Upload
lawrence-kok
View
831
Download
3
Tags:
Embed Size (px)
DESCRIPTION
IB Chemistry on Uncertainty calculation and significant figures
Citation preview
Click here and here for notes on sig figures
80 80.0 80.00 80.000 more precise
23.005g
Used in measurements Degree of precision Show digits believed to be correct/certain + 1 estimated/uncertain
Deals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, it’s accurate ( instrument may not be accurate)
Number sf necessary to express a measurement • Consistent with precision of measurement • Precise equipment = Measurement more sf • Last digit always an estimate/uncertain
Significant figures
All reads 80
least precise Certain 23.00
Uncertain 5
(15.831 ± 0.001)g (5 sig figures)
measurement 15.831g
All non zero digit (significant) 31.24 = 4 sf 563 = 3 sf 23 = 2sf
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
Zero right of decimal point and following a non zero digit (significant) 0.00500 = 3sf 0.02450 = 4sf 0.04050 = 4sf 0.50 = 2sf
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
Rules for significant figures
Smallest division = 0.1
Answer = 21.62 (4 sf) 21.6 2 (certain) (uncertain)
Certain = 21.6
Min = 21.61
Max = 21.63
Significant figures
1
(21.62 ±0.01)
Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01 Certain
21.6
2
3
Measurement = Certain digits + 1 uncertain digit
Uncertain = 21.62 ±0.01 4
5
1 Smallest division = 1
2 Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = ±0.1
3 Certain = 36
Certain 36
4 Uncertain = 36.5 ±0.1
5 Measurement = Certain digits + 1 uncertain digit
(36.5 ±0.1)
Answer = 36.5 (3 sf) 36. 5 (certain) (uncertain)
Max = 36.6
Min = 36.4
22 22
Smallest division = 10
Certain = 40
Min = 45
Max = 47
Significant figures
1
(46 ±1)
Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = ±1
2
3
Measurement = Certain digits + 1 uncertain digit
Uncertain = 46 ±1 4
5
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 = 3.4
4 Uncertain = 3.41±0.01
5 Measurement = Certain digits + 1 uncertain digit
Certain 40
Answer = 46 (2 sf) 4 6 (certain) (uncertain)
Certain 3.4
(3.41 ±0.01)
Answer = 3.41 (3sf) 3.4 1 (certain) (uncertain)
Max = 3.42
Min = 3.40
Smallest division = 0.05
Certain = 0.45
Min = 0.46
Max = 0.48
Significant figures
1
(0.47 ±0.01)
Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = ±0.005 (±0.01)
2
3
Measurement = Certain digits + 1 uncertain digit
Uncertain = 0.47 ± 0.01 4
5
Certain 0.45
Answer = 0.47 (2 sf) 0.4 7 (certain) (uncertain)
0.1
0.2
0.3
0.4
0.5
Measurement
Smallest division = 0.1
Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01
1
2
Certain = 5.7
Uncertain = 5.72 ± 0.01
(5.72 ±0.01)
Answer = 5.72 (3sf) 5.7 2 (certain) (uncertain)
3
4
Smallest division = 1
Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = ±0.1
Certain = 3
Uncertain = 3.0 ± 0.1
(3.0 ±0.1)
1
2
3
4
Answer =3.0 (2 sf) 3 0 (certain) (uncertain)
round up
round up
round up
round down
4.2 2.32 + 0.6157 7.1357
7.1
12.587 4.25 + 0.12 16.957
16.96
4.7832 1.234 + 2.02 8.0372
8.04
1.0236 - 0.97268 0.05092
0.0509
1.367 - 1.34 0.027
0.03
23.112233 1.3324 + 0.25 24.694633
24.69
1247 134.5 450 + 78 1909.5
1910
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
uncertain
round down
least number decimal places
round down
uncertain
least number decimal places
uncertain
least number decimal places
uncertain
least number decimal places
round down
uncertain
least number decimal places
uncertain
least number decimal places
least number decimal places
uncertain
68.7 - 68.42 0.28
7.987 - 0.54 7.447
0.3
round down
uncertain
least number decimal places
7.45
least number decimal places
uncertain
round up
2.300 x 103 + 4.59 x 103 6.890 x 103
6.89 x 103
round up
47.68 x 104 + 23.2 x 103
476.8 x 103 + 23.2 x 103 500.0 x 103
least number decimal places
500.0 x 103
5.000 x 105
Convert to same exponent least number decimal places
Scientific notation
round down
round down round down round up
round down round down
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)
72
round up
23.123123 x 1.3344 30.855495
30.855
least sf (5sf) 21.45 x 0.023 0.49335
least sf (2sf)
0.49
2.8723 x I.6 4.59568
least sf (2sf)
4.6
round up
