Ehrenfest equation

In case of second order transformation (Fig. 3 b), where the G(P, T) surfaces are in contact over a range of P and T the entropy changes continuously across the transformation. This transformation must obey the well-known Ehrenfest equations... 

Fig. 35. Examination of the validity of the Ehrenfest equations for the glass transition of the poly-(methylacrylate) shown in Fig, 34...

The evolution of the electronic wavefunction of the molecular system is determined by the appropriate Ehrenfest equation... 

We determine how a general operator evolves in time by determining the time evolution of the expectation value of the above time transformed operators. For doing this we utilize the Ehrenfest equation... 

We solve the Ehrenfest equation for each order of the perturbation and we collect for each order the necessary terms for calculating response functions at the given order. The only part within the Ehrenfest equation that is not covered by the ordinary response theory of quantum mechanical systems in vacuum is represented by the last term on the right hand side of Eq. (66) [80-83]. The contributions to the response functions arising from the last term are related to the presence of the structured environment coupled to the quantum mechanical subsystem, ft is the main contributor to changes in molecular properties when transferring fhe quantum subsystem from vacuum to the structured environment. 

Neglecting the last term on the right-hand side leads to the famous Ehrenfest equation for the pressure dependence of transition temperature in a second order transition at equilibrium. 

If the glass transition would be a true 2nd order phase transition, we would find zero for the volume difference AVj = in agreement with Ehrenfest Equation... 

The glass transition exhibits all the typical features of a second-order phase transformation, however, most of the experimental observations indicate that the Ehrenfest equations are obeyed only exceptionally and hence Tg is not an equilibrium quantity [9, 11]. First, the glass transition temperature is dependent... 

The ACp term is a significant parameter because it appears in the Ehrenfest equation [42]. Perhaps, in polymer blends, the intermolecular contribution to ACp plays a more important role than in many common homopolymers and copolymers. 

SO that it holds - dP + aA dT = - dP + ardT + (dV/dOr.pdC from which it follows that AE ,V = (dV/dQ. p/(dA/dQ, AayV = (dV/dOr.p (dS/dOr. / (dA/dQr.F and AC /T = p (dS/d0 rjd dA/d0r.p where Aar is the difrerence between the expansions of the equilibrium undercooled liquid and that of frozen glass (the same valid for the difference in volume compressibility, ASa and heat capacity, ACf). In practice, it follows that the classical form of Ehrenfest equation is supplemented by another term dependent on the degree of ordering, which states that if the properties of an undercooled liquid would be known, the properties of the glass close to this state can be derived, i.e., A(BCt/ccyjA(l/ay) = A(Cp/ay)A(Sfav), where T is the thermodynamic limit of glass transformation at Tg -> 7. ... 

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