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VAN DER WAALS EQUATION OF STATE

The Ideal Gas Law, PV = nRT, can be derived by assuming that the molecules that make up the gas have negligible sizes, that their collision with themselves and the wall are perfectly elastic, and that the molecules have no interactions with each other.
But this equation is applicable only for ideal gases i.e, gases at low pressure and high temperature.
The van der Waal's equation is a second order approximation of the equation of state of a gas that will work even when the density of the gas is not low.

The vanderwall equation is given by

Here a and b are constants particular to a given gas.

The parameter a is related to intermolecular attractive force between the molecules , and n/V is the density of molecules

· The term containing "a" is added to the pressure P. In other words the corrected pressure is slightly larger than the observed pressure.

· As the molecule moves toward the wall all of its gaseous neighbors are behind it, thus any intermolecular forces acting on the molecule are slowing it as it moves toward the wall.

· The molecule striking the wall does so at a velocity that is slightly less (determined by the size of the intermolecular forces) than it would without these forces.

· Or, the observed prssure is actually slightly less than the pressure would be without the intermolecular forces.

· To correct for this, a small positive pressure must be added to the observed pressure, hence the term with "a" in it is positive.

·

This is illustrated in the figure below:

When the density of the gas is low (i.e., when n/V is small and nb is small compared to V) the van der Waals equation reduces to that of the ideal gas law.
"b" is a correction for the real volume of the gas molecules.
The parameter b is related to the size of each molecule . The volume that the molecules have to move around in is not just the volume of the container V, but is reduced to ( V - nb ).
The "b" term is relatively easy to understand. Our gas is in a container with some volume "V". If the container is sufficiently large, the volume of the gas molecules is negligible, but if the volume of the gas molecules themselves starts to become significant with respect to the volume, we have a problem.
Under these conditions, we may wish to think of the actual free volume in the container as being the measured volume, V, minus the volume occupied by the molecules.
We calculate the volume occupied by the molecules by multiplying the number of moles of molecules, n, by their effective molar volume, b.

At High T, the gas molecules have a higher average kinetic energy (KE_avg) which overcomes the IMF.
At Low P, the gas molecules are spread further apart and can therefore avoid IMF.

P of a real gas < P of an ideal gas because the actual paths of gas molecules are curved (not straight) due to the IMF.

V of a real gas > V of an ideal gas because V of gas molecules is significant when P is high. Ideal Gas Equation assumes that the individual gas molecules have no volume.

[Image]

The blue curve shows the ideal gas equation: P=RT/V_m
The red curve shows the effect of the finite volume of the molecules: P=RT/(V _m-b)
The purple curve shows the Van der Waals equation. It incorporates the effects of both the finite

volume of the molecules, and the interaction between the molecules P=RT/(V _m-b)-a/V_m².