Error/Uncertainty

IB Chemistry,Uncertainty, Error Analysis, Standard Deviation
Uncertainty Calculation for Rate and Concentration of reaction.
Rate = 1/time, Time for X to disappear. ( Iodine Clock Reaction)
3 Methods for Uncertainty Calculation for Rate (0.10M) KI.
Average time is (5.28 + 4.75 + 4.47) / 3 =  4.83

1) % Uncertainty Method
  • Uncertainty time = Uncertainty stop watch + reaction time, ( 0.01 + 0.09 ) = ( 0.10 )
  • Time  = 4.83 ±( 0.10 )
2) Max-min range Method
  • Uncertainty time = (Max time - Min time)/2, = ( 5.28 - 4.47 )/2 = 0.41
  • Time = 4.83 ±0.41


1) %Uncertainty Method
Uncertainty time = (4.83 ± 0.10)
Rate = 1/ time, 1/4.83 = 0.207

2) Max-min range Method
Uncertainty time = ( 4.83 ± 0.41)
Rate = 1/time, 1/4.83 = 0.207


1) % Uncertainty Method

%Uncertainty time = (0.1/4.83) x100 %=2.07
%Uncertainty Rate = %Uncertainty time
%Uncertainty Rate = 2.07%
Rate = 0.207 ± 2.07 %
Rate = 0.207 ± 0.004




2) Max-min range Method
% Uncertainty time = (0.41/4.83) x 100% = 8.48%
% Uncertainty Rate = % Uncertainty time
%Uncertainty Rate = 8.48%
Rate = 0.207 ± 8.48%
Rate = 0.207 ± 0.017
3) Max /Min Rate Method

Max Rate = 1/Minimum time, 1/4.47 = 0.220
Min Rate = 1/Maximum time, 1/5.28 = 0.190
Ave rate = 1/ Ave time, 1/ 4.83 = 0.207
Uncertainty Rate = 0.207 ± ( 0.220 --- 0.190 )



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Uncertainty Cal for Conc, 30ml 0.1M KI (2 Methods used)
1) % Uncertainty Method
Pipette uncertainty  =  ± 0.4 and Total Volume used = 30ml

1) % Uncertainty Method
%Uncertainty Conc = %Uncertainty Vol of KI + %Uncertainty Vol Water
%Uncertainty Vol KI = (0.4/30) x100%=1.3%
%Uncertainty Conc = % Uncertainty Vol KI
%Uncertainty Conc = 1.3%





Calculation Absolute Uncertainty
Conc = 0.10 ± 1.3%
Conc = 0.100 ± 0.001



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2) Max/Minimum Method
Preparing 0.053M from 0.1M KI using pipette.
Pipette uncertainty = ± 0.4
Max Conc KI = Max Vol KI and Min Vol of water used
M x V (dil) = M x V (conc)
M x (15.8 + 14.2) = 0.1 x (15.8)
M = 0.1 x ( 15.8 )/( 15.8+ 14.2 )
M = 0.053

Max Conc KI=Max Vol KI + Min Vol water 
Max Vol KI = 15.8 ± 0.4 = 16.2
Min Vol water = 14.2 ± 0.4 = 13.8

M x V (dil) = M x V (conc)
Max Conc KI =  (0.1 x 16.2)/ ( 16.2 + 13.8 )
Max Conc = 0.054

Min Conc  = Min Vol KI and Max Vol water used
Min Vol KI = 15.8 ± 0.4 = 15.4
Max Vol water = 14.2 ± 0.4 = 14.6

M x V (dil) = M x V (conc)
M x (15.4 + 14.6)  = 0.1 x (15.4)
M = 0.1 x ( 15.4 )/( 15.4 + 14.6 )
M = 0.051
Min Conc = 0.051
Uncertainty Conc = 0.053 ± ( 0.054 --- 0.051)
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Uncertainty Cal, 2 fold Serial dilution using 3% H2O2
2 Methods using 
1st % Uncertainty Method
For 1.5%, Total % Uncertainty is 1.6%, Answer = 1.5 ± 1.6%, Propagation of error involved

