Chapter 3: Experimental Error
Chapter 3: Experimental Error Significant Figures • Significant Figures: the minimum number of digits needed to write a given value in scientific notation without loss of precision. o Zeros are significant when they occur in the middle of a number (101.01) or at the end of a number on the right-hand side of a decimal point (0.1060) o The last significant digit always has some associated uncertainty. o Interpolation: estimate all readings to the nearest tenth of the distance between scale divisions (on a 50-mL burette, which is graduated to 0.1-mL, read the level to the nearest 0.01-mL). Significant Figures in Arithmetic • Addition and Subtraction: express all numbers with the same exponent and align all numbers with respect to the decimal point; round off the answer according to the number of decimal places in the number with the fewest decimal places. o If the numbers to be added or subtracted have equal numbers of digits, the answer goes to the same decimal place (integers are always exact) as in any of the individual numbers. Example: 18.9984032 + 18.9984032 + 83.798 = 121.795, since 83.798 only has 3 numbers after the decimal point and rounding of the final answer. o If the first insignificant figure is below 5, we round the number down. o In the addition or subtraction of numbers expressed in scientific notation, all numbers should first be expressed with the same exponent. Example: 1.632 x 10^5 + 4.107 x 10^3 + 0.984 x 10^6 1.632 x 10^5 + 0.04107 x 10^5 + 9.84 x 10^5 = 11.51307 x 10^5 11.51 x 10^5 • Multiplication and Division: we are normally limited to the number of digits contained in the number with the fewest significant figures. o Example: (4.3179 x 10^12) x (3.6 x 10^-19) = 1.6 x 10^-6 o The power of 10 has no influence on the number of figures that should be retained. • Logarithms and Antilogarithms o If n = 10^a, then logn = a. Example: 2 is the logarithm of 100 because 100 = 10^2 and the logarithm of 0.001 is -3 because 0.001 = 10^-3. o In the equation above, the number n is said to be the antilogarithm of a. Example: the antilogarithm of 2 is 100 because 10^2 = 100. o A logarithm is composed of a characteristic and a mantissa. Characteristic: the integer part, whole number. Mantissa: the decimal part. o Number of digits in the mantissa of logx = number of significant figures in x. o Number of digits in antilogx (=10^x) = number of significant figures in mantissa of x. Example: log(5.403 x 10^-8) = -7.2674 Example: 10^6.142 = 1.39 x 10^6
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Example: log(0.001237) = -2.9076 Example: antilog(4.37) = 2.3 x 10^4 Significant Figures and Graphs: when drawing a graph on a computer, consider whether the graph is meant to display qualitative behavior of the data or precise values that must be read with several significant figures (it should at least have tick marks on both sides of the horizontal and vertical scales).
Types of Error • Systematic Error/Determinate Error: a consistent error that can be detected and corrected; arises from a flaw in equipment, design of an experiment or from an uncalibrated burette. • Ways to detect systematic error: o Analyze a known sample and your method should reproduce the known answer. o Analyze blank samples containing no analyte being sought; if you observe a nonzero result, your method responds to more than you intend. o Use different analytical methods to measure the same quantity; if results do not agree, there is error in one or more of the methods. o Round-Robin Experiment: different people in several laboratories analyze identical samples by the same or different methods; disagreement beyond the estimated random error is systematic error. • Random Error/Indeterminate Error: cannot be eliminated, but it might be reduced by a better experiment; arises from uncontrolled variables in the measurement and has an equal chance of being positive or negative (different people reading scales = subjective). • Precision: reproducibility of a result; if you measure a quantity several times and the values agree closely with one another. • Accuracy: nearness to the truth; if a known standard is available, accuracy is how close your value is to the known value. • Certified Reference Materials: can be used to test the accuracy of your analytical procedures. • Absolute Uncertainty: the margin of uncertainty associated with a measurement; if the estimated uncertainty in reading a calibrated burette is ± 0.02mL, we say that ± 0.02mL is the absolute uncertainty associated with the reading. • Relative Uncertainty: compares