The entropy of water

Entropy is an important thermodynamic property of a system and we depend on entropy for much of our understanding of physical, chemical, and biological phenomena. However, it is hard to calculate, even in computer simulations. The entropy of a system is highest in the gas phase and lowest in the solid phase. Entropy 

When atoms or molecules contain an attractive part in an intermolecular interaction, the magnitude of the potential energy due to interaction between the particles increases with an increase in density or lowering of temperature, and the entropy of the system, at fixed volume, decreases. While conceptually easy to understand, it is difficult to treat the attractive and repulsive parts together to obtain both enthalpy and entropy at the same level of approximation. Ironically, it is often easier to obtain the free energy of the system by using the new technique of statistical mechanics. 

A significant starting point is provided by the third law of thermodynamics, which states that the entropy of a perfectly crystalline solid is zero at the absolute zero of temperature. The third law allows one to estimate entropy by using the following thermodynamic relation 

Implementation of this exact relation depends on the accurate measurement of speeifie heat over the entire temperature range. The absolute value of entropy requires use of the third law of thermodynamics and the building up of entropy from the absolute zero of temperature. Important insight can be obtained if we eompare Eq. (19.1) with Eq. (19.2). The latter defines specific heat in terms of fluctuation in entropy, at constant temperature and pressure. 

The entropy of water is a sum of all its vibrations (intramolecular and intermolecular) and also rotations, represented as 

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The entropy of water vapour is usually referred to that of liquid at 0° C. as zero, and the entropy of unit mass of steam at 0° C. is therefore ... 

The contribution of this lack of regularity to the entropy of ice is thus R In 3/2 = 0.805 E. U. The observed entropy discrepancy of ice at low temperatures is 0.87 E. U., obtained by subtracting the entropy difference of ice at very low temperatures and water vapor at standard conditions, for which the value 44.23 E. U. has been calculated from thermal data by Giauque and Ashley,7 from the spectroscopic value 45.101 E. U. for the entropy of water vapor given by Gordon.8 The agreement in the experimental and theoretical entropy values provides strong support of the postulated structure of ice.9... 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

The three cases of chemisorption examined all showed localized adsorption, but there may be other examples in which a considerable mobility is possessed by the adsorbed molecules or atoms. Again, this would be more likely to occur at high temperatures. Entropy values can give indications of dissociation or, more particularly, of association when localized adsorption takes place. This was clearly noticeable in the values for the entropy of water on mercury. Likewise a knowledge of the changes in entropy during the course of an isotherm may be of use... 

Why is the entropy of water vapor so much higher than that of liquid water ... 

Further, if the difference in entropy between selenite and hydroxyl is the same on the surface as it is in solution, the value for the entropy of water (16.7 e.u.) indicates that (x — y) = 1—i.e., a molecule of water is displaced from the surface during exchange of SeOs2 for OH". The... 

The native state of a protein has many of its hydrophobic side chains shielded from water because they are packed in hydrophobic cores. Conversely, the denatured state has many of its hydrophobic side chains exposed to solvent. The water molecules stack around these in icebergs as they maximize their hydrogen bonds with one another (Chapter 11). This lowers the entropy of water, because the individual molecules have less freedom of movement, and lowers the enthalpy because more hydrogen bonds are made.2 Similarly, the hydrogen bond donors and acceptors in the polypeptide backbone of the denatured protein are largely exposed to solvent and tie down more water molecules.3 These water molecules are released as the protein folds, and the gain in entropy of water compensates considerably for the loss of conformational entropy. 

J. D. Dunitz19 has estimated the cost in entropy of tying up solvent water. The entropy of a water molecule of hydration in. a crystal or mineral is 42 J/mol/K (10 cal/mol/K), which represents the lower limit for a tightly bound molecule. The entropy of water in liquid water is 67-71 J/mol/K (16-17 cal/mol/K), which represents the upper limit for the least constrained water molecule in solution. Thus, the energetic cost of immobilizing a water molecule is between 0 and 8.3 kJ/mol (0 and 2 kcal/mol) at 25°C (298 K). 

If we want to calculate the entropy of a liquid, a gas, or a solid phase other than the most stable phase at T =0, we have to add in the entropy of all phase transitions between T = 0 and the temperature of interest (Fig. 7.11). Those entropies of transition are calculated from Eq. 5 or 6. For instance, if we wanted the entropy of water at 25°C, we would measure the heat capacity of ice from T = 0 (or as close to it as we can get), up to T = 273.15 K, determine the entropy of fusion at that temperature from the enthalpy of fusion, then measure the heat capacity of liquid water from T = 273.15 K up to T = 298.15 K. Table 7.3 gives selected values of the standard molar entropy, 5m°, the molar entropy of the pure substance at 1 bar. Note that all the values in the table refer to 298 K. They are all positive, which is consistent with all substances being more disordered at 298 K than at T = 0. 

In Fig. XI-6 we show the entropy of water in its three phases, as a function of pressure and temperature, computed as we have described above. We are struck by the resemblance of this figure to that giving the volume, Fig. XI-4 the entropy, like the volume, increases with increase of temperature or decrease of pressure. Lines of constant pres-... 

This is to be distinguished from the entropy of water in water for which, of course, free-volume values are available directly from measurements of sound velocity. 

What happens to the entropy of water in this process A mole of liquid water (18 g) has a volume of approximately 18 mL. A mole of gaseous water at 1 atm and 100°C occupies a volume of approximately 31 L. Clearly, there are many more positions available to the water molecules in a volume of 31 L than in 18 mL thus the vaporization of water is favored by this increase in positional probability. That is, for this process the entropy of the system increases ASsys has a positive sign. 

The transition state in Su2 solvolysis is more polar than the initial state and can therefore be expected to be more heavily solvated. However this additional solvation is probably less than for mechanism Sul (Ingold, 1953d Featherstone etcd., 1963) and the more negative AS in Su2 processes is presumably a consequence of the covalent participation of water in the transition state. The results in Table 5 are consistent with the solvation model if a water molecule loses much more of its entropy when it forms a partial covalent bond than when it solvates a charged centre, while the reduction in its heat capacity is similar for the two processes. More quantitative considerations, however, require the unlikely conclusion that nearly all the entropy of water is lost when it forms a partial covalent bond with the reaction centre, but the imperfections of the model may be responsible. A detailed discussion has already been given elsewhere (Kohnstam, 1962). 

Somewhat analogous considerations apply to the entropy of water vapor. The result derived from heat capacity measurements is again lower than the statistical value, and this can be accounted for by random orientation of the water molecules in the solid. The situation is complicated, however, by the distribution of hydrogen bonds in the ice crystal, and by other factors. In this instance, also, the crystal is not perfect, and so the entropy would not be zero at 0 K. The statistical value of the entropy is therefore the correct one to be used in thermodynamic calculations. 

One useful discriminator of structure and dynamics in liquid is obtained through entropy. However, experimental or theoretical estimation of the entropy of a liquid confined to a local region is quite hard. This has hampered our understanding of the order/disorder transition in the local region at the mesoscopic length scale. One such rare study concentrated on the estimation of the entropy of water molecules in the groove region and correlated with the observed dynamics. Note that calculation of... 

As mentioned earlier, it is difficult to obtain a quantitative measure of entropy. By using the 2PT method (the method will be described later in Chapter 19), one can obtain the entropy of water molecules in both major and minor grooves of DNA. One can also get a measure of the translational diffijsivity of those water molecules from the mean-square displacement or velocity autocorrelation function - all these are fortunately easily available with computer simulations. 

Note that the extensive HB network is compromised near both the hydrophilic and the hydrophobic surfaces, but differently. In the case of the hydrophilic surface, the enthalpic gain from the water-surface interaction compensates for the twin losses of enthalpy and the entropy of water arising from the molecular rearrangement imposed by the surface. However, for a hydrophobic surface, such a compensation is not present. Therefore, the chemical potential of a water molecule near a hydrophobic surface is higher than that in a bulk. 

The entropy of water in its own liquid state is 17 cahmol/K (at room temperature). Thus, insertion of each n-butane costs the solution about one molar entropy equivalent of pure water. This is a formidable cost. 

Physically, an external repulsive solute sphere that does not form any HB with the surrounding water molecules does not offer any enthalpie stabilization at all. Therefore, the surrounding water molecules reorder themselves to preserve as many HBs as possible. Therefore, the entropy of water deereases due to this ordering foreed on it by the solute. 

The entropy of water in the ideal gas state is about 31 eal K mor under ambient eonditions (T= 298 K and atmospheric pressure). On the other hand, in the... 

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