참값! 너 어디있니? Chaper 3 Experimental Error 행복 = 참값 ‘99. . 4 2 참값! 너 어디있니? 행복 = 참값 No way to measure the “ture value” of anything Yonsei Univ.

유효숫자 (Significant figures): Minimum number of digits required to express a value in scientific notation without loss of accuracy 92 500 X 적절한 표기법이 아님 9.25 x 104 3 유효숫자 9.250 x 104 4 유효숫자 9.250 0 x 104 5 유효숫자 * 지수 표기법이 가장 정당함 Yonsei Univ.

Rules of Significant Figures Nonzero digits are always significant 457 cm  3 sig. figs. Zeros between nonzero digits are always significant. 1005 kg  4 sig. figs. Zeros at the beginning of a number are never significant, they merely indicate the position of the decimal point. 0.02 g  1 sig. fig. Zeros that fall both at the end of a number and after the decimal point are always significant 0.0200 g  3 sig. fig. When a number ends in zeros but contains no decimal point, the zeros may or may not be significant. 130 cm  2 or 3 sig. figs. To remove this ambiguity, we can write in scientific notation 1.30 x 102 -> 3 sig. figs. 1.3 x 102 -> 2 sig. figs. Exact numbers are treated as if they have an infinite number of sig. Figs. What’s an exact number? Measurements are never exact!

Q? : How many digits in the answer? 3-2 연산과 유효숫자 Q? : How many digits in the answer? 덧셈과 뺄셈 Rule 1: Fewest decimal Places Ex 1) Molecular wt of KrF2 18.998 403 2 (F) +18.998 403 2 (F) +83.80 121.796 806 4 분자량 유효숫자 inside cover 121.80

Significant figures in arithmetic Addition and subtraction Keep number of decimal places in your answer the same as the number of decimal places in the number with the fewest. 1.2 + 1.41 = 2.6 Multiplication and division The number of sig. figs. is limited by the number of digits contained in the number with the fewest sig. figs. 30.26 x 1.5 = 45  4.5 x 101 Logarithms and Antilogarithms The number of digits in the mantissa should be equal to the number of sig figs in the number of which the log is being taken. The number of sig figs in an antilog should be equal to the number of digits in the mantissa.

Significant figures and graphs 3-3 유효숫자와 그래프 When a plot is being used to show the quantitative behavior of data, the rulings on a sheet of graph paper should be compatible with the number of sig. figs. of the coordinates. Correct Incorrect

Error and Uncertainty Error – difference between your answer and the ‘true’ one. Systematic Problem with a method All errors are of the same size, magnitude and direction Determinate errors. Random based on limits and precision of a measurement Indeterminate errors. Can be treated statistically. Blunders You screw up. Best to just repeat the work…

Determinate Errors Potential instrumental errors Variations in temperature Contamination of the equipment Power fluctuations Component failure All of these can be corrected by calibration or proper instrument maintenance.

Determinate Errors Method errors Slow or incomplete reactions Unstable species Nonspecific reagents Side reactions These can be corrected with proper method development.

Determinate Errors Personal errors Misreading of an instrument or scale Improper calibration Poor technique/sample preparation Personal bias Improper calculation of results These can be minimized or eliminated with proper training and experience.

Random Errors Each value you report is actually the result of many different factors and variables. Many of these variables are beyond your control and are random in nature. Example – Reading an electronic balance A sample that is reported as 1.0023 g actually weighs somewhere in the range of 1.0022 – 1.0024 g. You don’t know exactly what it is. The last digit is your most uncertain digit.

A simple example Calibration of a 100 mL pipette Steps in the procedure: Weigh a clean, dry container. Fill the pipette to the mark with water. Drain water into the container. Re-weigh the container. Determine volume based on weight and density of water.

A simple example Source of error Weight of the container. Every balance has a limit so you know that for an analytical balance you have minimum error of ±0.1 mg. But, this should be small compared to the other weight we will be measuring. Fill the pipette to the mark with water. How accurately did YOU judge the water level in the pipette?

A simple example Source of error Drain water into the container. How well did it drain? The angle of the pipette and the viscosity of the liquid will cause small variations. Re-weigh the container. Here we have another error introduce because of the limit of the balance. Now the error is associated with both the water and the container.

