Explanation of behaviour of real gases on the basis of van der Waals equation-
Unlike ideal gas equation, the van der Waals equation explains the behaviour of real gases under different conditions of temperature and pressure.
(i) At low pressures: At low pressures, volume V is very large and hence the correction term b (a constant of small value) can be neglected in comparison to very large value of V. Thus the van der Waals equation for 1 mole of a gas, i.e.,
\(\left(P + \frac a{V^2}\right)(V - b) = RT \quad ....(i)\)
may be written as \(\left( P + \frac a {V^2}\right)V = RT\)
or \(PV + \frac aV = RT\)
or \(PV = RT - \frac aV \quad ....(ii)\)
At extremely low pressure, since V is very large, the value of a/V is very small and hence can be ignored. Consequently, at very low pressures, the van der Waals equation is reduced to ideal gas equation, i.e.,
\(PV = RT\)
This explains why at extremely low pressures, the real gases obey the ideal gas equation.
The equation (ii) shows that PV is less than RT by an amount equal to a/V. As pressure increases, V decreases, a/V increases and ultimately PV becomes smaller and smaller. This explains the dip in the isotherms of most of the gases (e.g., CO and CH4).
(ii) At high pressures: At high pressures, volume V is quite small and hence the term b cannot be neglected in comparison to . Secondly, under these conditions although the term a/V2 is quite large, it is so small in comparison to high pressure P that it can be neglected. Thus the van der Waals equation is reduced to
\(P(V - b) = RT\)
\(PV - Pb = RT\)
\(PV = RT + Pb \quad .....(iii)\)
Thus PV is greater than RT by an amount equal to Pb. As the pressure increases, the factor Pb increases and hence PV increases. This explains why the value of PV, after reaching a minimum, increases with further increase of pressure.
(iii) At high temperatures: At any given pressure, if the temperature is sufficiently high, V is very large and hence the terms a/V2 and b can be neglected as in case (i) and thus the van der Waals equation reduces to PV = RT This explains why the real gases behave like ideal gas at high temperatures.
(iv) At low temperatures: At low temperatures both P and V are small, hence both pressure and volume corrections are appreciable, with the result the deviations are more pronounced.
(v) Exceptional behaviour of hydrogen and helium: Since hydrogen and helium have very small masses, the intermolecular forces of attraction are extremely small even at low pressures. In other words, the factor a / (V ^ 2) is negligible at all pressures. Hence the van der Waals equation is reduced to
P(V - b) = RT
PV = RT + Pb
This explains why hydrogen and helium show positive deviations only with increase in the value of P.