the size of the absolute uncertainty with the size of its associated measurement. o Relative Uncertainty = absolute uncertainty/magnitude of measurement o If the burette reading is 12.35 ± 0.02mL, then the relative uncertainty is 0.02mL/12.35mL = 0.002. o Percent Relative Uncertainty = 100 x relative uncertainty. o The percent relative uncertainty of 0.002 is 0.2%. Propagation of Uncertainty from Random Error • Most propagation of uncertainty computations that will be encountered deals with random error, not systematic error. • Addition and Subtraction: use absolute uncertainty. o Uncertainty in Addition and Subtraction: e4 = √(e1^2 + e2^2 + e3^2) o Example: for uncertainties of 3.06 (± 0.03, ± 0.02 and ± 0.02), the total
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uncertainty is equal to √(0.03)^2 + (0.02)^2 + (0.02)^2 = ± 0.04 absolute uncertainty. o For its relative uncertainty, 0.04/3.06 x 100 = ± 1%. Multiplication and Division: use percent relative uncertainty through conversion. o Uncertainty in Multiplication and Division: %e4 = √(%e1^2 + %e2^2 + %e3^2) o Example: 1.76 (± 0.03) x 1.89 (± 0.02) / 0.59 (± 0.02) = 5.64 ± e4
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To calculate e4, we have to convert absolute uncertainties into percent relative uncertainties (0.03/1.76 = ± 1.7%, 0.02/1.89 = ± 1.1% and 0.02/0.59 = 3.4%). Then find the percent relative uncertainty of the answer (%e4 = √(1.7^2) + 1.1^2 + 3.4^2) = 4%). Round off to the number of significant digits only in the answer. To convert relative uncertainty into absolute uncertainty, 4% x 5.6 = 0.04 x 5.6 = 0.2. The answer is 5.6 ± 0.02. Mixed Operations: first work out the uncertainties for addition/subtraction (absolute uncertainty) then the uncertainties of multiplication/division (percent relative uncertainty). Real Rule for Significant Figures: the first uncertain figure is the last significant figure. o Example: 0.002364 (± 0.000003) / 0.02500 (± 0.00005) = 0.1066 (± 0.0002); the answer is properly express with three significant figures even though the original data has four figures. o The answer’s decimal place has to coincide with the uncertainty. o In multiplication/division, keep an extra digit when answer lies between 1 and 2. Exponents and Logarithms: use relative uncertainty and not percent relative uncertainty in calculations involving logx, lnx, 10^x and e^x but percent relative uncertainty for powers and roots. o Uncertainty for Powers and Roots: y = x^a %ey= a(%ex) o Uncertainty for Logarithm: if y is the base 1- logarithm of x, then y = logx ey = (1/ln10)(ex/x) = 0.43429(ex/x). o Natural Logarithm (ln): ln of x is the number y whose value is such that x = e^y. o Uncertainty for Natural Logarithm: y = lnx ey = ex/x o Uncertainty for Antilog (10^x): y = 10^x ey/y = (ln10)ex = 2.3026 ex o Uncertainty for e^x: y = e^x ey/y = ex
Propagation of Uncertainty from Systematic Error • Uncertainty in Atomic Mass: The Rectangular Distribution o Atomic mass from different sources approximates a rectangular distribution where there is approximately equal probability of finding any atomic mass in between the uncertain range. o The standard uncertainty/standard deviation in the rectangular distribution is ± a/√3. • Uncertainty in Molecular Mass o For systematic uncertainty, we add the uncertainties of each term in a sum or difference.
Propagation of Systematic Uncertainty: Uncertainty in Mass of n Identical Atoms n x (standard uncertainty in atomic mass) 2C: 2(12.0107 ± 0.00046) = 24.0214 ± 0.00092, where 0.00046 x 2 = 0.00092. n x (uncertainty listed in periodic table)/√3 atomic mass of C = 12.0107 ± 0.0008/√3 = 12.0107 ± 0.00046. Multiple Deliveries from One Pipette: The Triangular Distribution o We approximate the volumes of a large number of pipettes by the triangular distribution; there is highest probability that the pipette will deliver said amount. o The standard uncertainty/standard deviation in the triangular distribution is ± a/√6. o The uncertainty is a systematic error so you use n x (standard uncertainty in triangular distribution). o 0.006mL is the standard deviation measured for multiple deliveries of water. o
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Example: log(0.001237) = -2.9076 Example: antilog(4.37) = 2.3 x 10^4 Significant Figures and Graphs: when drawing a graph on a computer, consider whether the graph is meant to display qualitative behavior of the data or precise values that must be read with several significant figures (it should at least have tick marks on both sides of the horizontal and vertical scales).
