Equations of state Clausius

Cj) Clausius Equation of State. It is a modification of vanderWaals equation, in which is introduced an expression l/T(V+c) to correct the pressure-correction term (a) for its variation with temperature. Clausius equation is given in Ref 8, p 161 in the form ... 

Symbolic determination of enthalpy departure function for the Clausius equation of state... 

Does a fluid obeying the Clausius equation of state have a vapor-liquid transition ... 

Dutta, M. The effect of molecular sice on vapour-liquid equilibria. Indian Chem, Eng. 13 (1971) 29. Elshayal, J. M., and Lu. B. C.-Y. Prediction of vapor-liquid-equilibria by means of a modified Clausius equation of state. Canad. J. chem. Engng. 51 (1973) 76. 

When the functional form of the correlation is suggested by theory, there is a great deal more confidence that the correlation can be extrapolated into regions of P that have no experimental data, and can be used for other families of compounds other than the training set S. Examples of theory-suggested functional forms include the van der Waals equation of state for gases, the Langmuir isotherm for adsorption and catalysis, and the Clausius-Clapeyron equation for the vapor pressure of liquids. 

Clausius (detonation) equation of state 4 D274 Clay 3 C328 Cleacid 3 C328... 

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

Redlich and Kwong proposed an equation of state which takes into account the temperature dependencies of the molecular attraction term in a manner similar to Clausius.14... 

BERTHELOT EQUATION. A form of the equation of state, relating the pressure, volume, and temperature of a gas, and the gas constant R. The Berthelot equation is derived from the Clausius equation and is of the form... 

Equations ol states of various types of systems are numerous. The Curie equation is the equation of state of a paramagnetic solid. The Beattie and Bridge-man equation. Berthelol equation. Clausius equation. Dieterice equation. Keyes equation, and Van der Waals equation are other examples in this category. 

Of course, the more conceptual analysis and criticism of these foundations attracted no attention at first because of the rich yield of results that could be compared with experiment. These results were obtained in fast succession by A. Krdnig (1856), R. Clausius (since 1857), and J. C. Maxwell (1859) through the kinetic interpretation of the equation of state, diffusion, heat conduction, and viscosity.4... 

In his own words, his incentive to his life s work came to me when, after my studies at university, I learned of a treatise by Clausius (1857) on the nature of the motion which we call heat. In this treatise. .. he showed how Boyle s law can very readily be derived on the assumption that a gas consists of material points which move at high velocity (van der Waals, The Equation of State of Gases and Liquids ). This treatise was a revelation, as it derived the ideal gas law from first principles and framed it in terms of atoms and molecules. Here was a theoretical rationalization for experimentally... 

Further, van der Waals recognized that real gases take up volume and atoms do interact with one another. For the German mathematical physicist Rudolf Clausius to obtain his results, he had to ignore both of these factors. Van der Waals found experimentally derived constants that allowed him to modify the ideal gas law to take into account real atoms. His equation of state,... 

Equation 44 can be written in a form closer to VDW by replacing V + Nc — Nb by V — Nb. Clearly this approach, which dates back to Clausius, will result in a reduction of the pressure. However, it is a poor substitute for using the correct hard-sphere equation of state. 

The free-volume concept dates back to the Clausius [1880] equation of state. The need for postulating the presence of occupied and free space in a material has been imposed by the fluid behavior. Only recently has positron annihilation lifetime spectroscopy (PALS see Chapters 10 to 12) provided direct evidence of free-volume presence. Chapter 6 traces the evolution of equations of state up to derivation of the configurational hole-cell theory [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971], in which the lattice hole fraction, h, a measure of the free-volume content, is given explicitly. Extracted from the pressure-volume-temperature PVT) data, the dependence, h = h T, P), has been used successfully for the interpretation of a plethora of physical phenomena under thermodynamic equilibria as well as in nonequilibrium dynamic systems. 

The temperature is tentatively lowered by a few degrees. The loss of enthalpy of the liquid then serves for vaporization. The volume released is replaced by vapour, as long as there is still liquid and the quantity of vaporized liquid is sufficient. Otherwise this quantity is the upper limit. As a consequence we obtain a new value for the pressure. By iteration the temperature is subsequently modified until the values for pressure and temperature lie on the vapour pressure curve (vid. Fig. 10.4). The latter can be determined from approximate equations [13] or the Clausius-Clapeyron relation [19]. The connection between temperature and pressure is ensured by the equation of state for gases. 

Concerning the caloric properties, the enthalpy of vaporization is linked to the vapor pressure curve via the Clausius-Clapeyron equation (Eq. (2.86)). Therefore, the quality is mainly determined by the ability of the ty-function in representing the vapor pressure data, which is usually sufficient (see above). The enthalpy of vaporization can easily be calculated with a cubic equation of state by subtracting the residua] parts of the enthalpy of vapor and liquid in the saturation state... 

The separation factors mainly depend on composition and temperature. The correct composition dependence is described with the help of activity coefficients. Following the Clausius-Clapeyron equation presented in Section 2.4.4 the temperature dependence is mainly influenced by the slope of the vapor pressure curves (enthalpy of vaporization) of the components involved. But also the activity coefficients are temperature-dependent following the Gibbs-Helmholtz equation (Eq. (5.26)). This means that besides a correct description of the composition dependence of the activity coefficients also an accurate description of their temperature dependence is required. For distillation processes at moderate pressures, the pressure effect on the activity coefficients (see Example 5.7) can be neglected. To take into account the real vapor phase behavior, equations of state, for example, the virial equation, cubic equations of state, such as the Redlich-Kwong, Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), the association model, and so on, can be applied. 

A second empirical equation of state is named after Clausius ... 

The vapor pressure of pseudocomponents is also an important property when an equation-of-state approach is not used. All other approaches to process thermo-dynamics require some form of vapor-pressure correlation. The vapor pressure for pure hydrocarbons has been extensively tabulated in many component databases such as DIPPR (Design Institute for Physical Property Research, American Institute of Chemical Engineers) and significant libraries are available in modern process modeling software. Several correlations are available in the literature for the vapor pressure of pseudocomponents. It is important to recall that the vapor pressure and heat vaporization are related through the Clausius-Clapeyron [17] Equation (Equation (1-46)). This relationship imposes a constraint if we wish the model to be thermodynamically consistent In general, most of the popular correlations for vapor pressure such as the Lee-Kesler [9,10] agree well with heat of vaporization correlations and maintain thermodynamic consistency. We present the Lee-Kesler vapor pressure correlation in Equation (1.47). 

Here A, B, and C are empirical parameters that are available for many fluids. Values for Antoine constants can be found in Appendix A. 1. The Antoine equation, an empirical equation, is strikingly similar to Equation (6.11). The Antoine equation brings back a similar theme to that discussed with equations of state (Chapter 4). When an empirical equation s form reflects the basic physics that it is trying to describe, it tends to work better. Why do you think the Antoine equation works better than the Clausius-Clapeyron equation in correlating saturation pressures There are several more complex correlations reported in the literature for as a function of T. These forms will not be covered in this text. 

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