# The Van der Waals Equation

How can deviations from ideal behaviour be compensated for in our model of the situation? Recall that our model produced p.Vm = RT (ideal gas equation)

One assumption was that the molecules had no size. This is clearly not true (but is a reasonable approximation at low pressures when the size of the molecules is very small compared to the molecular separation). We can adjust the equation to take account of this, and have our volume as (Vm – b), where b represents the volume occupied by the molecules themselves.

 Another assumption was that there were no interactions between particles, whereas there is a long-range attractive interaction to consider. This leads to another modification. A particle colliding with the wall of its container can only experience the attraction of molecules away from the wall (as the other molecules all lie within the container) and so the force with which it hits the wall will be reduced.

In addition, since molecules experience a small attraction away from the wall as they approach it, they will approach the wall more slowly (the additional attraction decelerates them) and so fewer molecules will hit the wall per unit time.

Both effects act to reduce the pressure, and they are each proportional to the number density (itself proportional to 1/Vm), thus we modify the pressure, p, to p + a/Vm2

Combining these modifications into our ideal gas equation, we have (p + a/Vm2).(Vm – b) = RT which easily re-arranges to

p = RT/(Vm – b) – a/Vm2

This very important equation is the van der Waals equation.

If we were to multiply out the van der Waals equation, we would get a cubic equation, and so plotting out a pV graph using the equation should give us a graph of cubic form. Does it?

 Indeed it does, but to make our cubic gaph fit the experimental pV graph for real gases we have to apply a procedure known as Maxwell construction.

Maxwell construction is most easily explained graphically:

 A horizontal line is drawn so that the areas on either side of the line are equal. Thus the experimental graph can be found from the Van der Waals cubic.