Change of Entropy

Since this reaction simultaneously proceeds for all elements, the change in the amount of substance of Xo, can be given by the following stoichiometric relation  

Here is referred to as the extent of reaction, and the reaction rate is given as the rate of the extent of reaction  

Let the amount of substance of a species X at the start of the reaction be o, then the amount at the extent is given by 

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The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

This expression states that there will be energy free to do work when Q exceeds AE. Expressed in another way work ean be done, that is an action can proceed, if AE - 0 is negative. If the difference between AE and Q is given the symbol AA, then it can be said that a reaction will proceed if the value of AA is negative. Since the heat term is the product of temperature T and change of entropy AS, for reactions at constant temperature then... 

At the same time, when we increase the separation of the charges from r to (r + Sr) isothermally, there is a change of entropy in the dielectric. For large values of r this amounts to... 

When solid AgCl is in contact with its saturated aqueous solution, we have found that, if additional ion pairs are transferred from the surface of the crystal to the solution, the total change of entropy is equivalent to 52.8 e.u Since the entropy of the solid is 23.0 e.u., we find that the partial molal entropy of AgCl in its saturated aqueous solution at 25°C is... 

After discussing in Chapter 1 a charged sphere in a dielectric, we saw in Sec. 14 that, when any pair of ions is added to a solvent, there will be a change of entropy in the co-sphere of each ion. If, for example, we knew the value of this change for the ion pair (Ag+ + Cl ) and likewise for the ion pair (Ag+ + I ), any difference between the two quantities could at once be ascribed to the difference between the co-spheres of the chloride ion Cl and the iodode ion I-, since the contribution from the Ag+ ion and its co-sphere would be the same in the two cases. 

The entropy per molecule of liquid water will therefore be 16.75 e.u., divided by Avogadro s constant. We have next to consider the change of entropy when a proton is added to a water molecule to give an (II30)+ ion. It is this quantity that is arbitrarily put equal to zero in Latimer s scale. We see at once that the value that must be allotted to the (HaO)+ ion in Latimer s list is 16.75 e.u. 

Just as the intrinsic energy of a body is defined only up to an arbitrary constant, so also the entropy of the body cannot, from the considerations of pure thermodynamics, be specified in absolute amount. We therefore select any convenient arbitrary standard state a, in which the entropy is taken as zero, and estimate the entropy in another state /3 as follows The change of entropy being the same along all reversible paths linking the states a and /3, and equal to the difference of the entropies of the two states, we may imagine the process conducted in the following two steps ... 

If now we have any reversible change which is not a cycle, there will be a change of entropy in the system, but this will have a compensating change outside the system. For suppose... 

For in all actual changes of such a system the entropy can only increase, so that if we consider a virtual change, and put (8S)u for the resulting change of entropy, then ... 

Equations (5), (6) define without ambiguity the change of entropy in any specified change of state. 

Then from (4), (6), (11), and (12) we find Theorem II. Ij an isopiestic p + dp is drawn to cut the three curves of transition (or their prolongations) meeting at a triple point, the central point of section corresponds u-ith the transition involving the greatest change of entropy. This theorem is due to Roozeboom (1901). 

Roozeboom, 1901) that the system of two phases which corresponds with the transformation invoicing the greatest change of entropy is in stable equilibrium under pressures lying on one side of the triple point, while the other two systems are in stable equilibrium under pressures lying on the other side of the triple jwint. 

With motion along the connodal curve towards the plait point the magnitudes Ui and U2, Si and S2, and ri and r2, approach limits which may be called the energy, entropy, and volume in the critical state. The temperature and pressure similarly tend to limits which may be called the critical temperature and the critical pressure. Hence, in evaporation, the change of volume, the change of. entropy, the external work, and the heat of evaporation per unit mass, all tend to zero as the system approaches the critical state ... 

As Gibbs and Helmholtz pointed out, however, this so-called Thomson Rule (which had also been proposed by Helmholtz in 1847) cannot be true in general, because the change of entropy cannot always be neglected. 

