The residual entropy

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 

The molar residual entropy (replace NhyN/ and use NJc=R) is therefore SjO) = Rlnf=3.4JK- mol  

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In either case each dimer has two possible orientations, and random disorder between these accounts for the residual entropy of the crystal (6.3JmoH of dimer). More recently ii)... 

The residual entropy can be calculated. For the para-hydrogen, 7 = 0 at T = 0 K. Since one-fourth of the hydrogen is para, the contribution to the... 

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. 

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 structures B and C have no odd electrons, and for resonance with them A must have the spins if the two odd electrons opposed. Hence the substance should be diamagnetic—as it has been observed to be.11 The residual entropy at low temperature,12 approximately Mn2 per mole of NjOj, can be explained as resulting from a disorder in the crystal, each NiOi rectangle having two possible orientations. 

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. 

Example 1. Calculate the residual entropy at T = 0 for carbon monoxide. 

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

Calculate change in entropy using Eq. (6-94) along with MCPS function for the mean heat capacity and SRB function for the residual entropy. 

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). 

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

Residual entropy (.S R) The following correlation can be used to estimate the residual entropy ... 

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]. 

According to Orr [210] the residual entropy is negligible as is, according to Temperley [211], the configurational entropy. Sawada [212] described the entropy of copolymerization randomness (unit placement in the chain) by the relation... 

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