
(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
Zeros after
decimal point
(significant)
4.580 = 4 sf
9.30 = 3sf
86.90000 = 7sf
3.040 = 4sf
67.030 = 5sf
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. 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
4.2
2.32
+ 0.6157
7.1357
8.04
least number
decimal places
uncertain
round down
round up
0.03
uncertain
least number
decimal places
68.7
- 68.42
0.28
0.0509
least number
decimal places
uncertain
7.987
- 0.54
7.447
uncertain
least number
decimal places
round up
round down
round up
0.3
16.96
7.1
1.367
- 1.34
0.027
1910
12.587
4.25
+ 0.12
16.957
uncertain
round down
round up
24.69
least number
decimal places
uncertain
least number
decimal places
2.300 x 103
+ 4.59 x 103
6.890 x 103
least number
decimal places
7.45
Convert to same exponent
x 104
476.8
47.68
+ 23.2 x 103
x 103
+ 23.2 x 103
500.0 x 103
round up
6.89 x 103
500.0 x 103
5.000 x 105
least number
decimal places

6. 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
least sf (2sf)
round up
23.123123
x
1.3344
30.855495
least sf (5sf)
21.45
x 0.023
0.49335
round down
round down
30.855
72
16.235
0.217
x
5
17.614975
least sf (1sf)
round up
4.52
÷ 6.3578
7.1093775
least sf (3sf)
923
÷ 20312
0.045441
least sf (3sf)
round down
0.0454
1300
x 57240
74412000
4.6
0.00435
x
4.6
0.02001
least sf (2sf)
round down
7.11
0.020
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
63000
6.3 x 105
I.3*103
x 5.724*104
7.4412 x 107
round down
74000000
7.4 x 107
Click here for practice notes on sig figures

7. Scientific notation
How many significant figures
Written as
a=1-9
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
80
How many significant figures
4.66 x
4660000
10 6
3 sf
4.660 x 10 6
5 sf
80.
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
3 ways to write 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
✔

8. Significant figures and Uncertainty 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
Their sf is not taken
(not a measurement)
least sf (3sf)
round down
41.6
Recording measurement using
uncertainty of equipment
Radius, r = (2.15 ±0.02) cm
Treatment of Uncertainty
Multiplying or dividing measured quantities Volume, V = 4/3πr3
% uncertainty = sum of % uncertainty of individual quantities
Radius, r = (2.15 ±0.02)
%uncertainty radius (%Δr) = 0.02 x 100 = 0.93%
2.15
% uncertainty V = 3 x % uncertainty r
% ΔV = 3 x % Δr
* For measurement raised to power of n, multiply % uncertainty by n
* Constant, pure/counting number has no uncertainty and sf not taken
Volume, V = 4/3πr3
4
Volume   3.14  2.153  41.60
3
0.02
100%  0.93%
2.15
Measurement raised to power of 3,
multiply % uncertainty by 3
%V  3  %r
%V  3  0.93  2.79%
Volume  (41.60  2.79%)
%r 
AbsoluteV 
2.79
 41.60  1.16
100
Volume  (41.60  1.16)
Volume  (42  1)

9. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 3.0 cm
Circumference, C = 2πr
C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495
2 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (2sf)
round up
19
Recording measurement using
uncertainty of equipment
Radius, r = (3.0 ±0.2) cm
Treatment of Uncertainty
Multiplying or dividing measured quantities Circumference, C = 2πr
% uncertainty = sum of % uncertainty of individual quantities
Radius, r = (3.0 ±0.2)
%uncertainty radius (%Δr) = 0.2 x 100 = 6.67%
3.0
% uncertainty C = % uncertainty r
% ΔC = % Δr
* Constant, pure/counting number has no uncertainty and sf not taken
Circumference, C = 2πr
Circumference  2  3.14 3.0  18.8495
0.2
100%  6.67%
3.0
%c  %r
%c  6.67%
Circumference  (18.8495  6.67%)
%r 
AbsoluteC 
6.67
18.8495  1.25
100
Circumference  (18.8495  1.25)
Circumference  (19  1)

