Vibration contribution to entropy

Structural features that make molecules more rigid reduce rotational and vibrational contributions to entropy. Thus the formation of a double bond or ring decreases the entropy even when the molecular weight is unchanged. The formation of comparatively rigid macromolecular structures from flexible polypeptide or polynucleotide chains also requires an entropy decrease, although this can be offset by increases in the entropy of the surrounding water molecules (see chapter 4). 

The partition function (2 has a translational, rotational and vibrational part, and so does entropy. The first two contributions can be calculated in a straightforward way, but the vibrational contribution to entropy requires a knowledge of the normal vibrational frequencies of the molecule or of the crystal. 

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... 

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. 

This requires knowledge of the vibrational frequencies, the same information as required for evaluation of the vibrational contribution to the entropy. 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

A direct quantitative comparison between AG theory and measurements requires the resolution of two issues. First, the excess entropy Sexc must be normahzed by the molar volume. We suggest that the lack of this normalization is partly responsible for previous claims [15, 49] that AG theory breaks down for small molecule fluids. Second, the vibrational contribution to which is absent in s, must be subtracted reliably. While the first correction can readily be introduced, the inclusion of the second correction requires further investigation [63, 240]. 

Evaluate the translational, rotational, and vibrational contributions to the entropy for 0.1 moles of the A127C135 molecule at 900°C and a pressure of 1 mBar. Assume a bond length of 2.13 A, vibrational frequency rotational symmetry number a = 1. 

Here, Cv h(T) and Svlh(T) are the vibrational contributions to the heat capacity and the entropy, respectively. Note that the slope of the replica symmetry-breaking parameter with respect to temperature is not unity as predicted by one-step replica symmetry breaking. Rather, the slope is governed by three factors the Narayanaswamy-Moynihan nonlinearity parameter x, the Kauzmann temperature, and the ratio of the Kauzmann temperature to the glass transition temperature. 

The combined translational, rotational and vibrational contributions to the molar heat capacity, heat content, free energy and entropy for 1,3,4-thiadiazoles are available between 50 and 2000 K. They are derived from the principal moments of inertia and the vapor-phase fundamental vibration frequencies (68SA(A)36l). 

Substitution of i q. (2.110) gives Vffscw which, when used in Eiq. (2.109), gives the vibration contribution to the entropy of an NSCW near ions (Table 2.19). 

It is only necessary to sum the contributions made by each mode to the entropy of the various reactants and products to obtain the vibrational contribution to the total entropy change of the reaction. The latter total is then... 

The vibrational frequency is 2360 cm. (Table IX), and hence i.e., 1.439w/T, is 11.4. Insertion of this result into equation (24.19) gives a value for Sv that is negligibly small. The vibrational contribution to the entropy of molecular nitrogen at 25 C may thus be taken as zero. 

The entropy of K2Se(cr) was estimated by Voronin [70VOR] to be 5°(K2Se, cr, 298.15 K) = 117.15 J K -mor using a semi-empirical relationship involving vibrational contributions to the entropy and by Mills [74M1L] to be S°(K2Se, cr. 

The dynamic problem of vibrational spectroscopy must be solved to find the normal coordinates as linear combinations of the basis Bloch functions, together with the amplitudes and frequencies of these normal vibrations. These depend on k, and therefore the problem must be solved for a number of k-points to ensure an adequate sampling of the Brillouin zone. Vibrational frequencies spread in k-space, just as the Bloch treatment of electronic energy gave a dispersion of electronic energies in k-space. The number of vibrational levels whose energy lies between E and fc +d E is called the vibrational density of states. Vibrational contributions to the heat capacity and to the crystal entropy can be calculated by appropriate integrations over the vibrational density of states, just like molecular heat capacities and entropies are obtained by summation over molecular vibration frequencies. 

Calculate the equilibrium number of Schottky defects n in an MO oxide at 1000 K in a solid for which the enthalpy for defect formation is 2 eV. Assume that the vibrational contribution to the entropy can be neglected. Calculate AG as a function of the number of Schottky defects for three concentrations, namely, n, all assumptions. Plot the resulting data... 

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