Free energy positive

Endergonic In a non-isolated system, a process that is accompanied by a positive change in free energy (positive AG) and therefore is thermodynamically not favored. Compare exergonic. 

The existence of an interfacial minimum in dA h(z) indicates that the alcohols are interfacially active. The free energy of transferring a solute molecule from the gas phase to the interface decreases with the chain length and is equal to —6.0, —6.9, —8.2 and —9.6 kcal/mol for methanol, ethanol, butanol and hexanol, respectively. The free energy of transferring the four alcohols from hexane to the interface, however, is almost eonstant, and equals to 4.5 + 0.3 keal/mol. This, in turn, implies that when the alcohols are in the minimum free energy positions, their hydroearbon portions are buried in hexane. 

The minimum free energy over the entire system is first subtracted off in order to prevent overflow/underflow problems that could arise from modest nonzero free energies (positive or negative). 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

S B/A is positive if spreading is accompanied by a decrease in free energy, that is, is spontaneous. 

For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results ... 

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

It is helpful to consider qualitatively the numerical magnitude of the surface tensional stabilization of a particle at a liquid-liquid interface. For simplicity, we will assume 6 = 90°, or that 7sa = 7SB- Also, with respect to the interfacial areas, J sA = SB, since the particle will lie so as to be bisected by the plane of the liquid-liquid interface, and. AB = rcr - The free energy to displace the particle from its stable position will then be just trr 7AB- For a particle of l-mm radius, this would amount to about 1 erg, for Tab = 40 ergs/cm. This corresponds roughly to a restoring force of 10 dyn, since this work must be expended in moving the particle out of the interface, and this amounts to a displacement equal to the radius of the particle. 

The leading order quantum correction to the classical free energy is always positive, is proportional to the sum of mean square forces acting on the particles and decreases with either increasing particle mass or mcreasing temperature. The next tenn in this expansion is of order This feature enables one to independently calculate the leading correction due to quanmm statistics, which is 0(h ). The result calculated in section A2.2.5.5 is... 

For the kind of transition above which the order parameter is zero and below which other values are stable, the coefficient 2 iiiust change sign at the transition point and must remain positive. As we have seen, the dependence of s on temperature is detemiined by requiring the free energy to be a miniimuii (i.e. by setting its derivative with respect to s equal to zero). Thus... 

Figure A3.8.1 A schematic diagram of the PMF along the reaction coordinate for an isomerizing solute in the gas phase (frill curve) and in solution (broken curve). Note the modification of the barrier height, the well positions, and the reaction free energy due to the interaction with the solvent.

The fluctuations of the local interfacial position increase the effective area. This increase in area is associated with an increase of free energy Wwhich is proportional to the interfacial tension y. The free energy of a specific interface configuration u(r,) can be described by the capillary wave Hamiltonian ... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

The simplest approach to understanding the reduced melting point in nanocrystals relies on a simple thennodynamic model which considers the volume and surface as separate components. Wliether solid or melted, a nanocrystal surface contains atoms which are not bound to interior atoms. This raises the net free energy of the system because of the positive surface free energy, but the energetic cost of the surface is higher for a solid cluster than for a liquid cluster. Thus the free-energy difference between the two phases of a nanocrystal becomes smaller as the cluster size... 

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