Concentration measurement error

Notice that the calculated k values differ from one another and range from 0.816 to 0.954 this is because of the concentration measurement error associated with each value. To reduce these results to one manageable value, it is reasonable to take the average and standard deviation of these four values.6 The resulting values are (0.893 0.069) year-1, which converts to a resident time of 410 days, which is virtually the same as the value we calculated previously for the green dye in this lake. 

The main sources of error which define the accuracy are counting statistics in tracer concentration measurements, the dispersion of the tracer cloud in the flare gas stream, and the stationarity of the flow during measurements. 

The relative measurement error in concentration, therefore, is determined by the magnitude of the error in measuring the cell s potential and by the charge of the analyte. Representative values are shown in Table 11.7 for ions with charges of+1 and +2, at a temperature of 25 °C. Accuracies of 1-5% for monovalent ions and 2-10% for divalent ions are typical. Although equation 11.22 was developed for membrane electrodes, it also applies to metallic electrodes of the first and second kind when z is replaced by n. 

Relationship Between Measurement Error in Potential and Relative Error in Concentration... 

Phase Doppler particle analyzers are essentially single-particle counters because they measure one particle at a time within a small sampling volume. This volume must be kept small to minimize the probabiUty of having more than one droplet in the volume at any given instant. This probabiUty increases as the concentration of droplets becomes greater, and there is more risk of measurement errors. 

There are two contradictory requirements here. The first is to keep the difference between Ci and C as small as possible so that it can be neglected. The second is to analyze these two only very slightly different concentrations with such precision that the difference will be significantly greater than the measurement error. This second need is for calculation of the rate of reaction, as shown in the first equation of this section. 

Typical approaches for measuring diffusivities in immobilised cell systems include bead methods, diffusion chambers and holographic laser interferometry. These methods can be applied to various support materials, but they are time consuming, making it onerous to measure effective dififusivity (Deff) over a wide range of cell fractions. Owing to the mathematical models involved, the deconvolution of diffusivities can be very sensitive to errors in concentration measurements. There are mathematical correlations developed to predict DeS as... 

Z is a coefficient which relates the concentration of the analyte in the unknown sample to the concentration in the calibration standard, where = bc. R is a residual matrix which contains the measurement error. Its rows represent null spectra. However, in the presence of other (interfering) compounds, the residual matrix R is not random, but contains structure. Therefore the rank of R is greater than zero. A PCA of R, after retaining the significant PCs, gives ... 

FIG. 6 Dependence of the square root of the SHG intensity ( I(2a>)) for membrane 2 without KTpCIPB (a) with KTpCIPB (b) on K+ ion concentrations in the adjacent aqueous solution containing KCl (O) and KSCN ( ), respectively. Inset The corresponding observed EMF to KCl and KSCN. The concentrations of ionophore 2 and KTpCIPB were 3.0 X 10 M and 1.0 x 10 M, respectively for both SHG and EMF measurements. The data points present averages for three sets of measurements. Error bars show standard deviations. (From Ref. 15.)... 

The major sources of error are the concentration measurements and attempts to improve the accuracy of the rate constant should focus on trying to improve the accuracy of the con-... 

Figure 3 shows the PCA score plot of the same data of figure 2 after the application of equation 4. The application of linear normalization to an array of linear sensors should produce, on the PCA score plot, one point for each compound, independent of its concentration, and achieve the highest possible recognition. Deviations from ideal behaviour, as shown in figure 3, are due to the presence of measurement errors, and to the non-linear relationship between sensor response and concentration. 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

Military and ammunition sites contaminated with explosives can cover substantial areas (Gerth et al. 2005). Soil contamination in these sites is often heterogeneous. Explosives are relatively non-volatile, and have low aqueous solubility. Sampling from sites within a few decimetres of one another can result in concentration differences of up to one hundredfold (Jenkins et al. 1996). For example, the coefficients of variation across samples taken from 11 abandoned sites in the USA were 248% for TNT and 137% for Hexogen (Crockett et al. 1998). As a result, sampling error greatly exceeds measurement error. Thus to obtain representative results... 

In the calibration problem two related quantities, X and Y, are investigated where Y, the response variable, is relatively easy to measure while X, the amount or concentration variable, is relatively difficult to measure in terms of cost or effort Furthermore, the measurement error for X is small compared with that of Y The experimenter observes a calibration set of N pairs of values (x, y ), i l,...,N, of the quantities X and Y, x being the known standard amount or concentration values and y the chromatographic response from the known standard The calibration graph is determined from this set of calibration samples using regression techniques Additional values of the dependent variable Y, say y., j l,, M, where M is arbitrary, are also observed whose corresponding X values, x. are the unknown quantities of interest The statistical literature on the calibration problem considers the estimation of these unknown values, x, from the observed and the... 

Figure 5 illustrates the cases in which the data are represented by a point (A 0), line (A=l) or plane (A=2). A is the number of product terms in Equation 2. Samples clustered in a point represent replicate analyses of a single sample in which there is no variation other than measurement error, and the product term in Equation (2) is 0. In these last two situations, the data vary about the mean, m, and the position of each object on the line or plane given by the peak coordinates. An example of data that would form a line are those based on an analysis of a range of concentrations of a single Aroclor (A=l). Data that could be represented in a plane result from the analysis of the fractional composition of two (or more) Aroclor mixtures (A=2). In Fiqure 5, designates the class number of the these hypothetical samples. 

Sample variabilities and the measurement error must be considered (risk analysis, cf. Section I.C) to avoid that an analyte signal yo will be measured outside the calibrated range. Thus, the range shall be chosen a little wider than the expected range of analyte concentrations. 

A diagram of a typical gas-phase (ozone-ethylene) chemiluminescent ozone analyzer is shown in Figure 6-10. The detector responds linearly to ozone concentrations between 0.003 and 30 ppm no interferences were initially observed. More recently, however, it has been established that, as the relative humidity goes from 0 to 60% and the temperature from 20° to 25° C, water vapor produces a small positive signal that results in an increase of about 8% in the ozone concentration measurement. This potential source of error can be minimized by using humidified, rather than dry, ozone in air streams when calibrating. 

One sees that the ion flow caused by a gas is proportional to the partial pressure. The linear equation system can be solved only for the special instance where m = g (square matrix) it is over-identified for m> g. Due to unavoidable measurement error (noise, etc.) there is no set of overall ion flow Ig (partial pressures or concentrations) which satisfies the equation system exactly. Among all the conceivable solutions it is now necessary to identify set 1 which after inverse calculation to the partial ion flows 1, will exhibit the smallest squared deviation from the partial ion currents i actually measured. Thus ... 

Designed Experiments Produce More Precise Models. In the context of linear regression, this is demonstrated by examining the statistical uncertainties of the regression coefficients. Equation 2.1 is the regression model where the response for the th sample (r ) of an instrument is shown as a linear function of the sample concentration (c.) with measurement error... 

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