subject: Der Waals Gas Equation [print this page] A number of equations of stage have been suggested to describe the P-V-T relationship in real gases. The oldest and best known equation is that of van der waals
The van der waals gas equation of state
In 1873, J.D. van der Waals proposed his famous equation of state for a non-ideal , he modified the ideal gas equation by suggesting that the gas molecules were not mass points but behave like rigid spheres having a certain diameter and that there exist intermolecular forces of attraction between them
The two correction terms introduced by van der Waals are discussed below
Correction due to volume of gas molecules
Correction due to intermolecular forces of attractions
Correction due to Volume of Gas Molecules
The ideal gas equation PV=nRT is derived on the assumption that the gas molecules are mass points, i.e. dont have finite volume. Van der Waals abandoned this assumption and suggested that a correction term nb should be substracted from the total volume V in order to get the ideal volume which is compressible
In order to understand the meaning of the correction term nb, let consider two gas molecules as unpenetrable and incompressible spheres, each of which has a diameter d.
Correction due to Intermolecular Forces of Attractions
In the derivation of the ideal gas equation, it was assumed that there are no intermolecular forces of attraction. Actually it is not so, in order to take into account the effects of intermolecular forces of attraction, let us consider a molecular lying somewhere in the midst of the vessel. These forces neutralize one another and there is not the molecule. So, it will strike the wall with a lower velocity and will exert a lower pressure than it would have done if there was no force of attraction. Fig 1
der waals gas equation
der waals gas equation
der waals gas equation as follows
p = pressure of the fluid
v = volume of the particles divided by the sum of particles
k = Boltzmann's constant
T = absolute temperature
a' = attraction between the particles
b' = average volume excluded from v by a particle.
Most textbooks give two different derivations[disambiguation needed]. One is the conventional derivation that goes back to van der Waals and the other is a statistical mechanics derivation. The latter has the major advantage that it makes explicit the intermolecular potential, which is neglected in the first derivation. The conventional Van der Waals equation is a mechanical equation of state, which cannot be used to specify all thermodynamic functions, while the statistical mechanical derivation yields the partition function for the system, which does allow all thermodynamic functions to be specified, including the mechanical equation of state.
Conventional derivation
Consider first one mole of gas which is composed of non-interacting point particles that satisfy the ideal gas law
p = frac{RT}{V_mathrm{m}}.
Next assume that all particles are hard spheres of the same finite radius r (the van der Waals radius). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace V by V - b, where b is called the excluded volume. The corrected equation becomes
p = frac{RT}{V_mathrm{m}-b}.
The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times that volume. To see this, we must realize that a particle is surrounded by a sphere of radius r = 2r (two times the original radius) that is forbidden for the centers of the other particles. If the distance between two particle centers were to be smaller than 2r, it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.
The excluded volume per particle (of average diameter d or radius r) is
