Vibration heat capacity

The vibrational heat capacity is the largest contribution to the total heat capacity and determines to a large extent the entropy. Analytical expressions for the entropy of the models described in the previous section can be derived. The entropy corresponding to the Einstein heat capacity is... 

Alternatively, when process (3) is slower than (4) or (5), but faster than (1) or (2), A will again relax by the route (3) followed by (4) or (5), but now (3) will be rate determining. This will give a linear variation of 1// A with x. B will relax independently, and more rapidly, via (4) and (3), with linear dependence of 1// B on x. There will thus be a double relaxation phenomenon with two relaxation times, PA involving only the vibrational heat capacity of A, and / B only that of B, both showing linear concentration dependence. This mechanism is analogous to the relaxation behaviour discussed in Section 3.1 for pure polyatomic gases, which show double dispersion because vibration-vibration transfer between modes is slower than vibration-translation transfer from the lowest mode. 

Several attempts have been made to make s more than just a fitting parameter without losing the simplicity of the model. For example Benson suggested that the number of effective oscillators s could be estimated from the molar vibrational heat capacity, s = [15], and Troe has similarly used the energy,... 

Molecular vibrations contribute to the heat capacity, but only when the temperature is high enough lor them to be significantly excited. For each vibrational mode, the equipartition mean energy is kT. so the maximum contribution to the molar heat capacity is R. However, it is very unusual for the vibrations to be so highly excited that equipartition is valid, and it is more appropriate to use the full expression lor the vibrational heat capacity which is obtained by differentiating eqn 17.28. The curve in Figure 17.12 of the... 

The total vibrational heat capacity is obtained by summing the last column (twice for the first two entries, since they represent doubly degenerate modes),... 

The vibrational heat capacity, which rises linearly with T and plays, therefore, an important part in the position of the minimum of the measured Cp values... 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

We see that, at low temperatures, the surface heat capacity Cp is proportional to T, as opposed to the dependence of the bulk-heat capacity. However, the model we have considered here, consisting of a surface layer of atomic thickness, is quite unrealistic it would be difficult to measure the heat capacity of a single atomic layer. In most cases the solid samples that can be used in experiments are small particles of variable surface/volume ratio or thin films many atomic layers thick. It would therefore be important to consider the heat capacity of such a sample and to see what contribution, if any, the surface makes to the total vibrational heat capacity. 

Thus it is only at high temperatures that the vibrational heat capacity attains the classical value, R. It is customary to define a characteristic temperature 9 = hv/k for each oscillator. Then ... 

A salt such as NaN03 has a vibrational heat capacity of 6R contributed by vibrations of Na and NO/" in the solid and an additional contribution from the vibrations within the nitrate ion, which are partly but not fully excited. Some metals, notably the transition metals, exhibit values of C greater than 3R at high temperatures this extra contribution comes from the heat capacity of the electron gas in the metal. 

For the case of a neat beam of a polyatomic molecule with rotational and vibrational heat capacities, and c, the above equation is written as... 

As the gas exits the nozzle, the temperature and gas densities drop rapidly. The rate of decrease depends only upon 7, the ratio of the constant pressure to constant volume heat capacity. For an ideal gas 7 = (C + R)/C. Its value thus ranges from a maximum of 5/3 = 1.67 for monatomic gases to 7 = 1 for a large molecule with a very large vibrational heat capacity. The beam and stagnation temperatures and gas densities are related by the equations (D.R. Miller, 1988) ... 

Equations (5.8) and (5.9) have been derived under the assumption that 7 is constant throughout the expansion. Although this is true for a monatomic gas, it is not true for the vibrational heat capacity of polyatomic gases which become smaller as the temperature drops. 

Fig. 12. Effects of relaxation processes, a) Frequency dispersion in CH. Due to the small vibrational heat capacity in CHt, a relatively small frequency shift is obtained, b) Broadening of the resonance curve in the pressure region where relaxation occurs...

To look into this further, we show in Fig 4, in part (a), the behavior of the heat capacity of polypropylene, in units of J/K.(mol of -CH2-CH2(CH3)- repeat units) (35,36) in comparison with that of the molecular liquid 3-methyl pentane (37) (divided by 2 to have the same mass basis as the polymer repeat unit) (38). It is seen that the liquid heat capacity of the hexane isomer (x 0.S) falls not much above the natural extrapolation to lower temperatures of the heat capacity per repeat unit of the polymer. This implies that the main effect of polymerization, as far as the change in heat capacity at Tg is concerned, is to postpone the glass transition until a much higher vibrational heat capacity has been excited. This not only reduces the value of ACp but has a disproportionate effect on the ratio Cp,i/Cp,g at Tg. This happens despite a lower glassy heat capacity in the polymer than in the molecular liquid at the same temperature. The latter effect is a direct consequence of the lower Debye temperature (and lower vibrational anharmonicity) at a given temperature for in-chain interactions in the polymer than for intermolecular interactions in the same mass of molecules. 

The curves of Figure lA are plotted with Cp (liquid) - Cp = ACp. The experimental data of ACp were, in addition, first normalized to die equilibrium difference of the liquid and vibrational heat capacities. Equation (8) allows then the transformation to y shown in Figure IB. Using the values of y of Figure IB, a plot of In x vs, /T can be drawn, as shown in Figure 2A cos y = (1/t)/[(1/x) + o>. Clearly, different... 

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