Isothermal compressibility equations

This coefficient normally is referred to simply as compressibility or gas compressibility. You must understand that the term compressibility is used to designate the coefficient of isothermal compressibility whereas, the term compressibility factor refers to z-factor, the coefficient in the compressibility equation of state. Although both are related to the effect of pressure on the volume of a gas, the two are distinctly not equivalent. 

The compressibility equation is the most commonly used equation of state in the petroleum industry. We will combine this equation with the equation which defines the coefficient of isothermal compressibility. Since z-factor changes as pressure changes, it must be considered to be a variable. 

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 ideal compression (in isothermal conditions) work can be calculated according to the following equation ... 

A virial equation is a power-series expansion of the compressibility factor isotherm in molar density from the zero-density ideal gas, z = 1 at p = 0, which is the common point of intersection of aU z isotherms. 

The energy equation isotherm for t = 0.11 is compared with the corresponding compressibility and pressure equation isotherms in Fig. 4. The three isotherms are very different. 

The calculation for supercritical isotherms is simpler. In this case there is no saturation pressure and the compressibility equation has one real root only. Therefore, steps 2 and 4 are skipped, otherwise the calculation is identical. 

Plot P= i< versus Hl and on the same graph plot P < versus Hv. This will give the vapor-liquid boundary. To add isotherms, first tabulate H at various pressures. Start at a pressure close to the ideal-gas state and calculate the enthalpy. If the compressibility equation has three real roots, use the largest one, since the calculation is done for the vapor. Increase the pressure in small steps and repeat the calculation until you reach P-. Past P = P use the smallest root, since the calculation is now done for the liquid. The isotherm is obtained by plotting the results of this calculation as P versus H. 

Consider the behaviour of the temperature dependence of isothermal compressibility (Equation 1.1.2-41) while the configurative point approaches from the side of lower temperatures T - T along the critical isochore (v = 0). 

Thus, the light scattering intensity is immediately related to fluctuations in the number of particles in the system (Equation 105) and the isothermal compressibility (Equation 107). If the radial function of density distribution is permitted to be spherically... 

Equations of state in the first category are for the empirical representation of volumetric data. The majority of equations in this group are the so-called isothermal compressibility equations (1-4). The purely empirical, original Tait equation and its modification by Tammon have been almost exclusively used for organics, including polymers (2,3,5,6). 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived ... 

The chapter on equation-of-state properties provides the basic approaches used for describing the high-pressure shock-compression response of materials. These theories provide the basis for separating the elastic compression components from the thermal contributions in shock compression, which is necessary for comparing shock-compression results with those obtained from other techniques such as isothermal compression. A basic understanding of the simple theories of shock compression, such as the Mie-Gruneisen equation of state, are prerequisite to understanding more advanced theories that will be discussed in subsequent volumes. 

Using Equation 2.84 to establish the theoretical limit of isothermal compression,... 

Isothermal compression takes place when the heat of compression is removed during compression and when the temperature of the gas stays constant. The characteristic equation is... 

The passage of a sound wave along a tube, so that no energy is dissipated by friction, is an example of a compressional wave of permanent type, and Newton applied his equation (1) to determine the velocity of sound in air. For this purpose he took e as the isothermal elasticity of air, which is equivalent to assuming that the temperature is the same in all parts of the wave as that in the unstrained medium. Since air is heated by compression and cooled by expansion, the assumption implies that these temperature differences are automatically annulled by conduction. Taking the isothermal elasticity, we have ... 

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