The residual entropy of ice

All the structural information discussed in the previous subsections is about the locations of oxygen atoms. Using X-ray crystallography, one cannot determine the locations of the hydrogen atoms. 

In 1933, Bernal and Fowler concluded from the available experimental data that water molecules in ice maintain their molecular identity, and that the hydrogen atoms are located at the corners of a regular tetrahedron, the center of which is occupied by an oxygen atom. They formulated the so-called two ice conditions (see Fig. 1.15). 

These two ice conditions reflect the experimental findings that when ice is formed from either the gas or the liquid phase, water retains its identity as a single molecule, i.e. one can identify single water molecules in the ice lattice. In addition, each pair of nearest-neighbor water molecules form hydrogen bonds in such a way that only one hydrogen is situated along the 0-0 

In 1935, Pauling calculated the approximate number of possible configurations for the hydrogen atoms that are consistent with the ice conditions. This calculation is now considered a classical example of a successful prediction based on an elementary probabilistic argument. Because of its historical importance and its didactic simplicity, we present here Pauling s solution to this problem. 

Consider a perfect ice structure containing N water molecules, i.e. N oxygen atoms are situated at N lattice points. The question is How many ways can we arrange the 2N hydrogen atoms on this lattice such that the two ice conditions are fulfilled  

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The equation for synchronized resonance with L = 4 and v - 2 gives R In 3/2 for the residual entropy of ice (14). This value differs by only 1.1% from that given by calculations that do not involve the approximations made in our simple treatment. It is likely that the accuracy of Eq. 4 is also reasonably high. 

Use your results from parts (a) and (b) to calculate the residual entropy of ice. The measured value is 3.4J/molK. 

Nagle JF (1966) Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice. J Math Phys 7 1484-1491... 

It is interesting to note that, for L = 4 and v = 2, eqn. (5) gives R n(3/2) for the residual entropy of ice this value differs by only 1% from that given by calculations not involving the approximations made here [48]. 

Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Amer. Chem. Soc. 57,2680-2684(1935). Nagle, J. F. Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice. J. Math. Phys. 7, 1484-1491 (1966). 

This result would not have been so remarkable had it not been related to the residual entropy of ice (see Appendix B). 

The agreement between the experimental and the theoretical values of the residual entropy of ice clearly indicates that the distribution of hydrogen atoms within the ice lattice is not unique. There are many possible configurations that are consistent with the two ice conditions. 

Photo 38 Linus Pauling lecturing on hydrogen bonding and ionic hydration, ca. 1975. In the upper right comer of the blackboard is the statistical factor (3/2), which is the basis of Pauling s famous calculation of the residual entropy of ice (SP 73). To the left of (3/2) is depicted a tetrahedrally coordinated water molecule. 

This number has been used to estimate the residual entropy of ice, SQ k nS2 = 0.805 e.u. 

The existence of this residual entropy of ice at very low temperatures was discovered by Giauque and Ashley (ref. 7), who preliminarily ascribed it to the persistence of rotation of ortho-water molecules (comprising 3/< of the total) about their electric-moment axes, giving an entropy of 3/ R In 2 = 1.03 E. U. 

The third law of thermodynamics says that the entropy of pure, perfect crystalline substance is zero at absolute zero. But, in actual practice, it has been found that certain chemical reactions between crystalline substance, do not have DS = 0 at 0°K, which indicates that exceptions to third law exist. Such exceptional reactions involve either ice, CO, N2O or H2. It means that in the crystalline state these substances do not have some definite value of entropy even at absolute zero. This entropy is known as Residual Entropy. At 0°K the residual entropies of some crystalline substances are... 

The first example is the results of the calculation of the residual entropy of the ordinary ice [126,127], This calculation shows how accurate the density of states can be obtained by multicanonical simulations from the reweighting formula of (4.24). 

When a crystal of ice is cooled to very low temperatures it is caught in some one of the many possible configurations but it does not assume (in a reasonable period of time) a uniquely determined configuration with no randomness of molecular orientation. It accordingly retains the residual entropy k In IF, in which k is the Boltzmann constant and W is the number of configurations accessible to the crystal. 

In all instances where this disorder was actually observed by X-ray or neutron diffraction methods, X and A are hydroxyl oxygen atoms, as in the ices Ih and Ic, in certain of the high pressure ices, and in the cyclodextrin hydrates, see Parts III and IV. In ice Ih the disorder gives rise to the well-known residual entropy of 0.82 0.05 cal deg-1 [114 to 117]. 

The total number of various proton configurations in ice-like systems is rather large. By analogy with well-known Pauling s formula we obtained the following expression for residual entropy of PWCs ... 

The Bernal-Fowler-Pauling statistical model for the proton arrangements in ice presents a very subtle problem in the actual evaluation of the number of possible configurations in a macroscopic crystal. This is not just an academic exercise for, if we suppose there are configurations, all of which are equally likely, then the entropy of the system due to this cause is k In Further, if these configurations become frozen at some temperature where the disordering is still essentially complete, the measured entropy of the ice crystal will still have the residual value In at o °K, entropy from all other sources having vanished. 

In actual calculation for equilibrium between ice and clathrate hydrate, the free energy of a whole clathrate hydrate, which is either empty or may contain spherical guest molecules, is calculated as the sum of the interaction energy at OK, the harmonic vibrational free energy, and the residual entropy neglecting the anharmonic term [47] as... 

A comparison quoted by Astonf is given in Table 15. It will be seen that the agreement is within the experimental error except in the fourth reaction which involves w ater. This discrepancy is attributed to lack of perfection in the ice crystal at very low temperatures. In fact a residual entropy of 0.806 cal mol can be accounted for... 

Consider a sample of ice of N HjO molecules. Each of the 2N H atoms can be either close to or relatively far from an O atom, resulting in 2 possible arrangements. However, of the 2 = 16 possible arrangements around a single O atom, only 6 have two short and two long bonds (Fig. 2.12) and hence are acceptable. Therefore W = (f) and the residual entropy is... 

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