10. Significant figures and Uncertainty in measurement
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
Their sf is not taken
(not a measurement)
least sf (3sf)
round down
24.8
Recording measurement using
uncertainty of equipment
Time, t = (2.25 ±0.01) cm
1
Displacement, s = gt 2
2
1
Displacement, s   9.8x2.25x2.25  24.80625
2
0.01
100%  0.4%
2.25
Measurement raised to power of 2,
multiply % uncertainty by 2
%s  2  %t
%s  2  0.4%  0.8%
Displacement  (24.80  0.8%)
%t 
Treatment of Uncertainty
1 2
Multiplying or dividing measured quantities Displacement, s = gt
2
% uncertainty = sum of % uncertainty of individual quantities
Time, t = (2.25 ±0.01)
%uncertainty time (%Δt) = 0.01 x 100 = 0.4%
2.25
% uncertainty s = 2 x % uncertainty t
% Δs = 2 x % Δt
* For measurement raised to power of n, multiply % uncertainty by n
Absolutes 
0.4
 24.80  0.198
100
Displacement  (24.80  0.198)
Displacement  (24.8  0.2)

11. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Length, I = 1.25 m
L
g
T  2
least sf (3sf)
T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
2, π and g – constant
Their sf is not taken
(not a measurement)
round down
2.24
Recording measurement using
uncertainty of equipment
T  2
Length, I = (1.25 ±0.05) m
T  2
L
g
1.25
 2.24
9. 8
0.05
100%  4%
1.25
1
Measurement raised to power of 1/2,
%T   %l multiply % uncertainty by 1/2
2
%T  2%
%l 
Treatment of Uncertainty
Multiplying or dividing measured quantities
T  2
L
g
% uncertainty = sum of % uncertainty of individual quantities
Length, I = (1.25 ±0.05)
%uncertainty length (%ΔI) = 0.05 x 100 = 4%
1.25
% uncertainty T = ½ x % uncertainty I
% ΔT = ½ x % ΔI
* For measurement raised to power of n, multiply % uncertainty by n
T  (2.24  2%)
AbsoluteT 
2
 2.24  0.044
100
T  (2.24  0.044)
T  (2.24  0.04)

12. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Area, A = I x h
Length, I = 4.52 cm
Height, h = 2.0 cm
4.52
2.0
9.04
x
least sf (2sf)
round down
9.0
Recording measurement using
uncertainty of equipment
Length, I = (4.52 ±0.02) cm
Height, h = (2.0 ±0.2)cm3
Area, A = I x h
Area  4.52 2.0  9.04
0.02
100%  0.442%
4.52
0.2
%h 
100%  10%
2.0
%A  %l  %h
%l 
Treatment of Uncertainty
Multiplying or dividing measured quantities
Area, A  Length, l  height, h
% uncertainty = sum of % uncertainty of individual quantities
Length, l = (4.52 ±0.02)
%uncertainty length (%Δl) = 0.02 x 100 = 0.442%
4.52
Height, h = (2.0 ±0.2)
%uncertainty height (%Δh) = 0.2 x 100 = 10%
2.0
% uncertainty A = % uncertainty length + % uncertainty height
% ΔA =
% ΔI
+
%Δh
%A  0.442%  10%  10.442%
Area  (9.04  10.442%)
AbsoluteA 
Area  (9.0  0.9)
10.442
 9.04  0.9
100

13. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Moles, n = Conc x Vol
Conc, c
= 2.00 M
Volume, v = 2.0 dm3
2.00
2.0
4.00
x
least sf (2sf)
round down
4.0
Recording measurement using
uncertainty of equipment
Conc, c
= (2.00 ±0.02) M
Volume, v = (2.0 ±0.1)dm3
Mole, n  Conc, c Volume, v
Mole  2.00 2.0  4.00
0.02
100%  1%
2.00
0.1
%v 
100%  5%
2.0
%n  %c  %v
%c 
Treatment of Uncertainty
Multiplying or dividing measured quantities
Mole, n  Conc, c Vol, v
% uncertainty = sum of % uncertainty of individual quantities
Conc, c = (2.00 ±0.02)
%uncertainty conc (%Δc) = 0.02 x 100 = 1%
2.00
Volume, v = (2.0 ±0.1)
%uncertainty volume (%Δv) = 0.1 x 100 = 5%
2.0
% uncertainty n = % uncertainty conc + % uncertainty volume
% Δn =
% Δc
+
%Δv
%n  1%  5%  6%
Mole  (4.00  6%)
Absoluten 
6
 4.00  0.24
100
Mole  (4.00  0.24)
Mole  (4.0  0.2)

14. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass, m = 482.63g
Volume, v = 258 cm3
Density = Mass
Volume
482.63
÷
258
1.870658
least sf (3sf)
round down
1.87
Recording measurement using
uncertainty of equipment
Mass, m = (482.63 ±1)g
Volume, v = (258 ±5)cm3
Treatment of Uncertainty
Multiplying or dividing measured quantities
Density, D =
Mass
Volume
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (482.63 ±1)
%uncertainty mass (%Δm) = 1
x 100 = 0.21%
482.63
Volume, V = (258 ±5)
%uncertainty vol (%ΔV) = 5 x 100 = 1.93%
258
% uncertainty density = % uncertainty mass + % uncertainty volume
% ΔD =
% Δm
+
%ΔV
Density, D =
Density, D =
Mass
Volume
482.63
=1.870658
258
1
100%  0.21%
482.63
5
%V 
100%  1.93%
258
%D  %m  %V
%m 
%D  0.21%  1.93%  2.14%
Density  (1.87  2.14%)
AbsoluteD 
2.14
1.87  0.04
100
Density  (1.87  0.04)

15. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass water = 2.00 g
ΔTemp
= 2.0 C
Enthalpy, H = mcΔT
x
2.00
4.18
2.0
16.72
c – constant
sf is not taken
(not a measurement)
least sf (2sf)
round up
17
Recording measurement using
uncertainty of equipment
Mass water = (2.00 ±0.02)g
ΔTemp
= (2.0 ±0.4) C
Treatment of Uncertainty
Multiplying or dividing measured quantities
Enthalpy, H  m  c  T
Enthalpy, H  2.00 4.18 2.0  16.72
Enthalpy, H  m  c  T
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (2.00 ±0.02)
%uncertainty mass (%Δm) = 0.02 x 100 = 1%
2.00
ΔTemp = (2.0 ±0.4)
%uncertainty temp (%ΔT) = 0.4 x 100 = 20%
2.0
% uncertainty H = % uncertainty mass + % uncertainty temp
% ΔH =
% Δm
+
%ΔT
0.02
100%  1%
2.00
0.4
%T 
100%  20%
2.0
%H  %m  %T
%m 
%H  1%  20%  21%
Enthalpy  (16.72  21%)
AbsoluteH 
21
16.72  3.51
100
Enthalpy  (16.72  3.51)
Enthalpy  (17  4)

16. Treatment of uncertainty in measurement
•
Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
Initial mass beaker, M1
= (20.00 ±0.01) g
Final mass beaker + water, M2 = (22.00 ±0.01)g
Addition/Subtraction/Multiply/Divide
•
Multiplying or dividing
Max %uncertainty is the SUM of individual %uncertainties
Addition/Subtraction
Add absolute uncertainty
Initial Temp, T1 = (21.2 ±0.2)C
Final Temp, T2 = (23.2 ±0.2)C
Enthalpy, H = (M2-M1) x c x (T2-T1)
Mass water, m = (M2 –M1)
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Diff Temp ΔT = (T2 –T1)
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Multiplication
Add % uncertainty
Mass water, m = (22.00 –20.00) = 2.00
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Mass water, m = (2.00 ±0.02)g
Mass water, m = (2.00 ±0.02)g
Treatment of Uncertainty
Multiplying or dividing measured quantities
Diff Temp ΔT = (23.2 –21.2) = 2.0
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Diff Temp, ΔT = (2.0 ±0.4)
ΔTemp = (2.0 ±0.4) C
Enthalpy, H  m  c  T
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (2.00 ±0.02)
%uncertainty mass (%Δm) = 0.02 x 100 = 1%
2.00
ΔTemp = (2.0 ±0.4)
%uncertainty temp (%ΔT) = 0.4 x 100 = 20%
2.0
% uncertainty H = % uncertainty mass + % uncertainty temp
% ΔH =
% Δm
+
%ΔT
Enthalpy, H  m  c  T
Enthalpy, H  2.00 4.18 2.0  16.72
0.02
100%  1%
2.00
0.4
%T 
100%  20%
2.0
%H  %m  %T
%m 
%H  1%  20%  21%
Enthalpy  (16.72  21%)
AbsoluteH 
21
16.72  3.51
100
Enthalpy  (16.72  3.51)
Enthalpy  (17  4)

