Conjugate generalized forces

In this equation Fj represents the conjugate generalized forces and the sum is taken over all of the generalized coordinates. The equation must reduce to Equation (4.1) when the number of moles of each component is held constant. 

The stress theorem is very useful in large macroscopic systems. In particular, since the structure of a crystal is completely specified by the size and shape of the unit ceil and positions of the atoms in the unit cell, the force and stress theorems give all the generalized forces conjugate to these variables. If the stress theorem is expressed in terms of relative coordinates and proper care is taken of the Coulomb interactions, then the stress theorem, Eqs. (8) and (8a), give the total stress in terms of the intrinsic bulk Hamiltonian and wave functions in the bulk of the crystal. Thus, the stress and force are sufficient to construct the complete equation of the state of any crystal. In general, the more complex and the lower the symmetry of the crystal, the more useful are the forces and stresses in the calculations. This is further described by Nielson in this proceedings and later in the present paper. 

The factor dR is the rate of change of volume of material B in dR. It follows that the quantity in square brackets in the integral over R in (9.91) is the generalized force that is work-conjugate to this local volume change. This field thus represents the local chemical potential of material B in the solution,... 

A particularly important result is that the matrix of phenomenological resistances (linear combinations of the Rj) which appears in the equations of motion relating the independent reaction rates to their conjugate thermodynamic forces in the stationary state is symmetric. However, the symmetry is of an algebraic nature the more stringent condition of differential symmetry does not hold in general. 

The general mechanistic features of the aldol addition and condensation reactions of aldehydes and ketones were discussed in Section 7.7 of Part A, where these general mechanisms can be reviewed. That mechanistic discussion pertains to reactions occurring in hydroxylic solvents and under thermodynamic control. These conditions are useful for the preparation of aldehyde dimers (aldols) and certain a,(3-unsaturated aldehydes and ketones. For example, the mixed condensation of aromatic aldehydes with aliphatic aldehydes and ketones is often done under these conditions. The conjugation in the (3-aryl enones provides a driving force for the elimination step. 

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