Plotting with Excel curve fitting

Curve jStting deals with finding an equation that best fits a set of data. There are a number of techniques that you can use to determine these functions. You will learn about them in your numerical methods and other future engjneerii classes. The purpose of this section is to demonstrate how to use Excel to find an equation that best fits a set of data which you have plotted. We will demonstrate the curve-fitting capabilities of Excel using the following example. 

Over the entire Q-range within experimental error the data points fall on the line and thus exhibit the predicted Q4 dependence. The insert in Fig. 7 demonstrates the scaling behavior of the experimental spectra which, according to the Rouse model, are required to collapse to one master curve if they are plotted in terms of the Rouse variable u = QV2 /wt. The solid line displays the result of a joint fit to the Rouse structure factor with the only parameter fit being the Rouse rate W 4. Excellent agreement with the theoretical prediction is observed. The resulting value is W/4 = 2.0 + 0.1 x 1013 A4s 1. 

The pH dependence of could be due to changes in A-B loop disorder rates, perhaps the chemical exchange phenomenon observed for NPl-ImH (Section ll,E,2,b), or to changes in ligand bond strength. The change in lies in the off-rates (Tables I-Ill) consistent with the loop disorder model. Plots of vs pH display an excellent fit with the equation for a titration curve (Fig. 21), indicating that the transition... 

The best fittings correspond to the following parameters k = 0.9, a = 0.5, and QP = 0.135 pC. In Fig. 7.49d, the experimental corrected charge-potential curves for this system have been plotted, along with the theoretical faradaic ones, calculated using Eqs. (7.116) and (7.138) and showing excellent agreement. 

Thereafter, the experimental data were fitted to the above model equation by a graphical superposition technique. The data and model curve were plotted separately as fraction adsorbed or desorbed against the log of the square root of time. The experimental curve was moved horizonally until the best fit was obtained, thereby determining the appropriate value at Dc/a. This method uses all of the data, as opposed to some approaches based on the values at early times which have been used by others (1, 2, 5). It was applied easily in this work because of the excellent agreement of the model with the data obtained. 

How well do the sedimentation coefficients and densities predicted by the model match the values actually observed for LDL Excellent agreement with the experimental points is shown by the solid curve of Fig. 2, which is a plot of the values for 525,1.20 given in Table II. However, this agreement was achieved by selecting a value for the partial specific volume of the cholesteryl esters to make the best fit, yielding the value of 1.058 ml/g for this this quantity. [If a value of 1.044 ml/g were employed for the partial specific volume of the cholesteryl esters, as was used by Sata et al. (1972), the values of 525,1.20 listed in Table II would have decreased by about 3,5%. The values of S[ in Table II would have dropped by 1 to 2 Svedbergs.]... 

Equation [8.89] is linear with respect to parameters 4> and 0, but nonlinear with respect to q, and therefore the data must be fitted to this calibration function using nonlinear least-squares regression (Section 8.3.8) it is emphasized that it is very important to ensure that the initial estimates for the unkown parameters should be reasonably close to the final best estimates (see the text box dealing with nonlinear regression). In the present example (Equation [8.89]) excellent initial estimates can be obtained experimentally (see below) but if this is not possible tricks can be employed to obtain reasonable first estimates. One way is to plot the experimental data for Ra VRsis s Qa VQsis" nd draw an approximate curve though the points by hand. Experimental data expected to be well represented by Equation [8.89] should extrapolate to a value of (Ra /Rsis ) = 0 as (Qa"/Qsis ) zero, and to (Ra VRsis") = as (Qa"/Qsis") becomes... 

Semi-quantitative analysis facilitates fast and simple multi-element measurements with limited precision. ICP-MS offers excellent semi-quantitative capabilities, as a result of the high ionisation efficiencies achieved for the majority of elements and the simplicity of the resulting mass spectra. Semi-quantitative determinations are mostly based on a comparison of response tables and the actual count rates of the sample. The response (intensity I in counts/s) of an analyte ion depends on the concentration of the analyte element, the isotopic abundance of the observed isotope, the ionisation efficiency, the atomic mass and the efficiencies of nebuUsation, ion transmission and ion detection in the mass spectrometer. In most ICP-MS instraments a plot of the atomic response, Ra, versus atomic mass yields a smooth response curve, which is fitted best by a third order polynomial (Figure 4.3). The atomic response is defined by equation (4.7) and is equivalent to the molar response divided by the Avogadro constant, IVa-... 

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