Kraemer equation

The Kraemer equation is another logarithmic equivalent of the Huggins equation ... 

Table 7.2 presents the equations for the extrapolation of the viscosity data to obtain the intrinsic viscosity. It is common to use the Huggins and Kraemer equations to determine the intrinsic viscosity. 

Both the Huggins and Kraemer equations lead to the same extrapolated value of [rj], so it has become common practice to use both equations on the same plot, as shown schematically in Figure 12-27. This, combined with the imposition of the requirement that k" - k - 1/2 increases confidence in the extrapolation, but you should be warned that not all data looks as good as that shown here. 

The constants of the two equations are connected by the relationships kfo = k - 0.5. Given that, for flexible polymers, the k values vary between 0.3 and 0.5, the slope of Kraemer equation is generally negative, with absolute values lower than the Huggins slope. This helps in the extrapolation procedure which is conveniently made, in order to reduce experimental uncertainties, by plotting in the same graph the viscosity data according to Eqs. (5) and (6), as shown in the example of Fig. 1. 

Fig. 1 Double extrapolation of viscometric data according to the Huggins and Kraemer equations reduced viscosity inherent viscosity.

The second result was obtained from expansion of the logarithm [ln(l +x + kux ) = x + ku - l/2).x for small x]. Dividing this last result by c gives the second extrapolation form, known as the Kraemer equation, having the same intercept but a different slope ... 

Figure 1.24 shows experimental determinations of the intrinsic viscosity. When both the Huggins and Kraemer equations provide the same intrinsic viscosity and Huggins coefficient, the higher-order terms in these equations can be safely ignored. On the other hand, if the Huggins and Kraemer plots are curved and do not give the same intercept, viscosity measurements need... 

Useful relationships are derived from these equations by working first with the Kraemer equation, rearranging and substituting from the Mark-Houwink equation, KMy for [r]], gives Equation (5.3),... 

Combining the Huggins and Kraemer equations and substituting for [rj] gives the relationship for thickening a specific base oil to a specific blend target, Equation (5.4),... 

ATs is a very complex constant, containing the constants for polymer interaction, and kf, from both the Huggins and Kraemer equations, K from the Mark-Houwink equation, as well as both the base oil and blend viscosities. This log-log relationship is verified for polymethacrylates. Fig. 5.3. 

Kraemer equation For dilute polymer solutions, an equation relating the inherent viscosity to intrinsic viscosity and concentration. It is... 

The Kraemer equation as well as the Solomon-Ciuta equation are strictly speaking only valid for a Huggins constant K =II5 [21], Thus in case of doubt a linear fit should be obtained initially from the Huggins equation. 

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