Experimental Errors in Measured Quantities

The probable random error in the mean of a set of repeated measurements can be determined statistically. 

When a variable is calculated by substitution of measured quantities into a formula, the estimated errors in the measured quantities can be propagated through the calculation. 

Another t5q)e of data reduction involves fitting a set of data to a formula. This can be done graphically or numerically by use of the least-squares (regression) procedure. 

identify probable sources of error in a physical chemistry experiment and classify the errors as systematic or random  

calculate the mean and standard deviation of a sample of numbers  

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The plots of the ionic radii, molar volume of the octahydrated sulphates, the basicity of the hydroxides, the predicted ionization energy and the hydration energies by Klemm (10) for the lanthanides showed (10,12) so weak a discontinuity near gadolinium that it raised the question whether or not the experimental errors in measuring these quantities are showing up as the break. To add some weight, Moeller and Kremers (15) in their review on the basic characteristics of Sc, Y and the lanthanides remarked (p. 119) ... 

An investigation of the possible experimental errors in measuring Ak/k has been started as well as measurements of other quantities which must be calculated correctly if one expects to calculate the reactivity of small samples. The first results are determinations of the ratio of for and the fissile materials in recent ZPPR or ZPR-3 cores. 

The determination of the chemical reaction rate is based on measurements of the concentrations, temperature and flow rates. Experimental errors in the measurements of these quantities are inevitable. 

The refraction of molecules may be calculated by the addition of the appropriate atomic refractions. It is necessary to add to the sum so obtained additional quantities corresponding to the multiplicity of the bonds involved. For a large number of molecules, the calculated and measured values of the molecular refraction agree, within the limits of experimental error. In molecules containing a conjugated system of double bonds, however, deviations from the additivity rule occur, the experimental value always being greater than the calculated. The difference, may... 

The experimental difficulties in measuring reaction rates are only too well known. Since at least one quantity for (4), namely 2o> derived from the experiments, the values of the rate at at least two temperatures must be known. If these temperatures are too close, a relatively small error in the measurement of the rate may lead to an incorrect determination of 00 a very great inaccuracy in the remaining factors. 

The discussed calculation procedure is not based on any extrathermodynamic assumptions and therefore the inaccuracy of the result obtained is determined only by the experimental errors of measuring a work function and the Volta potential difference. Furthermore, from the solvent surface potential x determined by any estimation method we can find the ideal solvent-electron interaction energy Vq = U — ex . Unlike U , V, is not a strictly thermodynamic quantity and the inaccuracy in determining it, besides experimental errors, is caused by the inaccuracy of model assumptions made for estimating x -... 

We also discuss the analysis of the accuracy of experimental data. In the case that we can directly measure some desired quantity, we need to estimate the accuracy of the measurement. If data reduction must be carried out, we must study the propagation of errors in measurements through the data reduction process. The two principal types of experimental errors, random errors and systematic errors, are discussed separately. Random errors are subject to statistical analysis, and we discuss this analysis. 

Many values that are obtained by measurement in a physical chemistry laboratory are used along with other values to calculate some quantity that is not directly measured. Such a calculation is call data reduction. An experimental error in a measured quantity will affect the accuracy of any quantity that is calculated from it. This is called propagation of errors. 

Here and elsewhere these standard deviations (S.D.) include contributions from both experimental error in the measured quantities and variability in the parameters which are assumed to be constant. 

Thus the quantity on the left evaluated for a series of polymer fractions differing only in chain length should be independent of M. Results shown in Table XLII for fractions of poly-(methyl methacry-late) and of polyisobutylene covering unusually wide ranges confirm this prediction within experimental error. It is borne out also by less extensive results of sedimentation measurements on several other systems. Introduction of the values of v, p, and rjo enables... 

The author gives an exampie of a study concerning a mixture of ethanol, toluene and ethyl acetate. The case is presented in the form of a Scheffe plan for which choice of compound quantities are not optimised to obtain a good matrix as shown in the matrix of effects correiation there is no point repetition in the middle of the matrix, which thus exciudes the quantification of the level of error of measurement that can only be estimated by the residual standard deviation of the regression. Finaliy, the author uses flashpoints of pure substances from partial experimental data. The available data give 9 to IS C for ethanol (the author 12.8), 2 to 9°C for toluene (5.56) and -4 to -2°C for ethyl acetate. 

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

In science, the word error has a very specific meaning it does not mean mistake . All experimental measurements will differ, to some degree, from the accurate or real value of the quantity being measured. The difference between the observed value of a physical quantity and the accurate value is called the error. It is very important to consider all the possible sources of errors in experimental measurements. Every experimental measurement reported should be accompanied by an estimate of the error - scientifically speaking, measurements without accompanying error estimates are worthless. 

When measuring KIE using isotope analysis for the product (Equation 7.17) three quantities are determined experimentally fL, Ros and Rp. For measurements of KIE using substrate (reactant) analysis (Equation 7.18) the corresponding quantities are fL, Ros and Rs. All these measurements, of course, are subject to experimental error. Equation 7.25 expresses the relative error of KIE in terms of the errors in these three experimental quantities ... 

Chapter 9 deals with the general principles of computational thermodynamics, which includes a discussion of how Gibbs energy minimisation can be practically achieved and various ways of presenting the output. Optimisation and, in particular, optimiser codes, such as the Lukas progranune and PARROT, are discussed. The essential aim of these codes is to reduce the statistical error between calculated phase equilibria, thermodynamic properties and the equivalent experimentally measured quantities. 

Other than the operations that emphasize certain aspects of the data, there are destructive influences that reduce our knowledge of the features of the data under investigation. These influences introduce deviations from the true values of the quantity being measured called experimental error, or simply error. Many of these degrading operations are selective, in that their... 

The remaining errors in the data are usually described as random, their properties ultimately attributable to the nature of our physical world. Random errors do not lend themselves easily to quantitative correction. However, certain aspects of random error exhibit a consistency of behavior in repeated trials under the same experimental conditions, which allows more probable values of the data elements to be obtained by averaging processes. The behavior of random phenomena is common to all experimental data and has given rise to the well-known branch of mathematical analysis known as statistics. Statistical quantities, unfortunately, cannot be assigned definite values. They can only be discussed in terms of probabilities. Because (random) uncertainties exist in all experimentally measured quantities, a restoration with all the possible constraints applied cannot yield an exact solution. The best that may be obtained in practice is the solution that is most probable. Actually, whether an error is classified as systematic or random depends on the extent of our knowledge of the data and the influences on them. All unaccounted errors are generally classified as part of the random component. Further knowledge determines many errors to be systematic that were previously classified as random. 

The Bashforth-Adams tables provide an alternate way of evaluating 7 by observing the profile of a sessile drop of the liquid under investigation. If, after all, the drop profiles of Figure 6.15 can be drawn using 0 as a parameter, then it should also be possible to match an experimental drop profile with the (3 value that characterizes it. Equation (85) then relates 7 to 0 and other measurable quantities. This method is claimed to have an error of only 0.1%, but it is slow and tedious and hence not often the method of choice in practice. 

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