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The Van der Waals equation is
more commonly seen as (n=number of mole)
where P is the pressure of the fluid, a is a measure of the attraction between the particles (in the two equations differing by a factor equal to the square of Avogadro's number), v is the volume of the fluid (in the 1st equation: per particle; in the 2nd equation: the total volume), b is the volume enclosed within the particles (in the 1st equation: per particle; in the 2nd equation: per mole), k is Boltzmann's constant, R is the molar gas constant, and T is the absolute temperature. A careful distinction must be drawn between the properties of the bulk fluid and the properties of the particles. In particular, in the first equation v refers to the volume of the bulk fluid (i.e. the volume of the container) divided by the number of particles, whereas b is the volume enclosed by a single particle (i.e. the volume bounded by the atomic radius) multiplied by the number of particles.
Above the critical temperature it is an improvement of the ideal gas law, and for lower temperatures the equation is also reasonable for the liquid state and the low-pressure gaseous state.
However, in the first-order phase transition range of (P,V,T) (between a liquid phaseIn the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. density, crystal structure, index of refraction, and so forth. The most familiar examples o and a gasFor other meanings see gas (disambiguation). A gas is one of the phases of matter. Gases are, like liquids, fluids: they have the ability to flow and do not resist deformation. Unlike liquids, however, unconstrained gases do not occupy a fixed volume, buteous phase) it does not exhibit that for a given temperature the vapor pressureThe vapor pressure is the pressure (if the vapor is mixed with other gases, the partial pressure) of a vapor. At any given temperature, for a particular substance, there is a pressure at which the vapor of that substance is in equilibrium with its liquid is constant for varying values of V, i.e. for various amounts of the material being in the veparous state.
The derivation of the Van der Waals equation begins with the equation of state of an ideal gas, which is composed of non-interacting point particles:
We now stop treating the fluid's constituent particles as point particles, instead modelling them as hard spheres with a small radius (the Van der Waals radiusThe van der Waals radius of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. Van der Waals radii are determined from measurements of atomic spacing between pairs of unbonded atoms in crystals. The va.) Denoting the volume of each sphere by b, we modify the equation of state to
The volume per particle, v, has been replaced by the "excluded volume" v - b, reflecting the fact that the particles cannot overlap. If the fluid is compressed, its pressure goes to infinityInfinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question as the total volume approaches the volume enclosed within the particles.
Next, we introduce a pairwise attractive force between atoms. This causes the average free energyIn thermodynamics, free energy is a measure of the amount of work that can be extracted from a system. In this sense, it measures not the energy content of the system, but the "useful energy" content. In different situations, free energy is related to int per particle to be reduced by an amount proportional to the fluid densityFor other meanings of density, see density (disambiguation Density (symbol: rho Greek: rho) is a measure of mass per unit of volume. The higher an object's density, the higher its mass per volume. The average density of an object equals its total mass div. However, the pressure obeys the thermodynamic relation
where f is the free energy per particle. The attraction therefore reduces the pressure by an amount proportional to 1/vē. Denoting the constant of proportionality by a, we obtain
which is the Van der Waals equation.
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