Harmonic entropy

The constant-volume heat capacity of the substance ( Cy) represents the variation of total energy U with T in the harmonic approximation (X is constant over T), and the integration of Cy over T gives the (harmonic) entropy of the substance ... 

It has already been stated that, theoretically, A atoms in a crystal have 3 A possible vibrational modes. Obviously, if we knew the energy associated with each vibrational mode at all T and could sum the energy terms in the manner discussed in section 3.1, we could define the internal energy of the crystal as a function of T, Cy could then be obtained by application of equation 3.27, and (harmonic) entropy could also be derived by integration of Cy in dTiT. 

Table 5.25 Parameters of the Kieflfer model for calculation of Cy and harmonic entropy for the 12 garnet end-members in the system (Mg, Fe, Ca, Mn)3(Al, Fe, Cr)2Si30i2. is the lower cutoff...

This is the entropy of each mode of Irequency to. We need to consider all the modes. In a solid, the total number of modes of vibration consisting of N atoms is 3N 6, giving rise to an energy spectrum. The total harmonic entropy of the solid is obtained by integration of Eq. (19.6) over this spectrum. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

The next step when computing configurational entropy is to calculate the vibrational contribution to the entropy Sv b- The most commonly employed technique used to accomplish this calculation is to assume that the configuration point of the liquid executes harmonic vibrations around its inherent structures (i.e., Svib Sharmb which is a description that can be expected to be accurate at low temperatures. The quantity Sharm for a given basin is then computed as117... 

As pointed out in Ref. [4], no entropy variation appears in the description given by the harmonic model, apart from the weak contribution arising from the frequency shifts of the oscillators. The applications of this model are then a priori restricted to redox reactions in which entropic contributions can be neglected. We shall see in Sect. 3 that the current interpretations of most electron transfer processes which take place in bacterial reaction centers are based on this assumption. 

The position of Ti and Zr is again important in this context. While the b.c.c. phase in these elements has long been known to indicate mechanical instability at 0 K, detailed calculations for Ti (Petty 1991) and Zr (Ho and Harmon 1990) show tiiat it is stabilised at high temperatures by additional entropy contributions arising from low values of the elastic constants (soft modes) in specific crystal directions. This concept had already been raised in a qualitative way by Zener (1967), but the... 

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

Note that there is nothing wrong widi Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively. 

Thus, the configurational entropy. S, (7. p) expresses the number of local potential energy minima and hence can be evaluated by potential energy landscape and thermodynamic integration methodology [53]. The vibration entropy was calculated within the framework of a harmonic approximation to each basin [165,166], an approximation that is valid at low temperatures [53]. Three notable results were obtained by Sastry [53] ... 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

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