## Sedimentation Equilibrium

As in the case of sedimentation velocity, methods for analysis of sedimentation equilibrium data can be divided into model-independent and model-dependent approaches. Model-independent methods are most useful to survey sample properties qualitatively, or for analysis of complex samples, i.e. polymeric mixtures, that cannot easily be described in a model-dependent analysis. In contrast, model-dependent analysis involves direct fitting of the sedimentation equilibrium concentration gradients to relevant physical models (e.g. single ideal species, noninteracting mixtures or a reversible association). This method provides the best-fit values and the associated statistical uncertainties in the fitting parameters (e.g. molecular mass, oligomer stoichiometry and association constants) and a statistical basis to discriminate among alternative physical models.

The simplest model-independent approach to obtain the molecular mass, M, is to plot ln c versus r2. According to eqn [5], the slope of this line is equal to M(1 — vp)rn2/2RT. Although linearity of this plot has been taken as evidence that a sample contains a single ideal species, this method can be quite insensitive to heterogeneity, particularly if the concentration gradient is shallow. Additionally, d(ln c)/dr2 can be calculated on a point-by-point basis to create a plot of the apparent weight-average molecular weight (Mw,app) versus concentration. For a homogeneous sample, Mw,app will be constant as a function of concentration. An increase in Mw,app with concentration indicates mass action-driven association. In this case, it is useful to overlay on the same plot data obtained from several samples over a range of loading concentrations and/or rotor speeds. For a reversibly self-associating system, all of the data will lie on a smooth curve, whereas for a noninteracting or slowly equilibrating system, the data will give rise to a family of nonsuperimposable curves. Other molecular weight averages (Mn, Mz) can also be obtained and can be useful in the analysis of associating systems or polymeric mixtures.

In model-dependent methods, a single experiment concentration gradient, or preferably, multiple concentration gradients, are fit to a physically relevant model using a nonlinear least-squares algorithm. In the simplest case of a single ideal species, data are fit to eqn [5]. For samples where there are more than one species in solution, or if thermodynamic nonideality is appreciable, it is necessary to fit the data to functions containing additional terms to incorporate sample heterogeneity, equilibrium association reactions or virial coefficients. Often it is difficult to distinguish between several models that fit the data equally well. In these cases, it is often useful to employ global methods in which multiple data sets that are collected over a wide range of sample loading concentrations and rotor speeds are simultaneously fit to a specific model. This global fitting approach helps to ensure that a unique solution is obtained and greatly reduces the statistical uncertainty in the parameters. Global nonlinear least-squares fitting of sedimentation equilibrium data was originally implemented in the NONLIN algorithm, and now several programs are available. In addition, equilibrium data are often fit using models programmed by the user within a general-purpose data analysis package.

## Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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