Clausius virial

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

Clausius-Mossotti equation). In this expression, V designates the mole volume and Ae, Be, Cf,... are the first, second, third,... virial dielectric coefficients. A similar expansion exists for the refractive index, n, which is related to the (frequency dependent) dielectric constant as n2 = e (Lorentz-Lorenz equation, [87]). The second virial dielectric coefficient Be may be considered the sum of an orientational and a polarization term, Be = B0r + Bpo, arising from binary interactions, while the second virial refractive coefficient is given by just the polarization term, B = Bpo at high enough frequencies, the orientational component falls off to small values and the difference Be — B may be considered a measurement of the interaction-induced dipole moments [73],... 

Clausius-Mossotti equation). The AE, BE,. .., are the first, second,. .., dielectric virial coefficients, given by... 

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... 

It is well known that for atomic gases at low densities the Clausius-Mossotti function can be related to the atomic polarizability via the following virial relation ... 

Empirical models for the induced trace have also been obtained from (nonspectroscopic) measurements of the second virial dielectric coefficient of the Clausius-Mosotti and Lorentz-Lorenz expansions [30]. Excellent surveys with numerous references to the historical as well as the modern dielectric research activities were given by Buckingham [27], Kielich [89], and Sutter [143] in 1972 see also a recent review with a somewhat more spectroscopic emphasis [11]. 

In the original derivation of the classical virial theorem given by Clausius, an expression corresponding to eqn (5.30) is also obtained. In the classical case one argues that the time average of d(f p)/dt vanishes over a sufficiently long period of time or that the motion is periodic to obtain the equivalent of eqn (5.31). ... 

That this is indeed the differential form of the customary virial theorem is readily seen by multiplying Eq. (26) throughout by x and then integrating over all x from —oo to +00. Some elementary integrations by parts recovers the usual (integral) virial theorem of Clausius, in, of course, now fully quantum-mechanical form [54]. 

The virial theorem of Clausius estabhshes the ratio of the average kinetic energy K to the average potential energy V of a mechanical system consisting of i particles [28, 33—36). The virial of the force F is equal to the total average kinetic energy K ... 

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. 

The isostetic enthalpy of adsorption, Affads, can be determined by IGC, based on the changes in retention times or retention volumes with column temperature, employing the Clausius-Clapeyron equation for calculations, or, alternatively, using the second adsorption virial coefficient... 

TUs is often called the viiial equation for the pressure since it can also be derived firom the virial theorem of classical mechanics of Clausius. The product ru (r) is the virial of the pair potential u(r). 

The corresponding virial expression for the isosteric heat can be directly obtained through the Clausius-Clapeyron equation for adsorption equilibrium [5,17,196] ... 

The virial theorem was first stated by Clausius in 1870 for the expectation value of the product between the position of each particle and the force acting on it. Indeed, substituting in Eq. 8.15 for the position of a particle, and using Hamilton s equation of motion, yields... 

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