ib chemistry on uncertainty, error analysis, random and systematic error

Every measurement – associated with an error No measurement is 100% precise or accurate.

Affects precision

2 Types of Errors

Systematic Error Random Error

Measurement

3 Types of Measurement

Not Precise + Not Accurate Precise + Not Accurate

NOT accurate

NOT precise

High systematic

High random error

Precise

low random error

high random error

Not accurate

High systematic error

Affects accuracy

high systematic

error

Measurement too high/ low • Instrument not calibrated • Faulty apparatus (zero error) • Incorrect measurement • Imperfect instrument • Procedure/method incorrect/predictable

Accurate

low systematic error

Measurement random • Instrument imprecise/uncertainty • Fluctuation reading burette/pipette • Small sample size/trials • Statistical fluctuation of

measurement/reading by someone/unpredictable

2 Types of Errors


Accuracy Measurement value close to correct value

Precise Measurement value close to each other

Accurate + Precise

Precise + Accurate

VS

VS

Heating expt Calorimetry expt

Prevent heat loss using insulator

Measurement too high/ low • Instrument not calibrated • Faulty apparatus (zero error) • Incorrect measurement • Imperfect instrument • Procedure/method incorrect • Predictable

Measurement random • Instrument imprecise/uncertainty • Fluctuation reading burette/pipette • Small sample size/trials • Statistical fluctuation of

measurement/reading by someone • Unpredictable

Affects precision

2 Types of Errors


Can be identified/eliminated

High random

error

Affects accuracy High systematic

error

By repeating more trials/average

Correct

value lower higher

Direction error – always one side (higher/lower)

Improve expt design

Calibrating equipment for zero error

Improve measuring technique

Can be reduced

Direction error – always random

Using precise instrument

Correct

value lower higher

✓ ✗

Cool down before weighing

)119(

)25.18495.18(

%)7.68495.18(

%6.6%

%%

%6.6%1000.3

2.0%

nceCircumfere

nceCircumfere

nceCircumfere

c

rc

r

Random and Systematic Error

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 = 6.6% 3.0 % uncertainty C = % uncertainty r % ΔC = % Δr

rnceCircumfere 2

8495.180.314.32 nceCircumfere

rnceCircumfere 2

25.18495.18100

6.6CAbsolute* Constant, pure/counting number has no uncertainty and sf not taken

Measuring circumference using a ruler

Correct value = 20.4 Expt value = 19 ±6.7%

%100)exp

(%

correct

correcttError


%Percentage Error = 6.7%

% Random Error

%Systematic Error

0.1%

%Random Error

6.6%

Small systematic error

Way reduce random error

%7.6%100)4.20

4.2019(%

Error

%)6.68495.18( nceCircumfere

Step/procedure correct

High random error

)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


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

198.080.24100

4.0sAbsolute

2

2

1, gtsntDisplaceme

2

2

1, gtsntDisplaceme

80.2425.225.28.92

1, xxsntDisplaceme

Measurement raised to power of 2,

multiply % uncertainty by 2

* For measurement raised to power of n, multiply % uncertainty by n


Measuring displacement using a stopwatch


Correct value = 23.2 Expt value = 24.8 ±0.8%

%100)exp

(%

correct

correcttError

%7.0%100)2.23

2.238.24(%

Error

%)8.080.24( ntDisplaceme

% Random Error


%Random Error 0.8%

% error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable but need to reduce random error

)04.024.2(

)044.024.2(

%)224.2(

%2%

%2

1%

%4%10025.1

05.0%

T

T

T

T

lT

l


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 l % ΔT = ½ x % ΔI

044.024.2100

2TAbsolute

24.28.9

25.12 T

g

LT 2

g

LT 2

* For measurement raised to power of n, multiply % uncertainty by n

Measurement raised to power of 1/2,

multiply % uncertainty by 1/2


Measuring period using a ruler


Correct value = 2.15 Expt value = 2.24 ±2%

%100)exp

(%

correct

correcttError

%2.4%100)15.2

15.224.2(%

Error

%)224.2( T

% Random Error


%Random Error = 2% %Systematic Error = 2.2%

% error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – way to reduce systematic error

)9.00.9(

%)1004.9(

%442.10%10%442.0%

%%%

%10%1000.2

2.0%

%442.0%10052.4

02.0%

Area

Area

A

hlA

h

l


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

10AAbsolute


Measuring Area using ruler



%100)exp

(%

correct

correcttError

%9%100)7.22

7.228.24(%

Error

%)1004.9( Area

% Random Error

%Percentage Error = 9%

%Random Error = 10%

% error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable – need to reduce random error

Reduce random error – HUGE (10%) – use precise instrument vernier calipers

Vernier caliper

hheightlLengthAArea ,,,

)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

Conc, c = (2.00 ±0.02) cm Volume, v = (2.0 ±0.1)dm3

Treatment of Uncertainty Multiplying or dividing measured quantity % uncertainty = sum of % uncertainty of individual quantity 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

VolConcnMole ,


Measuring moles using dropper and volumetric flask



%100)exp

(%

correct

correcttError

%10%100)63.3

63.34(%

Error

%)600.4( Mole

% Random Error


%Random Error = 6% %Systematic Error = 4%

% error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – improve on method/steps used.

