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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. • • A key feature of systematic error is that it is reproducible. Random error has an equal chance of being positive or negative. • 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 mL, we say that ±0.02 mL 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 0.000003 100 1.61045 102 0.002364 1.61045 10 2 2 0.00005 100 0.2 0.025 0.2 0.2 2 0.002 x 0.00946 = 0.00019 100 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/mL. To deliver 0.050 0 mol of HCl requires 4.18 mL 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 mL 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 mL solution leads to that 2% uncertainty. The arithmetic has the form a = b × c × d, for which %e2a = %e2b+%e2c+%e2d. If we know %ea, %ec, and %ed, we can find %eb by subtraction: %e2b = %e2a – %e2c – %e2d ) 0.050 0 (±2%) mol = Error analysis: