Temperature independent force constants

Our development has assumed temperature independent force constants. In real liquids, however, there is a small temperature dependence of frequencies and force constants due to anharmonicities, lattice expansion, etc. The incorporation of these effects into the theory is treated in later sections. 

Van der Linde and Robertson measured the secondary deuterium IEs for the acid dissociation constants of CD3NHj and (CD NHj 52 They found ApAa = 0.056 for the former and 0.12 for the latter, with the deuterated more basic. Surprisingly, these are temperature-independent between 5 and 45°C. Temperature independence requires that the IE lie in the entropy, not the enthalpy, so the zero enthalpy change was attributed to a fortuitous compensation of force constants. A large IE, easily measured, is seen with (CD3)3NH+, which is a stronger base than (CH3)3NH+, with a substantial ApAa, 0.206 at 0°C.53 The ApAa per D increases from 0.017 in methylamine to 0.019 in dimethylamine to 0.021 in trimethylamine. This increase was attributed to a lower IE in a strongly electron-demanding environment. 

The temperature independence of the CH frequency shifts is also reflected in the nearly constant attractive force parameters (see Table I). In fact, the frequency shifts predicted using the average attractive force parameter, Ca = 0.973, reproduce the experimental results to within 3% throughout the experimental density and temperature range. It thus appears that the attractive force parameter may reasonably be treated as a temperature and density independent constant. This behavior is reminiscent of that found for attractive force parameters derived from high pressure liquid equation of state studies using a perturbed hard sphere fluid model (37). 

Considering that dispersion and electrostatic forces are independent of temperature and assuming that the adsorption potential at a constant volume filling is also temperature-independent,... 

Persky and Klein have measured the temperature dependence of the isotope effect over the range — 30 °C—h 70 °C for the photochlorination of isotopic hydrogen molecules by comparing the rate of H2 chlorination with the rate for HD, D2, HT, DT and T2. Assuming a linear transition state, four force constants characterize the potential energy surface contours at the transition state configuration. Four independent isotope effect measurements are necessary to determine the four force constants for comparison with theoretical surface models hence, Persky and Klein have one isotope effect measurement which serves as a test of their method. 

The sound velocity in a fiber, and the sonic modulus calculated therefrom, are related to molecular orientation (De Vries ). As shown by Moseley ), the sonic modulus is independent of the crystallinity at temperatures well below the T (which means that the inter- and intramolecular force constants controlling fiber stiffness are not measurably different for crystalline and amorphous regions at these temperatures). An orientation parameter a, calculated from the sonic modulus, is therefore taken as a measure for the average orientation of all molecules in the sample, regardless of the degree of crystallinity. The parameter is called the total orientation , as contrasted to crystalline and amorphous orientation, from X-ray data. 

To simplify the problems represented by Eqs. (2 88) and (2 93), one of two possible assumptions is normally introduced. If the bounding surfaces are all at the same temperature, so that the only source of heat is viscous dissipation, it is often a good approximation to assume that the fluid is isothermal (i.e., the temperature is a constant, independent of spatial position or time). In this case, we do not need to consider (2 93) at all because it is an equation that is to be used to determine the temperature as a function of position and time. In addition, as noted in Chap. 2, the equation of motion also simplifies when the fluid is isothermal to either the form (2-89) or to (2 91) if the density in the system is also a constant, independent of position. When the temperature at the bounding surfaces is not constant, a different approximation known as the Boussinesq approximation is often used to simplify the problem. A detailed discussion of this approximation is postponed to a later point in the book. Here, we simply note the basic idea, which is that the material properties may be approximated as constants, provided that the temperature changes are not too large. In this case, the values of these properties can be evaluated at a representative temperature of the system, such as its mean value. An exception, as we shall see later, is that we sometimes cannot ignore the spatial variations of the density p in the body-force term of (2-88) even when the temperature changes are modest. Such density variations can produce motion... 

Quartz and piezoelectric ceramic crystals have more temperature independent constants than PVDF, so they are used for force and acceleration transducers. However, PVDF films can be used for large area flexible transducers. Their sensitivity to stress or strain allows the construction of pressure sensors (using the J33 coefficient), and accelerometers by mounting a seismic mass on the film. PVDF electrets are particularly suited for large area hydrophones (Fig. 12.21) that detect underwater signals. Their... 

To obtain an accurate estimate of thermodynamic properties for crystalline silica polymorphs, one needs an accurate description of the phonon density of states. Given the complexity of the problem, this is tractable only with several assumptions. For example, it is often assumed that the calculated phonon spectra are not strongly dependent on temperature. Certainly this will be a satisfactory assumption in the absence of any thermal expansion, and any changes in the interatomic potentials as a function of temperature. In this case, the force constants, and consequently the dynamical properties like phonon frequencies and density of states, will be independent of temperature. 

We simplify our considerations for the calculation of the van der Waals constant a furthermore in that we assume that the estimated interaction is valid up to a sharp boundary Rq of the molecule and oo for smaller distances. Without any doubt the higher multipoles and especially the exponentially decreasing repulsion forces of the first order manifest themselves in the medium distance, and apart fix)m this we know that the repulsion forces start gradually and that a sharply defined (i.e. temperature-independent) molecular radius does not exist. All the same our procedure reflects the same idealisations that we used previously in the discussion of this problem. For H atoms, where the force field is known, the situation can be checked in this respect and [we] find that the correct value deviates by 50% from the one calculated based on these simplifications. We are therefore not too unreasonable if we assume an uncertainty of 50% in our calculations, especially when we realise that the van der Waals equation itself allows for only a crude representation of the behaviour of real gases. But in my opinion it makes little sense to bring our presently very imprecise knowledge of molecular forces into agreement with more precise formulations of the equation of state. 

Structure and dynamics of molecules includes the geometric structure (interatomic distances and angles) as well as vibrational frequencies, force constants (see Force Fields A General Discussion), barriers to internal rotation, ionization energies, dipole moments, etc. These are intrinsic molecular properties, independent of temperature and pressure. 

Because of anharmonicity, which we discuss in Chap.5, the elastic constants and the atomic force constants are not independent of temperature. For the elastic constants, an increase of 10% or more in going from ordinary temperatures to 0 K is fairly typical. 

The often-cited Amontons law [101. 102] describes friction in tenns of a friction coefiBcient, which is, a priori, a material constant, independent of contact area or dynamic parameters, such as sliding velocity, temperature or load. We know today that all of these parameters can have a significant influence on the magnitude of the measured friction force, especially in thin-film and boundary-lubricated systems. 

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