Adiabatic bulk modulus

R. ZwanzigandR. D. Mountain, J. Chem. Phys. 43,4464 (1965) show that the modulus Goo and the isothermal compressibility are determined by similar integrals containing the pair correlation function and the interparticle potential for simple Lennard-Jones fluids. The adiabatic (zero frequency) bulk modulus Ko equals —y(0P/0P) j, which clearly is a kind... 

A fluid cannot support shear waves over any appreciable distance, so only longitudinal waves can propagate in a fluid. In this case the appropriate modulus is the adiabatic bulk modulus B. Extending to three dimensions with the Laplacian operator... 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

Thus a measurement of the ultrasonic velocity and density can be used to determine the adiabatic compressibility (or bulk modulus) of the material. For homogeneous solids measurements of the compression and shear velocities can be used to determine the bulk and shear moduli (see section 2.4). The Young s modulus of rod-like materials (e.g. spaghetti) can be determined by measuring the velocity of ultrasound. 

Most seismological constraints on mantle composition are derived by comparison of values of seismic wave velocities inferred for particular regions within the Earth to the values measured in the laboratory for particular minerals or mineral assemblages, with such comparisons being made under comparable regimes of pressure (P) and temperature (T). The primary parameters of interest, then, are the compressional (or P-) wave velocities (Vp) and the shear (or S-) wave velocities (Ej). These wave velocities are simply related to the density (p) and to the two isotropic elastic moduli, the adiabatic bulk modulus (Ks)... 

This isothermal bulk modulus (Kj) measured by static compression differs slightly from the aforementioned adiabatic bulk modulus (X5) defining seismic velocities in that the former (Kj) describes resistance to compression at constant temperature, such as is the case in a laboratory device in which a sample is slowly compressed in contact with a large thermal reservoir such as the atmosphere. The latter (X5), alternatively describes resistance to compression under adiabatic conditions, such as those pertaining when passage of a seismic wave causes compression (and relaxation) on a time-scale that is short compared to that of thermal conduction. Thus, the adiabatic bulk modulus generally exceeds the isothermal value (usually by a few percent), because it is more difihcult to compress a material whose temperature rises upon compression than one which is allowed to conduct away any such excess heat, as described by a simple multiplicative factor Kg = Kp(l + Tay), where a is the volumetric coefficient of thermal expansion and y is the thermodynamic Griineisen parameter. 

Bulk modulus can be treated from the adiabatic as well as the isothermal point of view. Phenomenologically adiabatic compression or expansion are processes where heat is neither lost to nor gained from the environment. If the process occurs under equilibrium conditions, then we have the thermodynamically tractable case at zero entropy change and we define the bulk modalui as... 

Do you expect the adiabatic bulk modulus of a material to be greater or less than the isothermal value ... 

The velocity of the wave is related to the density (p) and the elastic constant (Cm) of the medium through which it is propagating (in the equation shown below). The elastic constant is unique to the mode of propagation and to the material. For example, in liquids Cm is the adiabatic bulk modulus (B) ... 

Often the compressibility ( t ) is used rather than the bulk modulus (K). These quantities are simple reciprocals for isotropic materials, i.e., x = Ultrasonic methods for measuring elastic properties are inherently adiabatic processes. Compressibility, on the other hand, is frequently determined by hydrostatic pressurization techniques which are isothermal in nature. Conversion of isothermal values ( t) to adiabatic values ( s) is accomplished via the following relation ... 

Throughout this chapter the values for a, w and Cp given by Gschneidner (1964) were used to convert isothermal compressibilities to adiabatic values. In all cases, however, the reciprocal of adiabatic compressibility ( s), i-e. the bulk modulus (K), is quoted in this chapter. From the experimental point of view, the most common method used to determine the isothermal compressibility ( t) involves the measurement of volume change (AV/Vo) associated with applied pressure (P) to obtain a relative volume change versus pressure relation. Several... 

Bridgman (1954) obtained a room temperature adiabatic bulk modulus of 38.2 GPa (coverted by the writer from the isothermal value using eq. (8.5) and Gschneidner s (1964) values for a, Cp and a>). This result is somewhat less than Rosen s value of 44.5 GPa. Rosen s results are presented in the summary, table 8.22. 

Bulk modulus K is the ratio of change in pressure/volume compression (fractional change in volume), where K = V(APIAV). For solids, apphed stress is used instead of pressure, and A is a constant for a given material. When there is an adiabatic exchange of heat, we have bulk modulus at constant entropy [13]. When the bulk modulus is determined at constant temperature, we have the isothermal bulk modulus [13]. 

In comparing elastic constants measured acoustically with those obtained in a static (very low frequency) test, note that acoustic values are measured under adiabatic conditions, while static values are isothermal. The two t5q>es of bulk modulus measurements are related by the standard thermodynamic relation... 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

Due to the rapidity of pressure variations, hardly any heat is exchanged between the surroundings and the volume of air that is undergoing the pressure variations. It is essentially an adiabatic process. As a first approximation, we may assume the ideal gas law is valid for these rapid changes. In introductory physics texts it is shown that the velocity of sound, Csound> in a medium depends on the bulk modulus B and the density p according to the relation... 

As discussed below, sound propagation in heavy-fermion materials is believed to be adiabatic. When the bulk modulus is measured via ultrasonics, the appropriate parameter is the adiabatic bulk modulus Bg, which is related to the isothermal bulk modulus by a thermodynamic relation (Yoshizawa et al. 1986, Thalmeier 1988) ... 

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