Entropy vibrational contribution

Thus the entropy of localized adsorption can range widely, depending on whether the site is viewed as equivalent to a strong adsorption bond of negligible entropy or as a potential box plus a weak bond (see Ref. 12). In addition, estimates of AS ds should include possible surface vibrational contributions in the case of mobile adsorption, and all calculations are faced with possible contributions from a loss in rotational entropy on adsorption as well as from change in the adsorbent structure following adsorption (see Section XVI-4B). These uncertainties make it virtually impossible to affirm what the state of an adsorbed film is from entropy measurements alone for this, additional independent information about surface mobility and vibrational surface states is needed. (However, see Ref. 15 for a somewhat more optimistic conclusion.)... 

The entropy difference A5tot between the HS and the LS states of an iron(II) SCO complex is the driving force for thermally induced spin transition [97], About one quarter of AStot is due to the multiplicity of the HS state, whereas the remaining three quarters are due to a shift of vibrational frequencies upon SCO. The part that arises from the spin multiplicity can easily be calculated. However, the vibrational contribution AS ib is less readily accessible, either experimentally or theoretically, because the vibrational spectrum of a SCO complex, such as [Fe(phen)2(NCS)2] (with 147 normal modes for the free molecule) is rather complex. Therefore, a reasonably complete assignment of modes can be achieved only by a combination of complementary spectroscopic techniques in conjunction with appropriate calculations. 

From a thermodynamic perspective, Stillinger and Weber demonstrated that the total entropy of the liquid can similarly be divided into two additive terms, a configurational and a vibrational contribution.5,6 The configurational part Sc measures the number of structurally distinct basins of attraction on the PEL that the configuration point accesses at a given temperature, whereas the vibrational contribution Svib characterizes the number of states associated with intra-basin fluctuations. Thus, the AG relationship, when viewed from the PEL perspective, suggests that it is the thermodynamic availability of basins on the landscape that dominates the rate of liquid-state diffusive processes. 

These ideas have been employed to compute configurational entropy and hence test the AG relation via molecular simulation of several model systems.91-94,101,102 The approach used in those studies is conceptually simple. First, the total entropy of the fluid S is calculated by integration of standard thermodynamic relationships, for example, as discussed below. Then, the configurational contribution to the entropy Sc = S — Sv b, is approximated by subtracting from the total entropy an estimate for the vibrational contribution, Svib. 

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... 

Aminomethylpyridine (picolylamine) is an important ligand in respect to spin cross-over, [Fe(2-pic)3]Cl2 being the key compound." Fleat capacity measurements on [Fe(2-pic)3]Cl2 EtOH gave values of 6.14kJmol and 50.59 JK moC for the spin eross-over entropy the determined entropy was analyzed into a spin contribution of 13.38, an ethanol orientational effeet of 8.97, and a vibrational contribution of 28.24 JK mol. " This compound exhibits weak cooperativity in the solid state." The heat capacity of [Fe(2-pic)3]Cl2 MeOH is consistent with very weak cooperativity." [Fe(2-pic)3]Br2 EtOH shows a lattice expansion significantly different from that expected in comparison with earlier-established behavior of [Fe(2-pic)3]Cl2 EtOH." ... 

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. 

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. 

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). 

Spin-crossover phase transition of a manganese(IU) complex [Mn(taa)] was studied by variable-temperature laser Raman spectroscopy and it was found that the vibrational contribution in the transition entropy is not dominant in contrast to the cases of ordinary iron spin-crossover systems. The discovery of a dynamic disorder in the HS phase by means of dielectric measurements provided an alternative entropy source to explain the thermally induced spin-crossover transition. This dynamic disorder was attributed to the reorienting distortion dipoles accompanying the E e Jahn-Teller effect in HS manganese(III) ions. 

The excess entropy has contributions from both vibrational and configurational degrees of freedom. 

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). 

In these calculations the electronic contributions have been assumed to cancel, and vibrational assignments and internal rotation barriers were calculated according to Pitzer. Other assumptions have been discussed by the authcjrs. The vibrational contributions (not shown in table) nearly cancel each other near 300°K and can be neglected below 500°K in the calculation of AiS° for the reactions, so that the AaS° shown in the last two lines of Table XII.3, aside from symmetry changes, can be equated to the standard entropy of activation. 

This difficulty is much less in the computation of frequency factors which involve the computation of entropies of rotation and vibration. For an error of 10 in ky the error in aot must be 4.6 cal/mole-°K. The approximate methods discussed in this chapter for computing S ct are generally about this good because vibrational contributions, the most difficult to assess, are generally quite small. 

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). 

Calculate the librative and vibrational contributions for the entropy of an Na" ion in dilute solutions assuming 6 water molecules in the first shell. Why is it... 

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... 

For accurate results, should be the combined rotational and vibrational partition function derived from the actual energy levels of the molecule as obtained from spectroscopic measurements ( 16k). For most purposes, at ordinary temperatures, very little error results from the separation of Qi into the product of two independent factors, viz., Qr and Q., representing the rotational and vibrational partition functions, respectively. Because equation (24.18) involves Q< in logarithmic terms only, it follows that an expression of the same form can be used to give the separate rotational and vibrational entropies. Thus, if is replaced by Qr, the result is Sr, the rotational contribution to the entropy, and similarly the vibrational contribution , is obtained by using Q, for Qi in equation (24.18). The sum of Sr and Sv derived in this manner represents Si, which added to St, as given by equation (24.14), etc., yields the total entropy. 

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