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