CHEMISTRY 59 320 ANALYTICAL CHEMISTRY Fall 2010 Lecture
- Slides: 18
CHEMISTRY 59 -320 ANALYTICAL CHEMISTRY Fall - 2010 Lecture 4
Chapter 3 Experimental error 3. 1 Significant Figures The minimum number of digits needed to write a given value in scientific notation without loss of accuracy A Review of Significant Figures How many significant figures in the following examples? • 0. 216 90. 7 800. 0 0. 0670 500 • 88. 5470578% • 88. 55% • 0. 4911
The needle in the figure appears to be at an absorbance value of 0. 234. We say that this number has three significant figures because the numbers 2 and 3 are completely certain and the number 4 is an estimate. The value might be read 0. 233 or 0. 235 by other people. The percent transmittance is near 58. 3. A reasonable estimate of uncertainty might be 58. 3 ± 0. 2. There are three significant figures in the number 58. 3.
3. 2 Significant figures in arithmetic • Addition and subtraction The number of significant figures in the answer may exceed or be less than that in the original data. It is limited by the leastcertain one. • Rounding: When the number is exactly halfway, round it to the nearest EVEN digit.
• Multiplication and division: is limited to the number of digits contained in the number with the fewest significant figures: • Logarithms and antilogarithms A logarithm is composed of a characteristic and a mantissa. The characteristic is the integer part and the mantissa is the decimal part. The number of digits in the mantissa should equal the number of significant figures.
• Problem 3 -5. Write each answer with the correct number of digits. • • (a) 1. 021 + 2. 69 = 3. 711 (b) 12. 3 − 1. 63 = 10. 67 (c) 4. 34 × 9. 2 = 39. 928 (d) 0. 060 2 ÷ (2. 113 × 104) = 2. 84903 × 10− 6 • (e) log(4. 218 × 1012) = ? • (f) antilog(− 3. 22) = ? • (g) 102. 384 = ? • • (a) 3. 71 (b) 10. 7 (c) 4. 0 × 101 (d) 2. 85 × 10− 6 (e) 12. 6251 (f) 6. 0 × 10− 4 (g) 242
3 -3 Types of errors • Every measurement has some uncertainty, which is called experimental error • Random error, also called indeterminate error, arises from the effects of uncontrolled (and maybe uncontrollable) variables in the measurement. • Systematic error, also called determinate error, arises from a flaw in equipment or the design of an experiment. It is always positive in some region and always negative in others. • Random error has an equal chance of being positive or negative. • A key feature of systematic error is that it is reproducible. • It is always present and cannot be corrected. It might be reduced by a better experiment. • In principle, systematic error can be discovered and corrected, although this may not be easy.
Accuracy and Precision: Is There a Difference? • Accuracy: degree of agreement between measured value and the true value. • Absolute true value is seldom known • Realistic Definition: degree of agreement between measured value and accepted true value.
Precision • Precision: degree of agreement between replicate measurements of same quantity. • Repeatability of a result • Standard Deviation • Coefficient of Variation • Range of Data • Confidence Interval about Mean Value
You can’t have accuracy without good precision. But a precise result can have a determinate or systematic error. Illustration of Accuracy and precision.
Absolute and relative uncertainty: • Absolute uncertainty expresses the margin of uncertainty associated with a measurement. If the estimated uncertainty in reading a calibrated buret is ± 0. 02 m. L, we say that ± 0. 02 m. L is the absolute uncertainty associated with the reading.
3 -4 Propagation of Uncertainty from Random Error • Addition and subtraction:
• Multiplication and Division: first convert all uncertainties into percent relative uncertainties, then calculate the error of the product or quotient as follows:
The rule for significant figures: The first digit of the absolute uncertainty is the last significant digit in the answer. For example, in the quotient 100 0. 002 x 0. 00946 = 0. 00019 0. 002
3 -5 Propagation of uncertainty: Systematic error • It is calculated as the sum of the uncertainty of each term • For example: the calculation of oxygen molecular mass.
3 -C. We have a 37. 0 (± 0. 5) wt% HCl solution with a density of 1. 18 (± 0. 01) g/m. L. To deliver 0. 050 0 mol of HCl requires 4. 18 m. L of solution. If the uncertainty that can be tolerated in 0. 050 0 mol is ± 2%, how big can the absolute uncertainty in 4. 18 m. L be? (Caution: In this problem, you have to work backward). You would normally compute the uncertainty in mol HCl from the uncertainty in volume: But, in this case, we know the uncertainty in mol HCl (2%) and we need to find what uncertainty in m. L solution leads to that 2% uncertainty. The arithmetic has the form a = b × c × d, for which %e 2 a = %e 2 b+%e 2 c+%e 2 d. If we know %ea, %ec, and %ed, we can find %eb by subtraction: %e 2 b = %e 2 a – %e 2 c – %e 2 d )
0. 050 0 (± 2%) mol = Error analysis:
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