16.235 0.217 x 5 17.614975
least sf (1sf)
20
4.52 ÷ 6.3578 7.1093775
least sf (3sf)
7.11
round up
0.00435 x 4.6 0.02001
least sf (2sf)
0.020
6305 ÷ 0.010 630500
6.3 x 105
least sf (2sf)
63000
I.3*103 x 5.724*104 7.4412 x 107
923 ÷ 20312 0.045441
least sf (3sf)
0.0454
round down
1300 x 57240 74412000
least sf (2sf)
74000000 7.4 x 107
Click here for practice notes on sig figures
0.0000000001254
Scientific notation
Number too big/small How many significant figures
Written as
6,720,000,000
1010254.1
banotationScientific 10
a = 1 - 9 b = integer
91072.6
3 sf
4 sf
Speed of light
300000000 81000.3
3 sf
4660000
4.6600 x 10 6
4.66 x 10 6
4.660 x 10 6
Scientific notation
How many significant figures
3 sf
4 sf
5 sf
Size sand
Click here practice scientific notation Click here practice scientific notation
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
80. – 8.0 x 101 – (2sf) Digit 8 is certain It can be 79 to 81
81 or 8.1 x 101 80 or 8.0 x 101 79 or 7.9 x 101
80. 80
80.0 – 8.00 x 101 – (3sf) Digit 80 is certain It can be 79.9 or 80.1
80.1 or 8.01 x 101
80.0 or 8.00 x 101
79.9 or 7.99 x 101
80.0
3 ways to write 80
✔ More precise
round down
41.6
Volume, V = 4/3πr3
Radius, r = 2.15 cm
Significant figures and Uncertainty in measurement
least sf (3sf)
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
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Radius, r = (2.15 ±0.02) cm
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
4/3 – constant
π – constant
Their sf is not taken
(not a measurement)
60.4115.214.33
4 3 Volume
)142(
)16.160.41(
%)79.260.41(
%79.293.03%
%3%
%93.0%10015.2
02.0%
Volume
Volume
Volume
V
rV
r
16.160.41100
79.2VAbsolute
* Constant, pure/counting number has no uncertainty and sf not taken
Measurement raised to power of 3,
multiply % uncertainty by 3
* For measurement raised to power of n, multiply % uncertainty by n
Volume, V = 4/3πr3
Volume, V = 4/3πr3
)119(
)25.18495.18(
%)67.68495.18(
%67.6%
%%
%67.6%1000.3
2.0%
nceCircumfere
nceCircumfere
nceCircumfere
c
rc
r
round up
19
Circumference, C = 2πr
Radius, r = 3.0 cm
Significant figures and Uncertainty in measurement
least sf (2sf)
C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Radius, r = (3.0 ±0.2) cm
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
2 – constant
π – constant
Their sf is not taken
(not a measurement)
8495.180.314.32 nceCircumfere
25.18495.18100
67.6CAbsolute
* Constant, pure/counting number has no uncertainty and sf not taken
Circumference, C = 2πr
Circumference, C = 2πr
)2.08.24(
)198.080.24(
%)8.080.24(
%8.0%4.02%
%2%
%4.0%10025.2
01.0%
ntDisplaceme
ntDisplaceme
ntDisplaceme
s
ts
t
round down
24.8
Displacement, s = ½ gt2
Time, t = 2.25 s
Significant figures and Uncertainty in measurement
least sf (3sf)
s = 1/2 x 9.8 x (2.25)2 = 24.80625
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Time, t = (2.25 ±0.01) cm
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
g and ½ – constant
Their sf is not taken
(not a measurement)
198.080.24100
4.0sAbsolute
Displacement, s =1
2gt2
Displacement, s =1
2gt2
80625.2425.225.28.92
1, xxsntDisplaceme
Measurement raised to power of 2,
multiply % uncertainty by 2
* For measurement raised to power of n, multiply % uncertainty by n
round down
2.24
Length, I = 1.25 m
Significant figures and Uncertainty in measurement
least sf (3sf)
T = 2 x π x √(1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Length, I = (1.25 ±0.05) m
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
2, π and g – constant
Their sf is not taken
(not a measurement)
044.024.2100
2TAbsolute
g
LT 2
24.28.9
25.12 T
g
LT 2
g
LT 2
)04.024.2(
)044.024.2(
%)224.2(
%2%
%2
1%
%4%10025.1
05.0%
T
T
T
T
lT
l
* For measurement raised to power of n, multiply % uncertainty by n
Measurement raised to power of 1/2,
multiply % uncertainty by 1/2
)9.00.9(
%)442.1004.9(
%442.10%10%442.0%
%%%
%10%1000.2
2.0%
%442.0%10052.4
02.0%
Area
Area
A
hlA
h
l
round down
9.0
Area, A = I x h
Length, I = 4.52 cm Height, h = 2.0 cm
Significant figures and Uncertainty in measurement
least sf (2sf)
4.52 x 2.0 9.04
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
hheightlLengthAArea ,,,