Absolute Uncertainty for 1.5 ± 1.6% = (1.50 ± 0.02)%
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2nd, Max/ Min Method 
For 2 fold Serial dilution on 3% H2O2
M x V (dil) = M xV (conc)
M x (1.5 +1.5) = 3% x 1.5
M = 1.5%

Max Conc H2O2 when
Max Vol H2O2 used = 1.51
Min Vol water used = 1.49

Min Conc H2O2 when
Min Vol H2O2 used  = 1.49
Max Vol water used = 1.51



Max Conc H2O2                                         Min Conc H2O2                              

M x V (dil) = M x V (conc)                         M x V (dil) = M x V (conc)                           
M x (1.51 + 1.49)  = 3% x ( 1.51 )               M x (1.51 + 1.49)  = 3% x ( 1.49 )
M = 3% x ( 1.51 )/( 1.51 + 1.49 )                 M = 3% x ( 1.49 )/( 1.51 + 1.49 )
M = 1.51%                                                M = 1.49%
Max Conc = 1.51%                                    Min Conc = 1.49%
Uncertainty Conc H2O2 = 1.50 ± (1.51%---1.49%)
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Uncertainty Cal for Rate Using Vernier Sensor.
How changing conc of H2O2 affect the rate of decomposition of H2O2 (catalase) measured using pressure sensor?

Initial Rate for 1.5% = slope/ gradient of curve
Perform 3 trials at 1.5% and take Average rate
Average Rate by taking average for 3 trials
Uncertainty for Rate = Standard deviation
Use Excel to calculate Std deviation









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Video, adding uncertainty, error bar, Std deviation into Excel
Keynotes from video
  • Adding horizontal/vertical error bar/ uncertainty or Std deviation
  • Follow notes provided in powerpoint
Video on adding Std deviation into Excel

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IB Chemistry, Enthalpy, Uncertainty Calculation, Error analysis, Standard Deviation

Enthalpy change for displacement Zn + CuSO4

2 methods for Uncertainty Cal
1) % Uncertainty Method
Zinc excess, Copper limiting,
Conc CuSO4 = (1M± 0)
Vol = (20.0 ± 0.3) ml
Mol of Cu ions= M x V= 0.02 mol
△t = (68.0±0.2) - (25.1±0.2) = (42.9±0.4)
m = 20.0 ± 0.3ml

Q = mc△t
Q = 20.00 x 4.184 x △t
Q = 20.00 x 4.184 x 42.9
Q = 3589.8J ----------0.02 mol
Q = 3589.9/0.02------ 1mol
Q = 179.5kJ/mol


Extrapolation for Temp increase, 68.0 - 25.1= 42.9
%Uncertainty Q = % Uncertainty m + % Uncertainty △t
%Uncertainty m = (0.3/20) x 100% = 1.5%
%Uncertainty △t = (0.2+0.2) / 42.9 x 100% = 0.93%
%Total Uncertainty = (0.93 + 1.5) = 2.43%
Q = (179.5 ± 2.43%)
Q = (179.5 ± 4.36) = (180 ± 4)kJ/mol


2) Using Max/Min Method
Q = mc△t
Q = 20.00 x 4.184 x 42.9
Q = 3589.8J ----------0.02 mol
Q = 3589.9/0.02------ 1mol
Q = 179.5kJ/mol

Q = mc△t
△t = 42.9 ± (0.2+02) = (42.9±0.4)     max t= 42.9+ 0.4 = 43.3      min t=42.9- 0.4=42.5
m = 20.0 ± 0.3ml                             max m= 20.0+ 0.3=20.3     min m=20.0- 0.3= 19.7

Max  Q = Maximum  m and Maximum △t
Max  Q = m c△t = 20.3 x 4.184 x 43.3 = 3677.7J
Max Q = 3677.7J -----------0.02mol
Max Q = 3677.7/0.02-------1 mol
Max Q = 183.3kJ/mol

