Entropy as a function

The entropy of any chemical substance increases as temperature increases. These changes in entropy as a function of temperature can be calculated, but the techniques require calculus. Fortunately, temperature affects the entropies of reactants and products similarly. The absolute entropy of every substance increases with temperature, but the entropy of the reactants often changes with temperature by almost the same amount as the entropy of the products. This means that the temperature effect on the entropy change for a reaction is usually small enough that we can consider A Sj-eaction he independent of temperature. 

Fig. 3.6. Evolution of a WL simulation for the Lennard-Jones fluid at p = 0.88 and N = 110. The calculated quantity of interest is the dimensionless entropy, as a function of potential energy. The average statistical error is determined from the standard deviation offrom 10 independent runs. The modification factor curve (the dotted line) has also been averaged over these runs, and consequently appears smoother than would normally be the case...

The recent theoretical approach based on the information theory (IT) in studying aqueous solutions and hydration phenomena [62 66] shows such a direction. IT is a part of the system based on a probabilistic way of thinking about communication, introduced in 1948 by Sharmon and subsequently developed [114]. It consists in the quantitative description of the information by defining entropy as a function of probability... 

In lambda transitions, no discontinuity in enthalpy or entropy as a function of T and/or P at the transition zone is observed. However, heat capacity, thermal expansion, and compressibility show typical perturbations in the lambda zone, and T (or P) dependencies before and after transition are very different. 

Figure 7.12. Ordering entropy as a function of composition (a) exact values derived from the terminal entropy for a completely random solution and (b) using an empirical route via the entinipy The difference between these two curves is sufficient to generate spurious miscibility gaps (Inden 1991).

Figure 2-13 Schematic drawing of (a) density as a function of temperature, and (b) entropy as a function of temperature for glasses with different cooling rates and hence different glass transition temperature (Martens et al., 1987). The entropy of the undercooled liquid is estimated assuming constant heat capacity.

The function describing the change in entropy, as a function of temperature, involves the use of a prescription that contains a formula specific to a particular phase. At each phase transition temperature the function suffers a finite jump in value because of the sudden change in thermodynamic properties. For example, at the boiling point 7b the sudden change in entropy is due to the latent heat of evaporation (see Figure 2.8). 

Figure 2.8 A plot of the function describing the change in absolute entropy as a function of temperature. The discontinuities occur at phase changes...

It is often useful as w cll to have the entropy as a function of temperature and volume. We can find this by integrating the equations... 

Finally, reference is made to a series of articles by Griskey et al. (1966,1967) mentioning values for enthalpy and entropy as a function of temperature and pressure for a number of commercial plastics. 

Equation (1.90) is the total differential of the entropy as a function of the variables U and V only. To generalize this relation, we also consider the changes in the amounts of species. Using the mole amounts for the species, we have a general expression for the change of entropy from the Gibbs relation... 

Figure 6. Entropy as a function of solvent composition in alcohol-water mixtures ( ,), tert-butanol (O), ethanol.

For the implied infinitesimal changes of y with T the factor RTF remains constant. Generally, however, the derivative dy/dT also depends on T. From (4.3.22) S° will eventually be obtained as a function of x and T. As, for each T, F is accessible as a function of x, it is also possible to derive the surface excess entropy as a function of the monolayer composition. Accurate data are, as before, a prerequisite. From S the surface excess enthalpy = TS is obtainable. Alternatively, one can differentiate y /T with respect to the temperature, obtaining the enthalpy directly using the appropriate Gibbs-Helmholtz relation. 

The dominant contribution to the free energy of lengthy (rubbery) polymer chains is entropy. This is known to accoimt for rubber elasticity, which can be satisfactorily modelled by the entropy of the cross-linked pol3rmer chains alone. A simple illustrative model of copolymer self-assembly can be developed by extending rubber elasticity theory to include bending as well as stretching deformations, to calculate chain entropy as a function of interfacial curvatures in diblock aggregates. 

Figure 32. Entropy as a function of temperature for different modifications of water equilibrium (5), degassed structured (S), degassed with disordered structure (S). The heat capacities of water Cp and Cv as functions of T are plotted as well. (From Ref. 359.)...

Figure 3.03 Sketch of the communal entropy as a function of the fraction of liquidlike cells in the Cohen-Grest percolation formulation of the glass transition.

The subscripts eg and cs refer to end-group contribution and to elementary contribution of constitutive segments, respectively, and n is the number of segments per molecular chain. This model was applied satisfactorily to n-paraffins, but also to n-esters and n-ether. A linear variation of both the activation enthalpy and entropy as a function of n has been observed experimentally (22). One may designate as 7 the parameter that joins the intrinsic components describing the elementary contributions of the crystalline relaxation, i.e.,... 

This result can be applied to a long chain made of discrete links by using the correspondence given by (14.4.21). Moreover, as V(r) is arbitrary, we can consider the entropy as a functional of the monomer concentration. Thus, we obtain... 

Figure 10.2-6 Excess Gibbs energy, excess enthalpy, and excess entropy as a function of mole fraction for the benzene-2,2,4-trimethyl pentane system.

Figure A2.1.6. Entropy as a function of piston position / (the piston held by stops). The horizontal lines mark possible positions of stops, whose release produces an increase in entropy, the amount of which can be measured by driving the piston back reversibly. 

When Eq. (87) is used in Eq. (91), the entropy is expressed in terms of the Lagrange multipliers Xr. Using the approach of Agmon et al. (109), one can now regard the entropy as a function of trial values of the Lagrange multipliers, and seek its optimal value... 

You will note that we have the change in entropy as a function of the differences between normal lattice sites and vacancies. Since we know that Nl Nv, we can write for the entropy of mixing ... 

Equation (V.39) was used to obtain heat capacity values at various temperatures. Least squares analysis was then used to obtain an expression for the heat capacity as a function of temperature. From this expression, a similar expression was obtained for the standard entropy as a function of temperature. The latter expression was used to estimate a value for the standard entropy at 298.15 K ... 

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