17. Significant figures and Uncertainty in measurement
tI2
Energy  1/ 2
v
4.52 x 3.0 x 3.0 = 40.68
÷ 1.414
28.769
Recording measurement
using significant figures
Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s
least sf (2sf)
round up
29
Recording measurement using
uncertainty of equipment
Volt, v = (2.0 ± 0.2)
Current, I = ( 3.0 ± 0.6)
Time, t = (4.52 ± 0.02)
Treatment of Uncertainty
Multiplying or dividing measured quantities
tI2
Energy, E  1/ 2
v
% uncertainty = sum of % uncertainty of individual quantities
Time, t = (4.52 ±0.02)
%uncertainty time (%Δt) = 0.02 x 100 = 0.442%
4.52
Current, I = (3.0 ±0.6)
%uncertainty current (%ΔI) = 0.6 x 100 = 20%
3.0
Volt, v = (2.0±0.2)
%uncertainty volt (%Δv) = 0.2 x 100 = 10%
2.0
% ΔE = % Δt + 2 %ΔI + ½ %ΔV
* For measurement raised to power of n, multiply % uncertainty by n
tI2
Energy, E  1/ 2
v
4.52(3.0) 2
Energy, E 
 28.638
2.01/ 2
0.02
% t 
 100%  0.442%
4.52
0 .6
% I 
 100%  20%
3 .0
0.2
% v 
 100%  10%
2.0
1
%E  %t  2  % I   %v
2
%E  
0.02
0.6
1 0.2
100%    2  100%     100%
4.52
3.0
2 2.0
%E  0.442%  40%  5%  45.442%  45%
Energy, E  (28.638  45%)
AbsoluteE 
Energy, E  (29  13)
45
 28.638  13
100


18. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
(G + H )
Z
20 + 16 = 36
÷ 106
0.339
Speed, s =
G = (20 )
H = (16 )
Z = (106)
least sf (2sf)
round down
0.34
Speed, s 
Recording measurement using
uncertainty of equipment
G = (20 ± 0.5)
H = (16 ± 0.5)
Z = (106 ± 1.0)
✔
Addition
add absolute uncertainty
G+H = (36 ± 1)
Z = (106 ± 1.0)
(G  H )
Z
Speed, s 
(20  16)
 0.339
106
%(G  H ) 
Treatment of Uncertainty
Multiplying or dividing measured quantities
(G + H )
Speed, s =
Z
% uncertainty = sum of % uncertainty of individual quantities
(G + H) = (36 ±1)
%uncertainty (G+H) (%ΔG+H) = 1 x 100 = 2.77%
36
Z = (106 ±1.0)
%uncertainty Z (%Δz) = 1.0 x 100 = 0.94%
106
%uncertainty s = %uncertainty(G+H) + %uncertainty(Z)
% Δs = % Δ(G+H)
+
%Δz
*Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
%Z 
1.0
100%  2.77%
36
1.0
100%  0.94%
106
%S  %(G  H )  %Z
%S  2.77%  0.94%  3.71%
Speed, s  (0.339  3.71%)
AbsoluteS 
3.71
 0.339  0.012
100
Speed, s  (0.339  0.012)
Speed, s  (0.34  0.01)