Min Q = Minimum  m and Minimum △t
Min Q = mc△t = 19.7 x 4.184 x 42.5 = 3503.0J
Min Q = 3503.0J------------0.02mol
Min Q = 3503.0J/0.02-------1 mol
Min Q = 175.1kJ/mol
Using Max/Min Method                           Using %Uncertainty Method
Q = 179.5 ± ( 183.3--175.1)kJ/mol          Q = (179.5 ± 2.43%) = (180 ± 4)kJ/mol
Q = (183.3---175.1)                    Q = (184---176)
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Accurate way taking Uncertainty Moles Cu
1) Using %Uncertainty Method.
%Uncertainty moles of Cu used
Conc CuSO4 = (1M±0) , Vol = (20.0 ± 0.3) ml
Moles of Cu ions = M x V= 0.02 mol

%Uncertainty moles Cu = %Uncertainty in M + %Uncertainty in Vol 
%Uncertainty Cu = 0% + (0.3/ 20) x 100% = 0% +1.5% = 1.5%

Total %Uncertainty Q = %Uncertainty m + %Uncertainty △t + %Uncertainty mol Cu
Total %Uncertainty Q = 1.5% + 0.93% + 1.5% = 3.93%
Q = (179.5 ± 3.93%)= (179.5 ± 7.05) = (179 ± 7)
Q = ( 186 ----172 )kJ/mol

2) Using Max/Min Method
Conc CuSO4 = (1M±0) , Vol = (20.0 ± 0.3) ml
Moles of Cu ions = M x V = 0.02 mol

△t = 42.9 ± (0.2+02) = (42.9±0.4)     max t=42.9+0.4 = 43.3     min t=42.9- 0.4=42.5
m = 20.0 ± 0.3ml                                max m=20.0+ 0.3=20.3     min m=20.0- 0.3= 19.7


Max Vol = 20.3ml, Min Vol = 19.7ml        Molarity = (1M±0)
Max moles Cu = Max M x Max Vol = 1 x 20.3 = 0.0203
Min moles Cu = Min M x Min Vol = 1 x 19.7 = 0.0197
Uncertainty moles of Cu 0.020 ± (0.023--0.0197)

Max Uncertainty Q = mc△t = 20.3 x 4.184 x 43.3 = 3677.7J
Min Uncertainty Q = mc△t = 19.7 x 4.184 x 42.5 = 3503.0J
Max Q = 3677.7J ----------0.02mol
Min Q = 3503.0J-----------0.02mol
Uncertainty moles of Cu 0.02 ± (0.023--0.0197)
Max moles Cu = 0.023
Min moles Cu = 0.0197

Max Q = Max Uncertainty Q + Min Uncertainty mol Cu ( will give greatest error )
Min Q = Min Uncertainty Q + Max Uncertainty mol Cu ( will give least error )
Max Q = 3677.7---------------0.0197 mol Cu      Min Q = 3503.0---------------0.023mol Cu
Max Q = 3677.7/0.0197-------1 mol                   Min Q = 3503.0/0.023--------1mol
Max Q = 186.7kJ/mol                                        Min Q = 152.3kJ/mol
Q = 179.5 ± ( 186.7---152.3 )kJ/mol

%Uncertainty Method                                       Max/Min Method
Q = ( 186 ----172 )kJ/mol                                   Q = ( 186.7---152.3 )kJ/mol
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IA must be done with minimum 3 trials, for valid conclusion.
Displacement done with 3 trials.
Click HERE for Data
Displacement of Zinc + CuSO4
Zinc excess, Copper limiting,
Conc CuSO4 = (1M±0)
Vol = (20.0 ± 0.3) ml
Mol of Cu ions= M x V = 0.02 mol



Two Methods used
1st Method
Average Q for 3 trials and using Standard deviation as uncertainty