We see that the total change in entropy is a positive quantity for both these spontaneous processes, even though one process is exothermic and the other is endothermic. When this type of calculation is carried out for other processes, the same result is always obtained. For any spontaneous process, the total change of entropy is a positive quantity. Thus, this new state function of entropy provides a thermod3mamic criterion for spontaneity, which is summarized in the second law of thermodynamics ... 

Hence, we find a relation between K and the enthalpy of the reaction, instead of the free energy, and the expression for the equilibrium is in conflict with equilibrium thermodynamics, in particular with Eq. (32) of Chapter 2, since the prefactor can not be related to the change of entropy of the system. Hence, collision theory is not in accordance with thermodynamics. 

Figure 3.2 Change of entropy of a substance with temperature - the substance has undergone an allotropic change (t), fusion or melting (/) and vapourization (v) over the entire temperature range.

AG = change of Gibbs free energy AH = change of enthalpy T = absolute temperature AS = change of entropy... 

The theory of the thermodynamics of irreversible systems (Prigogine, 1979 Prigogine and Stengers, 1986) shows that the differential quotient of entropy with time (the change of entropy with time) can be expressed as the sum of products, the terms of which contain a force factor and a flow factor. In chemical systems, the... 

Starting from the second law of thermodynamics, it is possible to derive a principle according to which the change of entropy production in the neighbourhood of a stationary state is always negative if the flows in the system are kept constant and only the forces varied. As already mentioned, the entropy production reaches a minimum value in the stationary state of the system. If it is at a minimum, and a positive fluctuation occurs, the system reverts to the minimum, and a stable state is again reached. 

Further work is needed to determine in which regimes, if any, fluid bed behave as chaotic systems. Additional testing is needed to determine the sensitivity of important bed hydrodynamic characteristics to the Kolmogorov entropy, to quantitatively relate changes of entropy to... 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

Velocity inversion is a special case of velocity rotation of every particle, which corresponds to acceleration. Thus one can expect that velocity rotations and acceleration will also produce a change of the 7Y-function or a change of entropy. The effects of velocity rotations on the 7f-function have been considered in Ref. 22. 

With very short "blocks", snch as in the random-copolymer SBR, there is only a very small difference in entropy between the segregated and the homogeneous condition no or hardly any change of entropy npon segregation. As the block length increases, this difference, however, increases, so that (G decreases (see MT 9.1.4). Segregation is, therefore, less complete in mnlti-block copolymers (such as polyethers- polyesters). 

The number of ways of placing the units for initial position (corresponds to the initial entropy, So,t) and for final position (corresponds to entropy Si t) is almost the same because, in both cases, the monomer units cannot be located far from the template. It means that the volume accessible for a monomer molecule is reduced to the nearest neighborhood. In this case, So,t Si t and AS 0. The change of entropy concerns the propagation step of the template reaction. There is very little information in the literature about experimental results of entropy changes during template polymerization. 

The reaction is considerably endothermic and would not be feasible (changes of entropy are excluded from the calculation, but would not alter the conclusion). 

The process of contact adsorption can be viewed in the following way (Fig. 6.90) First, a hole of area of at least 1U —where r, is the radius of the bare ion—is swept free of water molecules in order to make room for the ion. At the same time, the ion strips itself of part of its solvent sheath and then jumps into the hole. During this process, the involved particles—electrode, ion, water molecules—break old attachments and make new ones (change of enthalpy, AH) and also exchange freedoms and restrictions for new freedoms (change of entropy, AS). 

These expressions demonstrate that the change of entropy and internal energy on deformation under these conditions is both intra- and intermolecular in origin. Intramolecular (conformational) changes, which are independent of deformation, are characterized by the temperature coefficient of the unperturbed dimensions of chains d In intermolecular changes are characterized by the thermal expansivity a and are strongly dependent on deformation. The difference between the thermodynamic values under P, T = const, and V, T = const, is vefy important at small deformations since at X - 1 2aT/(/,2 + X — 2) tends to infinity. 

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