Error Propagation Analytical Chemistry
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Definition Of Error In Analytical Chemistry
password Expand/collapse global hierarchy Home Textbook Maps Analytical Chemistry Textbook Maps Map: Analytical Chemistry 2.0 (Harvey) 4: Evaluating Analytical sources of error in analytical chemistry Data Expand/collapse global location 4.3: Propagation of Uncertainty Last updated 10:52, 25 May 2016 Save as PDF Share Share Share Tweet Share 4.3.1 A Few Symbols4.3.2 Uncertainty When Adding or Subtracting4.3.3 error propagation formula Uncertainty When Multiplying or Dividing4.3.4 Uncertainty for Mixed Operations4.3.5 Uncertainty for Other Mathematical Functions4.3.6 Is Calculating Uncertainty Actually Useful?Contributors Suppose you dispense 20 mL of a reagent using the Class A 10-mL pipet whose calibration information is given in Table 4.9. If the volume and uncertainty for one use of the pipet is 9.992 ± 0.006 mL, what is the volume and
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uncertainty when we use the pipet twice? As a first guess, we might simply add together the volume and the maximum uncertainty for each delivery; thus \[\mathrm{(9.992\: mL + 9.992\: mL) ± (0.006\: mL + 0.006\: mL) = 19.984 ± 0.012\: mL}\] It is easy to appreciate that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that with each use the indeterminate error is in the same direction and is as large as possible. At the other extreme, we might assume that the uncertainty for one delivery is positive and the other is negative. If we subtract the maximum uncertainties for each delivery, \[\mathrm{(9.992\: mL + 9.992\: mL) ± (0.006\: mL - 0.006\: mL) = 19.984 ± 0.000\: mL}\] we clearly underestimate the total uncertainty. So what is the total uncertainty? From the previous discussion we know that the total uncertainty is greater than ±0.000 mL and less than ±0.012 mL. To estimate the cumulative effect of multiple uncertainties we use a mathematical technique known as the propagation of uncertainty
is important to work as accurately and precisely as possible. Therefore, almost all analytical, volumetric glassware shows the error that is made when using the glassware, such that you can calculate the size of the error in the experiment. An propagation of error physics example is given in the picture below, which shows a close-up of a 100 error propagation chemistry ib mL volumetric flask. The error that you make when using this flask is ±0.1 mL. In the remainder of this section,
How To Calculate Uncertainty Chemistry
we will learn what this actually means and how it influences a final experimental result. (Source: Wikipedia) Question: is this a random or systematic error? More on volumetric glassware The error displayed on volumetric glassware http://chem.libretexts.org/Textbook_Maps/Analytical_Chemistry_Textbook_Maps/Map%3A_Analytical_Chemistry_2.0_(Harvey)/04_Evaluating_Analytical_Data/4.3%3A_Propagation_of_Uncertainty is the random error resulting from the production process. In the case of the volumetric flask above, this would mean that a collection of identical flasks together has an error of ±0.1 mL (in other words: the standard deviation is 0.1 mL). However, individual flasks from the collection may have an error of +0.05 mL or -0.07 mL (Question: are these systematic or random errors?). For accurate results, you http://webchem.science.ru.nl/chemical-analysis/error-propagation/ should constantly use different glassware such that errors cancel out. A second option is to calibrate the glassware: determine the volume by weighing. The error after calibration should be much smaller than the error shown on the glassware. Moreover, this error has now become random instead of systematic! Since this requires a lot of work each time you want to use volumetric glassware, we will from now on assume that errors shown on volumetric glassware are random errors. For example, each time when using the depicted volumetric flask properly, the volume will be 100 mL with an error of ±0.1 mL. Significant figures As a general rule, the last reported figure of a result is the first with uncertainty. Assume that we have measured the weight of an object: 80 kg. To indicate that we are not sure of the last digit,we can write 80 ± 1 kg. If we would have used a better scale to weigh the object, we might have found 80.00 ± 0.01 kg. Question: is the second result more precise or more accurate than the first? We can also display the error in a relative way. For instance, 80 ± 1 kg is identical to 80 ± 1.25%. The order of mag
simple piece of laboratory equipment, for example a burette or a thermometer, one would expect the number of variables contributing to uncertainties in that measurement to be fewer than a measurement which is the result of a multi-step process consisting http://www.csudh.edu/oliver/che230/textbook/ch05.htm of two or more weight measurements, a titration and the use of a variety of reagents. It is important to be able to estimate the uncertainty in any measurement because not doing so leaves the investigator as ignorant as though there were no measurement at all. The phrase "not doing so" perpetuates the myth that somehow a person can make a measurement and not know anything about the variability of the measurement. That doesn't happen very error propagation often. A needle swings back and forth or a digital output shows a slight instability, so the investigator can estimate the uncertainty, but what if a gross error is made in judgment, leading one to estimate an unrealistic "safe" envelope of uncertainty in the measurement? Consider the anecdote offered by Richard Feynman about one of his experiences while working on the Manhattan Project during World War II. Although this example doesn't address the uncertainty of a particular of error in measurement it touches on problems which can arise when there is complete ignorance of parameter boundaries: Some of the special problems I had at Los Alamos were rather interesting. One thing had to do with the safety of the plant at Oak Ridge, Tennessee. Los Alamos was going to make the [atomic] bomb, but at Oak Ridge they were trying to separate the isotopes of uranium -- uranium 238 and uranium 235, the explosive one. They were just beginning to get infinitesimal amounts from an experimental thing [isotope separation] of 235, and at the same time they were practicing the chemistry. There was going to be a big plant, they were going to have vats of the stuff, and then they were going to take the purified stuff and repurify and get it ready for the next stage. (You have to purify it in several stages.) So they were practicing on the one hand, and they were just getting a little bit of U235 from one of the pieces of apparatus experimentally on the other hand. And they were trying to learn how to assay it, to determine how much uranium 235 there is in it. Though we would send them instructions, they never got it right. So finally Emil Segrè said that the only possible way to get it right was for him to go down ther