A simple example Source of error Determine volume based on weight and density of water. The temperature will alter the density of the water. (How accurate is your temperature measurement?) There’s also limits to how well the density is know at any given temperature and will vary based on the purity of the water.

A simple example For this simple procedure, we have a minimum of five sources of error. Each can occur as a + or – error. Our total error becomes: ET = ± Ewt1 ± Evol ± Edrain ± Ewt2 ± Edensity And this was for a simple example!

A simple example (The solution?) Identifying sources of error can help you reduce some sources. But, you can never eliminate all source of error. Most sources will be random in nature. Therefore, we must rely on statistical treatment of our data to account for these errors.

정확도(Accuracy): 측정값의 참값에 대한 근접도 정확도(Accuracy) & 정밀도(Precision) 정확도(Accuracy): 측정값의 참값에 대한 근접도 정밀도 : X 정확도: O

정밀도(Precision): 측정값 간의 근접도 정밀도 : O 정확도: X

정밀.정확 정밀도 : O 정확도: O

Uncertainty Absolute uncertainty The margin of uncertainty associated with a measurement. An uncertainty of ±0.02 means that, when the reading is 13.33, the true value could be anywhere in the range 13.31 to 13.35. What’s the uncertainty in a buret reading?

The Vernier Scale A Vernier scale is a small, moveable scale placed next to the main scale of a measuring instrument It allows measurements to a precision of a small fraction of the smallest division on the main scale. 1.18 0.71

3-5 불확도의 전파 덧셈과 뺄셈 교재: 수식 (3-5) Ex) 1.76 (±0.03) +1.89 (±0.02) - 0.59 (±0.02) 3.06 (± ? ) Must be 0.03 < x <0.07 교재: 수식 (3-5)

Uncertainty Relative uncertainty Compares the size of absolute uncertainty with the size of its associated measurement. The percent relative uncertainty is simply the relative uncertainty multiplied by 100. What’s the percent relative uncertainty in reading a buret for a volume of 10 mL.

The Real Rule for Significant Figures The first uncertain figure of the answer is the last significant figure 0.09459 ± 0.0002  0.0946 ± 0.0002 the uncertainty occurs in the fourth decimal place so we round to the fourth decimal place. 0.1066 ± 0.1  0.1 ± 0.1 the uncertainty occurs in the first decimal place so we round to the first decimal place. Always carry at least one insignificant figure through all calculations to avoid rounding errors. If you have a result that might be used in additional calculations then report it with the first insignificant figure. For example: 1.23(5) where (5) is insignificant

Propagation of uncertainty If possible, uncertainty is expressed as the standard deviation or confidence interval. These parameters are based on a series of replicate measurements. These terms will be discussed in chapter 4. But, we don’t always have replicate measurements and it is then necessary to perform arithmetic operations on several numbers, each of which has an associated random error. Not simply a sum of the individual errors because some will be negative and some positive. There will be some cancellation of error. We can usually estimate the random error associated with a measurement. (i.e.. ±0.02 mL for each buret measurement, ±0.1 mg for each mass measurement)

Propagation of uncertainty Addition and subtraction For addition and subtraction use absolute uncertainty The volume deliverd by a buret is the difference between the final reading and the initial reading. If the uncertainty in each reading is ±0.02 mL, what is the uncertainty in the volume delivered? Suppose the initial reading is 0.05 mL and the final reading is 17.88 mL.

Propagation of uncertainty Multiplication and division For multiplication and division use percent relative uncertainty

Propagation of uncertainty Mixed Operations First deal with addition and subtraction using absolute uncertainties. Then deal with multiplication and division using relative uncertainties. And remember, the first uncertain figure of the answer is the last significant figure.

Propagation of uncertainty Exponents and Logarithms

Propagation of uncertainty Summary of rules for propagation of uncertainty (Table 3-1 in text)

Propagation of uncertainty (Final Words) But what if we have the time to take repetitive measurements? Then we don’t have to worry about propagation of uncertainty and instead can use statistics to determine the uncertainty in our measurement.

DONE WITH CHAPTER 3