Entropy translational motion contribution

The term F0 in Eq. (1) represents the contribution from the entropy of translational motion of free counter ions and salt ions within the network ... 

By far the largest contribution to a gas-phase species entropy comes from translational motion. Equation 8.98 provided a means to calculate this contribution ... 

First of all, in the solution of disconnected rods each rod has the freedom of an independent translational motion while in the case under consideration, only the chain as a whole (but not each segment) can move independently. Hence the contribution of the translational entropy to the free energy is equal to... 

Indicate whether each statement is true or false, (a) The third law of thermodynamics says that the entropy of a perfect, pure crystal at absolute zero increases with the mass of the crystal, (b) Translational motion of molecules refers to their change in spatial location as a function of time, (c) Rotational and vibrational motions contribute to the entropy in atomic gases like He and Xe. (d) The larger the number of atoms in a molecule, the more degrees of freedom of rotational and vibrational motion it likely has. 

A detailed study of crystals of macromolecules 20,21) and their melting under equilibrium conditions revealed that the entropy of fusion, ASf, is often about 7-12 J/(K mol) per mobile unit or "bead" (22). This entropy is linked mainly to the conformational disorder (A and mobility that is introduced on fusion. Sufficiently below the melting temperature, disorder and thermal motion in crystals is exclusively vibrational. While vibrations are small-amplitude motions that occur about equilibrium positions, conformational, orientational, and translational motions are of large amplittide. These types of large-amplitude motion can be assessed by their contributions to heat capacity (23), entropy (22), and identified by relaxation times of the nuclear magnetization 24), Orientational and positional entropies of fusion ASQ ent trans importance to describe the fusion of small molecules. They can be deriv from the many data on fusion of the appropriate rigid, small molecules of nonspheiical and spherical shapes [nonspheiical molecules Walden s rule (1908), ASf = AS j ent AStrans 20-60 J/(K mol) and spherical molecules Richards rule 0 97), ASf = trans = 2-14 J/(K mol)].. The contributions of ASQ ent melting of... 

It clearly illustrates the asymmetry between the interaction contribution and the entropy contribution. The linking of small molecules into maaomolecules converts almost all of the translational motions into internal conformational motions. The latter are responsible for the fact that polymer conformations tend to be very open. The number of interactions experienced by a maaomolecule of length N is still of the order N, whereas its thermal translational energy is of the order kT, independent of N. 

Rudolf Julius Emanuel Clausius (1822-1888), a German physicist and mathematician, was one of the founders of thermodynamics. By his restatement of Carnot s principle, he put the theory of heat on a sounder basis. His most important paper On the mechanical theory of heat (1850) first stated the ideas of the second law of thermodynamics. In 1865, he introduced the concept of entropy. He also contributed to the kinetic theory of gases by including translational, rotational, and vibrational molecular motions, and introduced the mean free path of a particle. Clausius deduced the Clausius-Cla-peyron relation - see Eq. (3.1.45) below - based on thermodynamic considerations. This law on phase transition had originally been developed in 1834 by Emile Clapeyron. 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The absolute entropies of small molecules can be calculated by statistical mechanical methods. Table 2.1 shows the results of such calculations for liquid propane. The largest contributions to the entropy come from the translational and rotational freedom of the molecule, and much smaller contributions from vibrations electronic terms are insignificant. Although exact calculations of this type become intractable for large biological molecules, the relative sizes of the contributions from different types of motions are similar to those in small molecules. Thus entropy is associated primarily with translation and rotation. This relationship is very different from enthalpy, in which electronic terms are dominant and translational and rotational energies are comparatively small. 

Table 3.1 displays the entropy changes of melting and vaporization for some pure substances. The entropy of vaporization is proportional to the ratio of the degree of randomness in the vapor and liquid phases. For a pure component, A.S. consists of translational, rotational, and conformational motion of molecules. The translational effect is the largest contribution to the entropy of vaporization. 

In case that the energy of a molecule can be represented as the sum of several terms (such as rotational, vibrational, electronic, and translational energy), the Boltzmann factor can be written as the product of individual Boltzmann factors, and the contributions of the various energy terms to the total energy of the system in thermodynamic equilibrium and to the heat capacity, entropy, and other properties can be calculated separately. To illustrate this we shall discuss the contributions of rotational and vibrational motion to the energy content, heat capacity, and entropy of hydrogen chloride gas. 

Typical entropy contributions from translational, rotational, and vibrational motions at 298 K [14]... 

All types of motion (i.e., translation, rotation, and vibration) contribute to the entropy of an ideal gas. The following equations that have been discussed in this... 

In the case of water, the situation is complicated because of the anisotropic nature of the potential. Thus, we have effective harmonic potential for translation, rotation, and librational motions. Each is characterized by a force constant and contributes to the partition function, free energy, and entropy. Furthermore, a water molecule can be categorized by the number of HBs it forms. Since these quantities can be considered as thermodynamic, they make a contribution as the entropy of mixing, also known as the cratic contribution. 

While accurate calculation of the entropy of pure water turned out to be a formidable task, it is, however, not hard to rationalize why the entropy of liquid water is much smaller than that in the ideal gas limit. First, the translational entropy is lower because of the excluded-volume effect. The volume available to a water molecule is defined by its neighbors and the specific volume in the Sackur-Tetrode equation is to be replaced by the free volume available to individual water molecules. When this is taken into account, we obtain a contribution of 11.8 caFK -mor from translational entropy. Second, the rotational contribution also gets reduced because of the restriction ftiat rotational motion experiences in liquid water due to hydrogen-bonding. The reduced value of the entropy... 

Fig. 17.2(e) shows the ratio of the vibrational contribution to the sum of translational and rotational contributions. Even at the highest temperature the vibrational contribution to the entropy change is less than 2.5% of the contributions from translational and rotational motion. The vibrational contribution is negligible at low temperature. 

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