Gibbs free energy first-order transitions

Ehrenfest3 refers to the type of phase transition described above as a first-order transition because there are discontinuities in the first partial derivatives of the Gibbs free energy at the transition point. For example,... 

In single-component systems (or pure substances), the chemical composition in all phases is the same. In multicomponent systems, the chemical composition of a given phase changes in response to pressure and temperature changes and these compositions are not the same in all phases. For single-component systems, first-order phase transitions occur with a discontinuity in the first derivative of the Gibbs free energy. In the transitions, T and p remain constant. 

The X transition in liquid helium shown in Figures 11.5 and 11.6 is a second-order transition. Most phase transitions that follow the Clapeyron equation exhibit a nonzero value of A5m and AYmi that is, they show a discontinuity in 5 and Fm. the first derivatives of the Gibbs free energy Gm- Thus, they are caHA A first-order transitions. In contrast, the X transition shows a zero value of A5m and AVm and exhibits discontinuities in the second derivatives of Gm, such as the heat capacity Cpm-... 

The standard enthalpy difference between reactant(s) of a reaction and the activated complex in the transition state at the same temperature and pressure. It is symbolized by AH and is equal to (E - RT), where E is the energy of activation, R is the molar gas constant, and T is the absolute temperature (provided that all non-first-order rate constants are expressed in temperature-independent concentration units, such as molarity, and are measured at a fixed temperature and pressure). Formally, this quantity is the enthalpy of activation at constant pressure. See Transition-State Theory (Thermodynamics) Transition-State Theory Gibbs Free Energy of Activation Entropy of Activation Volume of Activation... 

However, it is useful, to provide a thermodynamic definition of a first-order transition. Specifically, it is one in which there is a discontinuity in a first derivative of the Gibbs free energy. The advantage of this definition is the guidance it provides for the experimental study of phase transitions. A useful expression for the free energy in this regard is... 

As an alternative, it is obvious that the result (7.5.5) coincides with the mean field approach to describe the critical phenomena of fluids. It is evident that this model corresponds to the formation of Gibbs free energy curves such as shown in panel (c) of Fig. 7.5.2. It relates to the boundary at which a system executes a first order transition the minima correspond to the r]Q values given by Eq. (7.5.5). 

In a semicrystalline polymer, the molar volume V (and hence also the density p), and not just a, may manifest discontinuous changes at the melting temperature Tm of the crystalline phase. The enthalpy and entropy also typically manifest discontinuities at Tm. Such discontinuities observed at Tm in the first derivatives of the Gibbs free energy signify that melting is a first-order phase transition . 

Ptitsyn et al. (1968) argued that the collapse process was a first order transition (i.e. the first derivatives of the Gibbs free energy with respect to both the temperature and pressure are discontinuous at the transition). They used the terms coil and globule for the two different states and likened the transition to the first order condensation of a real vapour into liquid droplets when the forces of attraction are increased sufficiently. They were led to believe that experimental observation of the transition would be very difficult, if not impossible, because intermolecular condensation would overwhelm the intramolecular contraction, leading to phase separation. Until quite recently this pessimism seemed fully justified. 

The glass transition is a second-order transition, that is, the second derivative of the free energy function. In comparison, a first-order transition, such as the melting of a material, is the first derivative of Gibbs free energy. The equation used to determine the glass transition is ... 

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