First virial coefficient

The coefficients B and C are the second and third virial coefficients, respectively, the first virial coefficient being 1. 

When we make a plot of Z versus 1/V, we expect that the value of Z = 1 when 1/y = 0 or at infinite volume. The first virial coefficient B T) is the slope of the line at 1/y = 0. B T) is usually negative at very low temperatures due to the attractive forces. But when the temperature increases, B T) will turn positive, and the temperature at which B becomes zero is called the Boyle temperature, since that is the temperature where Boyle s law applies exactly (at infinite volume). 

These expansions are known as the virial expansions,10 and the coefficients B, B , C, C, and so forth, which are functions of temperature, are known as the virial coefficients. B(T) and B (T) are called the second virial coefficients. (The first virial coefficient is unity.)... 

The various virial coefficients can be determined from the P-V-T data for real gases. It is possible to calculate them from a knowledge of intermolecular potential energy. Unfortunately, however, this potential is not accurately known and hence the calculation of higher virial coefficients is a formidable problem in theoretical chemistry. The first virial coefficient is, evidently, equal... 

The first virial coefficient A fulfils the limiting condition ... 

R = gas constant and T = absolute temperature). The so-called first virial coefficient, Bi, is simply equal to the reciprocal of the molecular weight, M. The evaluation of the second virial coefficient, Bi, has become one of the principal theoretical developments in recent years. In his theory of polymer solutions Flory (1953) has shown that Bi is proportional to a term (1 — 6/T), which vanishes T = 0. Accordingly 0 may be considered as the ideal temperature at which the above equation is reduced to the well-known van t Hoff s law, i.e.,... 

The coefficients B, C, . .. are called the second virial coefficient, the third virial coefficient, and so on. The first virial coefficient is unity. The virial coefficients (except the first) depend on the temperature and the identity of the fluid. 

In first approximation, we can content ourselves with the three first virial coefficients... 

Remember that ideal gases should have no inter-particle interactions. This gives the first virial coefficient, R, and the ideal gas law. Unfortunately, the atoms or molecules of real gases do interact. The energy E of one mole of a real gas is the sum of the kinetic and intermolecular potential energies of all its molecules, E = K + v. 

The first virial coefficient term The first term (/,) in equation (10.16) has an osmotic character since its contribution to the force between the plates, obtained from — dJJdh), is 2 RT r%/h = RTc2, where C2 = mean polymer concentration between plates. Dolan and Edwards (1974 1975) obtained a similar term but, due to an algebraic error, they presented the term with the opposite sign, implying not a repulsive interaction but rather an attraction. 

The most general approach (Pitzer and Brewer, 1961, p. 326) would be to start with the D-H limiting law (Equation 15.26) and add a power series of virial coefficients, as for gases. A simpler approach, begun by Scatchard in 1936, and used by Pitzer and Brewer (1961, pp. 326, 578 and Appendix 4), is to define a deviation function B (called B to distinguish it from the first virial coefficient, which in a sense it replaces) as the difference between observed and predicted activity coefficients for an electrolyte such as NaCl. This is... 

The first virial coefficient /(/) is some function of the ionic strength and is not 0 as it would be for an ideal solution, but is in fact a version of the Debye-Htickel equation, which represents departure from ideality in very dilute solutions. The following term is a function of the interactions of all pairs of ions, and the third term a function of the interactions of ions taken three at a time. The second coefficient. Ay, is a function of ionic strength, but the third coefficient ju-y - is considered to be independent of ionic strength and equals zero if /, j, and k are all anions or cations. Later extensions to the model published by Pitzer and co-workers allow for an ionic strength dependence to the third coefficient. Pitzer (1987) and Harvie and Weare (1980) note that higher virial coefficients are required only for extremely concentrated solutions, so the series is stopped at the third coefficient. 

The temperature dependence of several first virial coefficients is calculated for the Lennard-Johns (12,6) model potential (Equation 1.7 0). Figure 1.41 contains reduced virial coefficients (where 6 is the excluded volume of hard spheres, see... 

The first term on the right in equation (1) is the first virial coefficients. The first virial coefficients i.e. the Debye-Hiickel limiting law, f(I), is a function only of ionic strength to express the long-range ion-interaction pwtential energy of one pair of ions in solution and not on individual ionic molalities or other solute properties. 

The dependence i7(Cm) then agrees exactly with the pressure-concentration dependence of an ideal gas. Dissolved polymers do not constitute ideal solutions in this strict sense, even if all higher order virial coefficients vanish, because the first virial coefficient is not unity but given by... 

EoS based on perturbation theory and computer simulation results validity of the Barker-Henderson theory Derivation of the BGY, PY, and HNC integral equations in 2D Fourth virial coefficient Fifth virial coefficient Quantum corrections to the third and fourth virial coefficients quantum corrections to the Helmholtz free energy Results at very high densities EoS based on computer simulation results and on the five first virial coefficients No influence of number of particles for states far from the phase transitions values of RDF... 

Virial theory Five first virial coefficients [195] Reddy et al., 1986... 

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