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The virial expansion is a model of thermodynamic equations of state.It expresses the pressure P of a gas in local equilibrium as a power series of the density.This equation may be represented in terms of the compressibility factor, Z, as = + + + This equation was first proposed by Kamerlingh Onnes. [1]
Besides the well-known Pitzer-like equations, there is a simple and easy-to-use semi-empirical model, which is called the three-characteristic-parameter correlation (TCPC) model. It was first proposed by Lin et al. [ 22 ] It is a combination of the Pitzer long-range interaction and short-range solvation effect:
In these equations, the subscript A is for analyte (the solution) and T is for the toluene with the Rayleigh ratio of toluene, R T being 1.35×10 −5 cm −1 for a HeNe laser. As described above, the radius of gyration, R g, and the second virial coefficient, A 2, are also calculated from this equation.
The equation corrects the inaccurate van der Waals repulsive term that is also applied in the Peng–Robinson EOS. The attractive term includes a contribution that relates to the second virial coefficient of square-well spheres, and also shares some features of the Twu temperature dependence.
The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties. In galaxy dynamics, the mass of a galaxy is often inferred by measuring the rotation velocity of its gas and stars, assuming circular Keplerian orbits. Using the virial theorem, the velocity dispersion σ can be
The virial pressure can be derived, using the virial theorem and splitting forces between particles and the container [4] or, alternatively, via direct application of the defining equation = (,,) and using scaled coordinates in the calculation.
This is the virial equation of state and describes a real gas. Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature (or when c = 1 V m {\textstyle c={\frac {1}{V_{m}}}} or P ...
The above equation is obtained by replacing the spatial and temporal derivatives in the previous first order hyperbolic equation using forward differences. Corrector step: In the corrector step, the predicted value u i p {\displaystyle u_{i}^{p}} is corrected according to the equation