Latent entropy

Values for many properties can be determined using reference substances, including density, surface tension, viscosity, partition coefficient, solubihty, diffusion coefficient, vapor pressure, latent heat, critical properties, entropies of vaporization, heats of solution, coUigative properties, and activity coefficients. Table 1 Hsts the equations needed for determining these properties. 

The entropy change AS/ - and the volume change AV/ - are the changes which occur when a unit amount of a pure chemical species is transferred from phase I to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition ... 

Let [1], [2], [8] be any three modifications of a substance which can exist together in equilibrium at a triple point, and let t i, r2, r3 be their specific volumes su s2, s3, their entropies per unit mass. The gradients of the p-T curves at the triple point are given by the latent-heat equations ... 

The quantity (Sp - Sa) is the entropy change in the phase change and hence (Sp - Sa) = AS = L/T, where L is the latent heat per mole associated with the phase change at temperature r. Taking this into account, the above relationship can be expressed as... 

The limitation of the storage capacity is, as mentioned before, caused by the limitation of entropy change AS within the storage (see Figure 4). For sensible and latent heat storage (so-called direct thermal energy storage) this is defined by the specific heat... 

Some interesting aspects of the interface kinetics appear only when temperature and latent heat are included into the model, if the process of heat conductivity is governed by a classical Fourier law, the entropy balance equation takes the form Ts,= + x w where s = - df dr. Suppose for simplicity that equilibrium stress is cubic in strain and linear in temperature and assume that specific heat at fixed strain is constant. Then in nondimensional variables the system of equations takes the form (see Ngan and Truskinovsky, 1996a)... 

Hv/Te Molal latent heat of vaporization at te divided by Te. (Equal to the molal entropy of vaporization at te.)... 

We can also write the transition entropy ASa for this phase transition in terms of the associated latent heat AHa and transition temperature T(r as... 

Heat of formation of the liquid substance at 25°C —A//f Heat of formation of the gaseous substance —AHf Latent heat of evaporation (A//v) at 25°C Latent heat of evaporation (AHv) at boiling point Entropy (in relation to absolute zero) of the liquid substance at 25°C Entropy of formation of the liquid at 25°C Gibbs free energy of formation AFf of the liquid substance at 25°C 45.7 kcal/mole 37.0 kcal/mole 8.7 kcal/mole 8.1 kcal/mole 59.08 cal/mole/degree —118.1 cal/mole/degree —10.5 kcal/mole... 

LATENT HEAT. Hem tunned by a substance or system without tin accompanying rise in temperature during a change ol state. As examples, the latent heat of fusion is the amount of heat necessary to concert a unit mass of a substance trnin the solid stale to the liquid stale at the same temperature, the pressure being that to allow coexistence of the two phases. A considerable pari of the latent heat arises from the entropy increase consequent on the greater disorder of the liquid state. The latent heat of sublimation is the amount of heat necessary to convert a unit mass of a substance from the solid state to the gaseous stale at (he same temperature, the pressure being that to allow coexistence of the two phases. 

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). 

The transition at 19° C involves an expansion of 0.0058 cm3/g (Clark and Muus). Sincethe transition temperatureincreaseswith pressure by about 0.013° C per atmosphere (Beecroft and Swenson), the latent heat is about 3.2 cal/g. These values are for the crystal and would be reduced in proportion to the crystalline content. The transition at 30° C is only about one-tenth as large. The over-all increase in entropy at these transitions is about 0.0108 cal deg-1g-1. The portion due to the increase in volume is (a// ) A V, where a is the volumetric coefficient of thermal expansion and / is the compressibility. Since the compressibility of the crystal is not known, this quantity is somewhat uncertain. Using the average of the values of a (Quinn, Roberts, and Work) and p (Weir, 1951) for the whole polymer above and below the transitions, it appears that (a/P)A V is about 0.0041 cal deg 1g 1. The entropy of the transition corrected to constant volume is, therefore, about 0.0067 cal deg g-1. 

The above conclusion is unfortunate for the case of polymeric solutes, because then-entropies of dissolution are unusually small. The repeat units can not become as disordered as can the corresponding monomer molecules since they are constrained to be part of a chain-like structure. Such disordering is particularly difficult if the chain is stiff. Thus, in this situation dissolution is even less likely. Crystalline polymers are also more difficult to dissolve than are their amorphous counterparts since the enthalpy of dissolution also contains a large, positive contribution from the latent heat of fusion. 

In second-order transitions, there is no latent heat, nor discontinuous changes in volume or entropy. 

There has been a wealth of activity based on the idea that glassy dynamics is due to some underlying thermodynamic transition [1-25], If a glass former shows a jump in some an appropriate order parameter without the evolution of latent heat, then such a system is said to exhibit a random first-order transition [94,95]. Models of this kind, which include the p-spin glasses [110], and the random energy model [111], do not have symmetry between states but do have quenched random long-range interactions and exhibit the so-called Kauzmann entropy crisis. 

Solve Eq. (6.72) for the latent heat and divide by T to get the entropy change of vaporization ... 

Fig. 2.4 Free-energy changes at transitions (a) first-order transition (b) change in S at constant T and, consequently, latent heat (c) second-order transition (d) continuous change in entropy and so no latent heat (discontinuity in d2G/dT2).

According to the thermodynamio laws the latent heat is determined by the product of the absolute temperature and the entropy change AS, relating to the reaotion ... 

If the pressure is above the triple point, the solid will first melt, then vaporize. In this case, wo can proceed in a similar way. On melting, the entropy increases by the entropy of fusion, determined from the latent heat of fusion and temperature of fusion by the relation... 

It is interesting to note that a relation between the latent heat of vaporization of the solid, the heat of fusion, and the heat of vaporization of the liquid, at the triple point, arises from the fact that Eqs. (3.3) and (3.6) must give identical values for the entropy of the gas at the triple point. 

We can think of two limiting sorts of transitions one in which the transition always occurs at the same temperature independent of pressure, the other where it is always at the same pressure independent of temperature. These would correspond to vertical and horizontal lines respectively in Fig. XI-3. In Clapeyron s equation dP/dT = L/TAV, these correspond to the case dP/dT = oo or 0 respectively. Thus in the first case we must have AV = 0, or the two phases have the same volume, in which case pressure does not affect the transition. And in the second case L = 0, or AS = 0, there is no latent heat, or the two phases have the same entropy, in which case temperature does not affect the transition. Put differently, increase of pressure tends to favor the phase of small volume, increase of temperature favors the phase of large entropy. 

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