Temp change for 3 trials
△t1 =( 26.2 ± 0.2)- (21.7± 0.2)=(4.5± 0.4)
△t2 = (26.1 ± 0.2)- (21.6± 0.2)=(4.5± 0.4)
△t3 = (25.8 ± 0.2)-(21.7± 0.2)=(4.1± 0.4)
Q1 = mc△t1 = 20.0 x 4.184 x 4.5 = 376.5
Q2 = mc△t2 = 20.0 x 4.184 x 4.5 = 376.5
Q3 = mc△t3 = 20.0 x 4.184 x 4.1= 343.1

Average Q = (376.5 + 376.5 + 343.1)/3 = 365.4J    Uncertainty Q = Standard Deviation Q= (19.28)
Average Q = 365.4J----------0.02 mol                      Uncertainty Q = 19.28-----------0.02 mol
Average Q = 365.4/0.02------1 mol                        Uncertainty Q = 19.28/0.02-----1 mol
Average Q = 18.3kJ/mol                                       Uncertainty Q = 0.96kJ/mol
Average Q = (18.3 ± 0.96)kJ/mol                           

2nd Method 
Taking average Temp change
Click HERE for data

Average Temp data is taken.
Displacement for Zinc + CuSO4
Zinc excess, Copper limiting,
Conc CuSO4 = (1M± 0)
Vol = 20.0 ± 0.3 ml
Mol of Cu ions= M x V = 0.02 mol
△t = (25.7 ± 0.2) - (21.6 ± 0.2)= (4.1 ± 0.4)





By extrapolation using average temp
△t = (25.7 ± 0.2) - (21.6 ± 0.2)= (4.1 ± 0.4)



Q = mc△t
Q = 20.0 x 4.184 x △t
Q = 20.0 x 4.184 x 4.1
Q = 343.1J ---------0.02 mol
Q = 343.1/0.02------1 mol
Q = 17.155kJ/mol

Using %Uncertainty Method 
Temp increase = (25.7 ± 0.2) - (21.6 ± 0.2)= (4.1 ± 0.4)
%Uncertainty Q = % Uncertainty m + % Uncertainty △t
%Uncertainty m = (0.3/20) x 100% = 1.5%
%Uncertainty △t = (0.4 / 4.1) x 100% = 9.75%
%Total Uncertainty = (1.5 + 9.75) = 11.25%
Q = (17.155± 11.25%)= (17.15±1.92)kJ/mol

More accurate way is to consider Uncertainty for moles Cu
Conc CuSO4 = (1M± 0), Vol = (20.0 ± 0.3) ml
Mol of Cu ions= M x V

%Uncertainty moles Cu = %Uncertainty in M + %Uncertainty in Vol 
%Uncertainty Cu = 0% + (0.3/ 20) x 100% = 0% +1.5% = 1.5%

%Total Uncertainty Q = %Uncertainty m + %Uncertainty △t + %Uncertainty mol Cu
%Total Uncertainty Q = 1.5% + 9.75% + 1.5% = 12.75%
Q = (17.15± 12.75%) = (17.15± 2.18)kJ/mol

You can also use Max/min Method as shown above.
In short, 
  • Make sure you perform 3 trials and compute the average or use std deviation
  • Use %Uncertainty or Max/min Method of your choice
  • Use lots of common sense for error treatment
  • The only certainty in life is continual uncertainty
  • Have fun with uncertainty
Click HERE to view more on uncertainty and error analysis
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Uncertainty Calculation for Rate of reaction, Enthalpy Change and Ideal Gas Equation.


Uncertainty Calculation for RMM using Ideal Gas Equation.
Ideal gas law equation,  PV = nRT
Notes for Ideal Gas Equation.


