CHEMISTRY ANALYTICAL CHEMISTRY Fall 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? % 88.55%

The needle in the figure appears to be at an absorbance value of 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 or by other people. The percent transmittance is near 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 least- certain 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) = (b) 12.3 − 1.63 = (c) 4.34 × 9.2 = (d) ÷ (2.113 × 10 4 ) = × 10 −6 (e) log(4.218 × ) = ? (f) antilog(−3.22) = ? (g) = ? (a) 3.71 (b) 10.7 (c) 4.0 × 10 1 (d) 2.85 × 10 −6 (e) (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. 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. 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. 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

Illustration of Accuracy and precision. You can’t have accuracy without good precision. But a precise result can have a determinate or systematic error. You can’t have accuracy without good precision. But a precise result can have a determinate or systematic error.

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 x =

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 mol of HCl requires 4.18 mL of solution. If the uncertainty that can be tolerated in 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 %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 )

Error analysis: (±2%) mol =