Sector-shaped centerpiece

So far the equations that have been developed are independent of the shape of the cell. In order to make Equation 17 more useful, it is necessary to relate Ci( — 0) to the initial concentration of component i, Coi. To do this one applies conservation of mass, which states that in a closed system (the solution column in the ultracentrifuge cell) the total amount of solute is constant at all times. Since mass equals concentration times volume, it is necessary to know the volume of the centerpiece, and this depends on the shape of the centerpiece. Two types of centerpieces are currently used. One is a sector-shaped centerpiece, and the other is a six-channel, equilibrium [or Yphantis (24)] centerpiece. Top views of both centerpieces are shown in Figure 1. In either centerpiece, one side is reserved for solvent and the other for solution. 

For the sector-shaped centerpiece the conservation of mass equation can be expressed as (for 6 in radians)... 

For sector-shaped centerpieces the substitution of Equation 17 into Equation 18a leads to... 

Equations 26 and 27 are the equations needed for obtaining MWD s when sector-shaped centerpieces are used. 

Evaluation of the Weight Fraction of Solute. Sector-Shaped Centerpieces. For sector-shaped centerpieces Scholte (13, 14) starts his analysis with Equation 27 which he writes as... 

Non-Sector-Shaped Centerpieces. Although Scholte (13, 14) set up his method originally for equations applicable to sector-shaped center-pieces, this restriction can be disposed of readily. For non-sector-shaped centerpieces start with Equation 31 written as... 

This set of equations can be solved for fa by linear programming methods since these equations meet the same criteria that were set up for the sector-shaped centerpiece case. Again one can obtain the fi for all four series once the series in Mt have been chosen. For data that are not too precise, Scholte suggests using three series in M instead of four series. Here 21/s is used for the second series, and 2% for the third series (13,14). 

These values can be used with the appropriate, conservation of mass equation to obtain (crm)id. Once this is known, the ideal concentration distribution can be obtained. For sector-shaped centerpieces the conservation of mass equation is written as... 

In order to use Donnelly s (11,12) method, we must convert the observed values of cr and d In c/d(r2) in nonideal solutions to ideal values. Then we can use the ideal value of d In c/d(r2) to calculate MWD s as has been described previously here and elsewhere (11, 12). One must also remember to use Equation 49 for non-sector-shaped centerpieces. 

How to Evaluate Bls The final item to dispose of is how to evaluate Bls- Methods for evaluating BLs using sector-shaped centerpieces will be outlined first. 

Sector-Shaped Centerpieces. Method 1. Do the Archibald experiment and calculate MWappft at rm and r6 using standard procedures (32, 34). Here MWapp/t is the value of the apparent weight-average molecular weight at rm or r6. Extrapolate these values of Mw pp)t to zero time to get MWapp° details for this extrapolation have been given by Fujita et al. (32). 

Thus BLs can be obtained from the limiting slope of a plot of 1/Mw pp° vs. Co. The Archibald method is restricted to sector-shaped centerpieces. 

Method 3. For sector-shaped centerpieces, Fujita showed that (30) if A was the same for each initial concentration c0, then... 

Non-Sector-Shaped Centerpieces. For non-sector-shaped center-pieces one is restricted at present to Method 2. Here the equation for Mw,pp(ce l) is given by... 

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

Equations 15 and 16 were first obtained by Kegeles and his co-workers (16, 20, 21) and they are valid only for cells with sector-shaped center-pieces. Because of convective disturbances in the solution column of the ultracentfifuge cell during the transient state when nonsector-shaped cells are used, it is customary to do Archibald experiments in cells with sectorshaped centerpieces which avoid this problem. For details on the Archibald method—pitfalls, extrapolation to zero time—one should consult the papers of LaBar (17) and Fujita et al. (22). 

The centerpiece is the central component of the cell assembly. It is shaped to minimize convection. The thickness of the centerpiece represents the thickness of the fluid column through which the light will pass. An aluminum centerpiece with a thickness of 1.2 cm and a 2V20 sector angle was used in this study. 

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