Bulk isothermal compressibility

A MC study of adsorption of living polymers [28] at hard walls has been carried out in a grand canonical ensemble for semiflexible o- 0 polymer chains and adsorbing interaction e < 0 at the walls of a box of size C. A number of thermodynamic quantities, such as internal energy (per lattice site) U, bulk density (f), surface coverage (the fraction of the wall that is directly covered with segments) 9, specific heat C = C /[k T ]) U ) — U) ), bulk isothermal compressibility... 

These are the 2D analogs of the bulk isothermal compressibility and bulk modulus, respectively. The scaling concepts introduced above clearly make sense in terms of the compressibility/elasticity as well. For polymers where the interface is a good solvent, the lateral modulus, sometimes called static di-lational elasticity, is small whereas it becomes larger as the interface becomes poorer. 

The above discussion leads to the consideration of another empirical rule involving the (bulk) isothermal compressibility. It is defined as... 

The limiting compression (or maximum v value) is, theoretically, the one that places the film in equilibrium with the bulk material. Compression beyond this point should force film material into patches of bulk solid or liquid, but in practice one may sometimes compress past this point. Thus in the case of stearic acid, with slow compression collapse occurred at about 15 dyn/cm [81] that is, film material began to go over to a three-dimensional state. With faster rates of compression, the v-a isotherm could be followed up to 50 dyn/cm, or well into a metastable region. The mechanism of collapse may involve folding of the film into a bilayer (note Fig. IV-18). 

The monolayer stability limit is defined as the maximum pressure attainable in a film spread from solution before the monolayer collapses (Gaines, 1966). This limit may in some cases correspond directly to the ESP, suggesting that the mechanism of film collapse is a return to the bulk crystalline state, or may be at surface pressures higher than the ESP if the film is metastable with respect to the bulk phase. In either case, the monolayer stability limit must be known before such properties as work of compression, isothermal compressibility, or monolayer viscosity can be determined. 

Equations of state for solids are often cast in terms of the bulk modulus, Kp, which is the inverse of the isothermal compressibility, Kp, and thus defined as... 

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... 

The slope of surface pressure isotherms is a measure of their compressibility the steeper it is, the more difficult it is to compress the monolayer. Recall [2.11.4], where the isothermal bulk compressibility was defined as -(31n V/3p)j,. By analogy we introduce the two-dimensional isothermal compressibility through... 

In eqs Al-1—Al-3, k is the Boltzmann constant, T is the absolute temperature, Np is the number of particles of species / in the volume v, is the chemical potential per molecule of species a, Va is the partial molar volume per molecule of species a, kf is the isothermal compressibility, and Ca is the bulk molecular concentration of component a (ca = Na/v). The derivative (dfiJdNp)T,v y f is taken rmder isothermal—isochoric conditions and with Ny = constant for any y B o/3 represents the cofactor... 

Statistical mechanics gives relationships between the distribution functions and the bulk properties of fluids. The total internal energy of a fluid is given by the energy equation, the pressure is given by the virial equation, and the isothermal compressibility is given by the compressibility equation, see e. g.. Ref. 11. Through the Kirkwood-Buff formulas (0,... 

The bulk modulus K is defined as the reciprocal of the isothermal compressibility, and Young s modulus E is defined as the ratio of longitudinal tensile stress and longitudinal tensile strain ... 

Diffraction experiments at high pressures provide information concerning the compression-induced changes of lattice parameters and, thus, sample volume. In pure phases of constant chemical composition and in the absence of external fields, the thermodynamic parameters volume V, temperature T and pressure P are related by equations of state, i.e. each value of a state variable can be defined as a function of the other two parameters. Some macroscopic quantities are partial differentials of these equations of state, e.g. the frequently used isothermal bulk modulus Bq of a phase at a defined temperature and zero pressure 5q = — Fq (9P/9F) for T= constant and P = 0, with the reciprocal of Bq V) being the isothermal compressibility k. Equations of state can also be formulated as derivatives of thermodynamic functions like the internal energy U or the Helmholtz free-energy F. However, for practical use the macroscopic properties of solids are often described by means of semi-empirical equations, some of which will be discussed in more detail. 

The constant temperature process is a case when n=l, which is equivalent to isothermal compression, the constant pressure process n = 0 and the constant volume process n = Generally, it is impractical to build sufficient heat transfer equipment into the design of most compressors to convey the bulk of the heat of compression. Therefore most machines tend to operate along a poly tropic path that approaches the adiabatic. Most compressor calculations are based on the adiabatic curve [3]. 

Much of the work on the compressibility and bulk modulus of liquids reported in the literature was motivated by problems in mass hydraulic flow, such as raising a hydraulic fluid to a pressure in the range 68.9-137.8 MPa (10,000-20,000 Ib/in ) and circulating it through the hydraulic system. In this type of problem most of the emphasis is on the isothermal compressibility of the fluid. 

Because the free energy of formation of a surface is always positive, a particle that consists only of surfaces (that is, platelets or droplets of atomic dimensions) would be thermodynamically unstable. This is also apparent from the Kelvin equation [Eq. 3.70], which states that a particle that falls below a certain size will have an increased vapor pressure and will therefore evaporate. There must be a stabilizing influence, however, that allows small particles of atomic dimensions to form and grow a common occurrence in nature. This influence is given by the free energy of formation of the bulk condensed phase. In this process, n moles of vapor are transferred to the liquid phase under isothermal conditions. This work of isothermal compression is given by... 

When considering the potential effect of pressure on a system, it is useful to recognize the magnitude of pressure required to significantly alter molecular and bulk properties. The isothermal compressibility, k (or its reciprocal K, the bulk modulus) (Eq. 2), gives an indication of the sensitivity of a system to pres-... 

For the evaluation of Gcav several formulas are available, based on the shape and size of the solute and on different parameters of the solvent surface tension, isothermal compressibility, and geometrical data of the molecules. The first three formulas here mentioned are of empirical nar ture and follow almost the same philosophy of the continuum dielectric, neglecting the discrete nature of the solvent molecules but making use of experimental bulk parameters. The last formulation, on the contrary, derives from a theory based on a discrete model of fluids (the Scaled Particle Theory, SPT), even if the final expression of Gcav depends again on bulk solvent parameters only. 

The isothermal compressibility, written as or k, is the reciprocal of the bulk modulus. Beware of the... 

The bulk phase diagrams of pure hydrocarbons and mixtures are well known from the experiments. In the work by Sage et al. [3], the bubble point pressures of methane + n-butane mixtures are determined experimentally from the discontinuity of isothermal compressibility of constant-composition mixture at the point of phase transition. The composition of vapor phase is determined in that work from the residual specific volume of gas. Later experiments employ phase recirculation techniques [4] to achieve vapor-Uquid equilibrium [5, 6], and the phase compositions are analyzed by more advanced methods such as gas chromatography. 

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... 

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