# Question #ae572

Sep 10, 2017

From kinetic theory if gases, we have,

$p V = R T$ is the equation of state for 1 mole of an ideal gas.

However, there are two fundamentally flawed assumptions in the kinetic theory which lead to an equation of state which holds only under high temperature limits. These assumptions include,

1) Molecules of the gas are so small that they occupy a negligible volume.

2) Molecules exert no forces on each other except collisions.

None of these assumptions is justified.

The first one can be corrected as follows,

Even though each molecules has a very negligible volume, in an aggregate of a few moles of molecules, the volume is not negligible. Hence the total volume $V$ of the container is not available for motion of molecules, the $V$ in the ideal gas equation is replaced by,

${V}_{e f f} = V - b$ where $b$ is the correction term known as co-volume.

Now coming to intermolecular forces, we are not aware of the nature of the forces other than that they are attractive in general (untill ofcourse there is a collision where the force acts repulsively).

The pressure of the gas which is due to force exerted by molecules on the walls of the container is then to be modified with a correction term which accounts for the intermolecular forces as well.

Now, for a molecule in the container with a fixed concentration of molecules, the intermolecular force is proportional to number of molecules around it in the gas, which means that the intermolecular force depends on the concentration $\frac{N}{V}$. But for $N$ constant, the force and hence the pressure correction term is,

${p}_{c} \propto \frac{1}{V}$

Also the pressure on the walls is proportional to the concentration of molecules and hence inversely to $V$.

Therefore we must have,

${p}_{c} \propto \frac{1}{V} ^ 2$
$\implies {p}_{c} = \frac{a}{V} ^ 2$
Where $a$ is a constant.

The effect of the intermolecular force is to reduce the effective pressure on the walls since the forces are attractive and inward.

Therefore, the effective pressure,

${p}_{e f f} = p - \frac{a}{V} ^ 2$

Therefore, the modified equation of state is,

${p}_{e f f} {V}_{e f f} = R T$
$\implies \left(p + \frac{a}{V} ^ 2\right) \left(V - b\right) = R T$

Where $a$ and $b$ are constants known as Van der Waal's constants.

This is the Van der Waal's equation of state.