04.90.252.4 Area
9.004.9100
442.10AAbsolute
Area, A = I x h
)2.00.4(
)24.000.4(
%)600.4(
%6%5%1%
%%%
%5%1000.2
1.0%
%1%10000.2
02.0%
Mole
Mole
Mole
n
vcn
v
c
round down
4.0
Moles, n = Conc x Vol
Conc, c = 2.00 M Volume, v = 2.0 dm3
Significant figures and Uncertainty in measurement
least sf (2sf)
2.00 x 2.0 4.00
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Conc, c = (2.00 ±0.02) M Volume, v = (2.0 ±0.1)dm3
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
vVolumecConcnMole ,,,
00.40.200.2 Mole
24.000.4100
6nAbsolute
vVolcConcnMole ,,,
)04.087.1(
%)14.287.1(
%14.2%93.1%21.0%
%%%
%93.1%100258
5%
%21.0%10063.482
1%
Density
Density
D
VmD
V
m
round down
1.87
Density = Mass Volume
Mass, m = 482.63g Volume, v = 258 cm3
Significant figures and Uncertainty in measurement
least sf (3sf)
482.63 ÷ 258 1.870658
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3
Density,D =Mass
Volume
Density,D =482.63
258=1.870658
04.087.1100
14.2DAbsolute
Treatment of Uncertainty Multiplying or dividing measured quantities % 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 =Mass
Volume
)417(
)51.372.16(
%)2172.16(
%21%20%1%
%%%
%20%1000.2
4.0%
%1%10000.2
02.0%
Enthalpy
Enthalpy
Enthalpy
H
TmH
T
m
round up
17
Enthalpy, H = mcΔT
Mass water = 2.00 g ΔTemp = 2.0 C
Significant figures and Uncertainty in measurement
least sf (2sf)
2.00 4.18 x 2.0 16.72
Recording measurement using significant figures
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 % 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
51.372.16100
21HAbsolute
TcmHEnthalpy ,
c – constant
sf is not taken
(not a measurement)
TcmHEnthalpy ,
72.160.218.400.2, HEnthalpy
)417(
)51.372.16(
%)2172.16(
%21%20%1%
%%%
%20%1000.2
4.0%
%1%10000.2
02.0%
Enthalpy
Enthalpy
Enthalpy
H
TmH
T
m
Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + water, M2 = (22.00 ±0.01)g
Treatment of uncertainty in measurement
Initial Temp, T1 = (21.2 ±0.2)C Final Temp, T2 = (23.2 ±0.2)C
Treatment of Uncertainty Multiplying or dividing measured quantities % 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 51.372.16
100
21HAbsolute
TcmHEnthalpy ,
TcmHEnthalpy ,
72.160.218.400.2, HEnthalpy
Adding or subtracting • Max absolute uncertainty is the SUM of individual uncertainties
Addition/Subtraction/Multiply/Divide
Mass water, m = (M2 –M1) Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Multiplying or dividing • Max %uncertainty is the SUM of individual %uncertainties
Diff Temp ΔT = (T2 –T1) Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Enthalpy, H = (M2-M1) x c x (T2-T1)
Addition/Subtraction
Add absolute 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
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
Multiplication
Add % uncertainty
Mass water, m = (2.00 ±0.02)g
round up
29
Significant figures and Uncertainty in measurement
least sf (2sf)
4.52 x 3.0 x 3.0 = 40.68 ÷ 1.414 28.769
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s
2/1
2
v
ItEnergy
Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Time, t = (4.52 ± 0.02)
2/1
2
,v
ItEEnergy
2/1
2
,v
ItEEnergy
%10%1000.2
2.0%
%20%1000.3
6.0%
%442.0%10052.4
02.0%
v
I
t
vItE %2
1%2%%
%1000.2
2.0
2
1%100
0.3
6.02%100
52.4
02.0% E
%45%442.45%5%40%442.0% E
%)45638.28(, EEnergy
)1329(, EEnergy
13638.28100
45EAbsolute
* For measurement raised to power of n, multiply % uncertainty by n
638.280.2
)0.3(52.4,
2/1
2
EEnergy
Z
HGsSpeed
)(,
round down
0.34
Significant figures and Uncertainty in measurement
20 + 16 = 36 ÷ 106 0.339
Recording measurement using significant figures
Recording measurement using uncertainty of equipment
Treatment of Uncertainty Multiplying or dividing measured quantities % 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
G = (20 ) H = (16 ) Z = (106)
G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0)
Speed, s =(G+H )
Zleast sf (2sf)
Speed, s =(G+H )
Z
339.0106
)1620(,
sSpeed
%77.2%10036
0.1)(% HG
%94.0%100106
0.1% Z
ZHGS %)(%%
%71.3%94.0%77.2% S
%)71.3339.0(, sSpeed
)012.0339.0(, sSpeed
012.0339.0100
71.3SAbsolute
Addition
add absolute uncertainty G+H = (36 ± 1) Z = (106 ± 1.0)
✔
*Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties
)01.034.0(, sSpeed