Dropper, volumetric

flask

Ways to reduce error

Random error (6%) More precise instrument -pipette

Systematic error (4%) Calibration of instrument

)04.087.1(

%)1.287.1(

%1.2%93.1%21.0%

%%%

%93.1%100258

5%

%21.0%10063.482

1%

Density

Density

D

VmD

V

m

Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3

Volume

MassDDensity ,

870658.1258

63.482, DDensity

04.087.1100

1.2DAbsolute

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

Volume

MassDDensity ,


Measuring density using mass and measuring cylinder



%100)exp

(%

correct

correcttError

%5%100)78.1

78.187.1(%

Error

%)1.287.1( Density

% Random Error


%Random Error = 2.1% %Systematic Error = 2.9%

% error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occurs


Random error (6%) Precise instrument mass balance

Systematic error (4%) Use different method like displacement can

Displacement can Precise balance

)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


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 ,

TcmHEnthalpy ,

72.160.218.400.2, HEnthalpy


Measuring Enthalpy change using calorimeter/thermometer



%100)exp

(%

correct

correcttError

%50%100)44.33

44.3372.16(%

Error

% Random Error


%Random Error =21% %Systematic Error = 29%

% error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occurs – reduce this error

Random error (21%) Precise Temp sensor


Systematic error (29%) Reduce heat loss using styrofoam cup

%)2172.16( Enthalpy

Temp sensor

012.0339.0100

7.3SAbsolute

Z

HGsSpeed

)(,


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 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0)

Z

HGsSpeed

)(,

339.0106

)1620(,

sSpeed

%77.2%10036

0.1)(% HG

%94.0%100106

0.1% Z

ZHGS %)(%%

%7.3%94.0%77.2% S

%)7.3339.0(, sSpeed

)012.0339.0(, sSpeed

Addition

add absolute uncertainty G+H = (36 ± 1) Z = (106 ± 1.0)

✔

*Adding or subtracting- Max absolute uncertainty is the SUM of individual uncertainties


Measuring speed change using stopwatch



%100)exp

(%

correct

correcttError

%)7.3339.0( Speed


%Random Error = 3.7%

% error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable – need to reduce random error

% Random Error

No systematic error Steps/method are reliable.


Random error (3.7%) Precise time sensor

%3%100)330.0

330.0339.0(%

Error

precise time sensor


Treatment of Uncertainty Multiplying or dividing measured quantities % uncertainty = sum of % uncertainty of individual quantity Time, t = (4.52 ±0.02) %uncertainty temp (%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 x %ΔI + ½ x %ΔV

Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Temp, 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%%

%45%1000.2

2.0

2

1%100

0.3

6.02%100

52.4

02.0% E

%)45638.28(, EEnergy

)1329(, EEnergy

13638.28100

45EAbsolute

638.280.2

)0.3(52.4,

2/1

2

EEnergy




%100)exp

(%

correct

correcttError

%50%100)092.19

092.19638.28(%

Error

% Random Error


%Random Error = 45% %Systematic Error = 5%

%)45638.28(, EEnergy Reduce random error – HUGE (45%) Precise instrument.

% error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – small compared to random error

Temp sensor

)2212(

)8.1212(

%)88.0212(

%88.0%8.0%08.0%

%%%

%8.0%1006.50

4.0%

%08.0%10000.25

02.0%

Enthalpy

Enthalpy

Enthalpy

H

TmH

T

m

Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + CuSO4 M2 = (45.00 ±0.01)g

Treatment of uncertainty

Initial Temp, T1 = (20.0 ±0.2)C Final Temp, T2 = (70.6 ±0.2)C

Treatment of Uncertainty Multiplying or dividing measured quantities % uncertainty = sum of % uncertainty of individual quantities Mass, m = (25.00 ±0.02) %uncertainty mass (%Δm) = 0.02 x 100 = 0.08% 25.00 ΔTemp = (50.6 ±0.4) %uncertainty temp (%ΔT) = 0.4 x 100 = 0.8% 50.6 % uncertainty H = % uncertainty mass + % uncertainty temp % ΔH = % Δm + %ΔT

86.1212100

88.0HAbsolute

TcmHEnthalpy ,

TcmHEnthalpy ,

29.5025.0

29.56.5018.400.25,

4

moleCuSO

HEnthalpy

Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties

Addition/Subtraction/Multiply/Divide

Mass CuSO4 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 CuSO4 m = (45.00 –20.00) = 25.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass CuSO4 m = (25.00 ±0.02)g

Diff Temp ΔT = (70.6 –20.0) = 50.6 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (50.6 ±0.4)

ΔTemp = (50.6 ±0.4) C

Multiplication

Add % uncertainty

Mass CuSO4 m = (25.00 ±0.02)g

Continue next slide

212025.0

129.51 4 moleCuSO

Expt on enthalpy change of displacement between Zinc and copper sulphate

25 ml (1M) (0.025mole) CuSO4 solution added to cup. Initial Temp, T1 taken. Excess zinc powder was added. Final Temp T2 was taken. Calculate ΔH for reaction.

%)88.0212( Enthalpy


Mass CuSO4 = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C


Measuring Enthalpy change using calorimeter/thermometer


Correct value = 250 Expt value = 212 ±0.8%

%100)exp

(%

correct

correcttError

% Random Error


%Random Error =0.88% %Systematic Error = 14.1%

% error fall outside> than % uncertainty (%Random error) • Small random error cannot account for % error • Systematic error occurs – reduce this error

Small random error Equipments OK


Reduce heat loss use styrofoam cup

Systematic error (14.2%)

Extrapolate to higher temp (Temp correction)

Assumption wrong • Heat capacity cup is significant • Specific heat capacity CuSO4 is not 4.18 • Thermometer has measurable heat capacity • Density solution not 1.00g/dm3

Enthalpy = (212±0.88%)

Stir the solution to distribute heat

stirrer

✗

%15%100)250

250212(%

Error

ib chemistry on uncertainty, error analysis, random and systematic error

Education

error systematic error

random error random

percentage error

error fall

otherhigh random error

error exp t

systematic error correct

high random error way