P - absolute pressure of the gas in Pa /Nm-2
V - volume in m3
n - number of moles
T - absolute temperature.(Kelvin)
R = gas constant, 8.314 J K-1 mol-1
Calculate the RMM of a gas, given
M1, mass empty flask, = (25.385 ±0.001)g
M2, mass flask filled with gas, = (26.017 ±0.001)g
M3, mass flask with water, = (231.985 ±0.001)g
Temperature = 32.0C = (273.0 +32.0) = (305.0 ±0.1)K
Atmospheric pressure = 101,000 Nm-2 /Pa
PV = nRT,     n = mass/ RMM
PV =  RT x mass/RMM 
RMM = RT x mass / PV

Mass of gas = (M2 - M1) = (26.017 - 25.385)g = 0.632g
Vol of gas = Vol of flask = Vol of water = Mass of water = (231.985 - 25.385) = 206.6 x10-6 m3
RMM = RT x mass / PV 
RMM = 8.314 x 305.0 x 0.632 / 101,000 x 206.6 x10-6
RMM (expt value) = 76.80    RMM (actual value) = 80.00


Percentage Error 
= RMM (actual) - RMM (expt) /RMM (actual) x100%
= (80.00 - 76.80)/80.00 x 100%
= 3.20/80 x 100% = 4% (Error)
    
Percentage Uncertainty due random error ( Temp, Mass and Vol )
% Uncertainty Temp = (0.1/305.0 x 100)% = 0.0327% = 0.33%
% Uncertainty Mass gas = (0.002/0.632 x 100)% = 0.32%
% Uncertainty Vol = (0.002/206.6 ) x 100% = 0.000968% = 0.001%
Total % Uncertainty due to random error = ( 0.33 + 0.32 + 0.001 )% = 0.651% = 0.65%
Conclusion:
  • 4% error is more than the total error due to all random error (T, Vol and Mass)
  • Systematic error have to account for the 3.35% difference which might be due to experimental procedure.
  • RMM = (76.8 ± 0.65%) in percentage uncertainty
  • RMM = (76.8 ± 0.5) in absolute uncertainty, max/min range is ( 76.3----77.3)
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    Simple Experimental setup
    Determination of (RMM) of a gas ( CO2 or butane ) using ideal gas equation.
    Steps to follow
    1. Weigh a dry 100ml volumetric flask with stopper to 0.001g
    2. Remove stopper
    3. Fill it with CO2/butane gas with glass delivery tube, stopper it and reweigh
    4. Repeat step 2 and 3 until mass of flask is constant
    5. Fill the flask with water, insert a stopper, and reweigh it.
    Click Here for IA on DCP and CE
    Detail Experimental Setup
    Click Here to view detail procedure setup.
    Source from Chemistry SLSS website
    Click HERE for more sample ideal gas calculation
    Click HERE for more info on ideal gas equation
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    Uncertainty Calculation on Rate of reaction

    IB Chemistry, IB Biology on Uncertainty calculation, error analysis and standard deviation on rate of reaction 
    Conclusion:
    • 3 ways to calculate uncertainty for rate of reaction
    • 1st method - using % percentage uncertainty method( most common)
    • 2nd method - using Max/Min method
    • 3rd method - using standard deviation method
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    Uncertainty Calculation on Enthalpy Change 
    (Displacement Reaction)

    Conclusion:
    • 2 ways to calculate Uncertainty for Enthalpy Change (ΔQ)
    • 1st method - using % percentage uncertainty method
    • 2nd method - using Max/Min method

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    IB Chemistry, IB Biology on Uncertainty Error Calculation, Random vs Systematic error, Precision vs Accuracy

    Conclusion and keywords from slides:
    • Addition and Subtraction - Add absolute uncertainties
    • Multiplication and Division - Add percentage uncertainties
    • Uncertainty calculation for Rate of reaction and Conc of solution can be calculated using - % Uncertainty Method or  Max/Min Method
    • Accuracy vs Precision
    • Random Error vs Systematic error

    Click here for final checklist and assessment rubric before final submission.
      Thank to all pictures, sources and Stephen Taylor for his assessment rubric used for the above post
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