The entropy of evaporation

As a conclusion from the Hildebrand/Trouton Rule, the definition of a standard vapor phase in a standard state with a well known amount of disorder can be made. This definition can be used as a starting point for modeling diffusion coefficients of gases and liquids. 

The change in entropy AS for a reversible isothermal expansion of an ideal gas from its initial volume Vj to a volume V2 is AS - R In(V2IVj) and therefore V2IVj = exp(ASIR). By setting V2IVt equal to the ratio between the molar volume Vq = 

8 dm3 mol-1 of an ideal gas under standard conditions (T = 298.15 K,/ = 1 bar) and assigning a volume V°L to one mole of a liquid at 7/, then VyV°L = exp(ASVIR) = exp(AHJRTh). Where AHv stands for the molar enthalpy of evaporation at the Hildebrand temperature, Th, and ASv is the molar entropy of evaporation. By using ASV = 

9 JK mol-1 the value V°L = 0.91 cm3mol is obtained. An interpretation of the Hildebrand/Trouton Rule is that this free volume, V°L, allows for the freedom of movement of molecules (particles) necessary for the liquid state at the temperature Th. The explanation of the constant entropy of evaporation is that it takes into account only the translational entropy of the vapor and the liquid. It has to be pointed out that V°L does not represent the real molar volume of a liquid, but designates only a fraction of the corresponding molar volume of an ideal gas Vy derived from the entropy of evaporation. The real molar volume VL of the liquid contains in addition the molar volume occupied by the molecules V0. As a result the following relations are valid VL -V°L + V0 and Vc=Vq + V0. However, while V] V0 and VL is practically independent of the pressure, V0 VaG in the gaseous phase. Only in the critical phase does VCIVL = 1 and the entropy difference between the two phases vanishes. 

---

When the pressure/ is expressed in atmospheres, then at the boiling point 7 the pressure/ = i and thus C R = AHjTb = AS. In this last expression we meet the latent heat of evaporation at constant pressure, divided by the boiling point temperature on the absolute scale according to Trouton s rule this quotient has an approximately constant value actually about 22 for normal liquids. This means, therefore, that the entropy of evaporation (at i atm.) also amounts approximately to 22 cal/mole degree (alkali halides 24 cal/mole degree). 

It should, however, be borne in mind that the theoretical argument can in fact tell us something about the heat of evaporation but not about the boiling point itself, except when the entropy of evaporation (AH/T = Trouton constant, p. 88) shows no differences for related compounds. This is almost correct in the above-mentioned case of cis- and trans-dichloroethylene, namely 21.64 and 21.52 cal/mol. deg., respectively. 

We notice that A s is simply the entropy of evaporation for p = l atm. and is identical with the standard entropy change (2lfS ). 

This may be simplified if we assume the entropies of evaporation from the solution to obey Trouton s rule, and put... 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

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 ... 

For convenience and in accordance with a familiar formulation of the third law of thermodynamics, let us take our starting point for entropy measurements such that the entropy of the crystal is zero at the extremely low temperature involved. Starting with the crystal let us then form by reversible evaporation one mole of vapor at the vapor pressure. The entropy of the gas thus formed will evidently be... 

In order to clarify these ideas, we need to compare the irreversible entropy productions (or the exergy destruction) in cycles that utilize regenerative heating of compressed air, thermal recuperation in the form of evaporation and superheating of the methanol fuel, and chemical recuperation through either reforming or cracking reaction with methanol. The next section presents such a comparison in a simplified form to illustrate the utility of thermodynamic analyses. 

As shown in the previous section a common feature of all systems in the liquid state is their molar entropy of evaporation at similar particle densities at pressures with an order of magnitude of one bar. Taking this into account a reference temperature, Tr, will be selected for systems at a standard pressure, p° = 105 Pa = 1 bar, having the same molar entropy as for the pressure unit, p = 1 Pa at T = 2.98058 K. As can easily be verified, the same value of molar entropy and consequently the same degree of disorder results at p if a one hundred-fold value of the above T-value is used in Eq. (6-14). This value denoted as Tw = 298.058 K = Tr is used as the temperature reference value for the following model for diffusion coefficients. The coincidence of Tw with the standard temperature T = 298.15 K is pure chance. 

The second law of thermodynamics states that for all spontaneous processes the entropy of the universe will always increase. This is often misunderstood to mean that the entropy of all parts of the system will increase. For example, if a small container of water is placed into a freezer, it will spontaneously freeze. Although the entropy of the water in the container decreased, a number of processes had to occur for that change to take place. The processes that occurred in the freezer that allowed the water to freeze (such as the movement of the compressor, the evaporation and condensation of the refrigerant, and the warming of the air around the container), all combine to produce a net increase in the entropy of the universe. 

It can be seen that, once an assumption is made for the value of M, the only quantity still unknown in the above equation is a, the evaporation coefficient, which must have a finite value equal to or less than 1. If it is assumed that a is constant but unknown, then the vapor pressure at any given temperature is proportional to the vaporization rate, and the enthalpy of vaporization may be found from the Clausius-Clapeyron type treatment. If a value is assigned to a, then vapor pressure values and the entropy of vaporization can be calculated as well. If the entropy of vaporization found in this way is a reasonable value, then the assumed value of a receives support. The latter procedure has been adopted here, and a value of unity has been taken for a. The reasons for choosing this value are ... 

Lennard-Jones and Devonshire, with the potential energy function —ar "+br" ( 41.VIIC) calculated the molar entropy of evaporation ... 

Postponing model interpretations until the next sections, let us reconsider the relative constancy of S". It is Interesting to note that a similar trend is observed in the molar entropy of evaporation, For all the liquids mentioned in fig. 

Intuitively, surface and interfacial tensions may be expected to be related to a number of physical characteristics of the liquid or the liquid-vapour transition. Two of these are the enthalpy and entropy of evaporation, discussed in sec. 2.9. Other parameters that come to mind are the molar volume V, the isothermal compressibility and the expansion coefficient. The combination of certain powers of such parameters and y sometimes leads to products with interesting properties, like temperature independence or additivity. Severed of such scaling rules have been proposed over the past century, mostly with limited quantitative success. A few of these wUl now be discussed. 

---