Derivatives of the Specific Entropy

The mixed quadratic derivatives of the specific entropy must be identical according to the law of Schwarz  

it has been shown that is related to the temperature derivative of the specific entropy. Analogously, a similar expression for cp can be evaluated  

Comparison with the total differential of the specific enthalpy gives 

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The particular partial derivatives of the specific entropy are explained in Sections C-A2 and C-A6, respectively. 

For the first order transformation the G(P, T) surfaces of the phases, being in an internal equilibrium, intersect over a range of P and T (Fig. 3a). Thus in a first order transformation a finite jump in the first derivative of the specific free energy G the specific entropy and the specific volume V is observed. 

Regardless of the relative importance of polar and nonpolar interactions in stabilizing the cyclohexaamylose-DFP inclusion complex, the results derived for this system cannot, with any confidence, be extrapolated to the chiral analogs. DFP is peculiar in the sense that the dissociation constant of the cyclohexaamylose-DFP complex exceeds the dissociation constants of related cyclohexaamylose-substrate inclusion complexes by an order of magnitude. This is probably a direct result of the unfavorable entropy change associated with the formation of the DFP complex. Thus, worthwhile speculation about the attractive forces that lead to enantiomeric specificity must await the measurement of thermodynamic parameters for the chiral substrates. 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

In other words, every chemical reaction takes place without change in entropy at the absolute zero. From this it follows that the entropy of a compound is equal to the sum of the atomic entropies. The assumption made by Planck in addition to Nernst s theorem, viz. that the entropy of all substances vanish like the specific heats at the absolute zero, is sufficient but not necessary for the derivation of the heat theorem. 

Finally, we note that once we have the molecular properties of the molecules, we can calculate all the thermodynamic quantities of the system, such as the Gibbs energy, entropy, enthalpy, etc., Note also that the equation of state does not depend on the specific properties of the system, only on the total number of the particles in the system, at a given P, T. The same is true for the derivatives of the volume with respect to pressure and temperature. 

Given that (see Fig. 9.8) at the glass transition temperature, the specific volume Vs and entropy S are continuous, whereas the thermal expansivity a and heat capacity Cp are discontinuous, at first glance it is not unreasonable to characterize the transformation occurring at Tg as a second-order phase transformation. After all, recall that, by definition, second-order phase transitions require that the properties that depend on the first derivative of the free energy G such as... 

The transition from a glass to a rubberlike state is accompanied by marked changes in the specific volume, the modulus, the heat capacity, the refractive index, and other physical properties of the polymer. The glass transition is not a first-order transition, in the thermodynamic sense, as no discontinuities are observed when the entropy or volume of the polymer is measured as a function of temperature (Figure 12.2). If the first derivative of the property-temperature curve is measured, a change in the vicinity of is found for this reason, it is sometimes called a second-order transition (Figure 12.2). Thus, whereas the change in a physical property can be used to locate Tg, the transition bears many of the characteristics of a relaxation process, and the precise value of can depend on the method used and the rate of the measurement. 

The result states that the specific entropy in the tank remains constant at all times it is, therefore, equal to its value at the beginning of the venting process. This provides the additional equation for the integration of the energy balance equation. We return to eq. (6.7 ) and work on the derivative d(MU)/dM. First, we note that this derivative expresses changes in the tank that occur at constant specific entropy. 

By eliminating the contribution of the temperature derivative of the mixed-phase fraction, the entropy peak effect is eliminated, in a justified way. The resulting entropy curve resembles the results obtained from specific heat measurements when compared to results from magnetic measurements, as seen in Refs. (Liu et al., 2007) and (Tocado et al, 2009), among others. 

We next consider the temperature derivatives of the fi ee energy, such as entropy and specific heat. The temperature appears through the interaction parameter x (T) and association constant X(T). Because it is evident that the former does not lead to any singularity, we consider the derivatives with respect to the latter. 

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