Woessner equation

The first quantitative estimate of the rotational diffusion tensor for simple molecules was accomplished by Grant et al. [163], By solving the Woessner equations, they were able to show e.g. for trans-decalin that the molecule rotates preferentially like a propeller, i.e. about the axis perpendicular to the plane of the molecule. The values given as a measure of the rotational frequencies do not correlate with the moments of inertia, but instead with the ellipticities of the molecule as defined [163]. They are accessible from the ratios of the interatomic distances perpendicular to the axes of rotation, and can be adopted as a measure of the number of solvent molecules that have to be displaced on rotation about each of the three axes. 

For less symmetric molecules one has to resort to computer programs [164] to solve the Woessner equations. The orientation of the rotational diffusion tensor is usually defined by assuming that its principal axes coincide with those of the moment of inertia tensor. This assumption is probably a good approximation for molecules of low polarity containing no heavy atoms, since under these conditions the moment of inertia tensor roughly represents the shape of the molecule. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

Woessner s equations thus permit prediction of spin-lattice relaxation times for the dipole-dipole mechanism, which can be of help in the assignment of 13C NMR spectra. Moreover, the calculations described can be applied to the problem of internal molecular motion. 

Woessner (157) has studied the relaxation in heavy water at 9.5 megacycles and notes that the data cannot be described by an Arrhenius equation. Instead, the curve appears to break around 40°-45°C. 

In many systems, the diffusion of small molecules is not free but is restricted. Examples of systems with restricted diffusion are molecules in cellular compartments, fluids between long flat plates, lamellar and vesicular systems, water-filled pores in rocks, and small molecules in colloidal suspensions. If the time during which the molecular diffusion is monitored in the experiment (the time between rf pulses in the continuous gradient experiment and A-6/3 in a pulse gradient diffusion experiment) is much longer than the time for a molecule to travel to a boundary, then the calculated D will be too small (Woessner, 1963). In such cases of restricted diffusion, appropriate equations must be derived for the particular geometry of the constraint (Stejskal, 1965 Wayne and Cotts, 1966 Robertson, 1966 Tanner and Stejskal, 1968 Boss and Stejskal, 1968). In favorable cases, the diffusion measurement can yield not only D but an estimate of the restraining dimension as well. 

Corey and coworkers , in a synthesis of prostaglandins, prepared diene 34 by alkylation of the lithiodithiane 32 with 2-bromomethyl-l,3-butadiene (equation 41). A synthesis of jasmone (35), in an overall yield of 50%, has been reported by Ellison and Woessner in which the bisdithianylethane 33 was sequentially alkylated, followed by hydrolysis and cyclization (equation 42). A similar route for preparation of 4-hydroxy-2-cyclopenten-l-ones has been reported . This method appears to provide a general route to 1,4-diketones via 1,3-dithianes. 

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