Types of Error • Systematic Error/Determinate Error: a consistent error that can be detected and corrected; arises from a flaw in equipment, design of an experiment or from an uncalibrated burette. • Ways to detect systematic error: o Analyze a known sample and your method should reproduce the known answer. o Analyze blank samples containing no analyte being sought; if you observe a nonzero result, your method responds to more than you intend. o Use different analytical methods to measure the same quantity; if results do not agree, there is error in one or more of the methods. o Round-Robin Experiment: different people in several laboratories analyze identical samples by the same or different methods; disagreement beyond the estimated random error is systematic error. • Random Error/Indeterminate Error: cannot be eliminated, but it might be reduced by a better experiment; arises from uncontrolled variables in the measurement and has an equal chance of being positive or negative (different people reading scales = subjective). • Precision: reproducibility of a result; if you measure a quantity several times and the values agree closely with one another. • Accuracy: nearness to the truth; if a known standard is available, accuracy is how close your value is to the known value. • Certified Reference Materials: can be used to test the accuracy of your analytical procedures. • Absolute Uncertainty: the margin of uncertainty associated with a measurement; if the estimated uncertainty in reading a calibrated burette is ± 0.02mL, we say that ± 0.02mL is the absolute uncertainty associated with the reading. • Relative Uncertainty: compares the size of the absolute uncertainty with the size of its associated measurement. o Relative Uncertainty = absolute uncertainty/magnitude of measurement o If the burette reading is 12.35 ± 0.02mL, then the relative uncertainty is 0.02mL/12.35mL = 0.002. o Percent Relative Uncertainty = 100 x relative uncertainty. o The percent relative uncertainty of 0.002 is 0.2%. Propagation of Uncertainty from Random Error • Most propagation of uncertainty computations that will be encountered deals with random error, not systematic error. • Addition and Subtraction: use absolute uncertainty. o Uncertainty in Addition and Subtraction: e4 = √(e1^2 + e2^2 + e3^2) o Example: for uncertainties of 3.06 (± 0.03, ± 0.02 and ± 0.02), the total
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uncertainty is equal to √(0.03)^2 + (0.02)^2 + (0.02)^2 = ± 0.04 absolute uncertainty. o For its relative uncertainty, 0.04/3.06 x 100 = ± 1%. Multiplication and Division: use percent relative uncertainty through conversion. o Uncertainty in Multiplication and Division: %e4 = √(%e1^2 + %e2^2 + %e3^2) o Example: 1.76 (± 0.03) x 1.89 (± 0.02) / 0.59 (± 0.02) = 5.64 ± e4
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To calculate e4, we have to convert absolute uncertainties into percent relative uncertainties (0.03/1.76 = ± 1.7%, 0.02/1.89 = ± 1.1% and 0.02/0.59 = 3.4%). Then find the percent relative uncertainty of the answer (%e4 = √(1.7^2) + 1.1^2 + 3.4^2) = 4%). Round off to the number of significant digits only in the answer. To convert relative uncertainty into absolute uncertainty, 4% x 5.6 = 0.04 x 5.6 = 0.2. The answer is 5.6 ± 0.02. Mixed Operations: first work out the uncertainties for addition/subtraction (absolute uncertainty) then the uncertainties of multiplication/division (percent relative uncertainty). Real Rule for Significant Figures: the first uncertain figure is the last significant figure. o Example: 0.002364 (± 0.000003) / 0.02500 (± 0.00005) = 0.1066 (± 0.0002); the answer is properly express with three significant figures even though the original data has four figures. o The answer’s decimal place has to coincide with the uncertainty. o In multiplication/division, keep an extra digit when answer lies between 1 and 2. Exponents and Logarithms: use relative uncertainty and not percent relative uncertainty in calculations involving logx, lnx, 10^x and e^x but percent relative uncertainty for powers and roots. o Uncertainty for Powers and Roots: y = x^a %ey= a(%ex) o Uncertainty for Logarithm: if y is the base 1- logarithm of x, then y = logx ey = (1/ln10)(ex/x) = 0.43429(ex/x). o Natural Logarithm (ln): ln of x is the number y whose value is such that x = e^y. o Uncertainty for Natural Logarithm: y = lnx ey = ex/x o Uncertainty for Antilog (10^x): y = 10^x ey/y = (ln10)ex = 2.3026 ex o Uncertainty for e^x: y = e^x ey/y = ex
Propagation of Uncertainty from Systematic Error • Uncertainty in Atomic Mass: The Rectangular Distribution o Atomic mass from different sources approximates a rectangular distribution where there is approximately equal probability of finding any atomic mass in between the uncertain range. o The standard uncertainty/standard deviation in the rectangular distribution is ± a/√3. • Uncertainty in Molecular Mass o For systematic uncertainty, we add the uncertainties of each term in a sum or difference.
Propagation of Systematic Uncertainty: Uncertainty in Mass of n Identical Atoms n x (standard uncertainty in atomic mass) 2C: 2(12.0107 ± 0.00046) = 24.0214 ± 0.00092, where 0.00046 x 2 = 0.00092. n x (uncertainty listed in periodic table)/√3 atomic mass of C = 12.0107 ± 0.0008/√3 = 12.0107 ± 0.00046. Multiple Deliveries from One Pipette: The Triangular Distribution o We approximate the volumes of a large number of pipettes by the triangular distribution; there is highest probability that the pipette will deliver said amount. o The standard uncertainty/standard deviation in the triangular distribution is ± a/√6. o The uncertainty is a systematic error so you use n x (standard uncertainty in triangular distribution). o 0.006mL is the standard deviation measured for multiple deliveries of water. o
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