Real Gases: Effects of Intermolecular Forces and Molecular Volume Deriving Van der Waals Equation

Ideal gases follow the relation PV over nRT equals one. Recall Boyle’s law, which states that when the amount and temperature of a gas are held constant, increasing the pressure will invariably decrease the volume to maintain a constant ratio.

But when that ratio is plotted as a function of pressure for one mole of several real gases, it equals the ideal value “one” only at low pressures. As the pressure increases, the curves deviate significantly from ideality.

At low pressures, the combined volume of gas particles is negligible relative to the container volume — like a pea inside a basketball. Therefore, the volume available to ideal gas particles equals the total container volume.

At higher pressures, the gas density is much greater. Thus, the combined volume of the gas particles becomes significant — like a pea inside a ping-pong ball.

Therefore, the assumption of the kinetic molecular theory that gas particles occupy negligible volume is invalid at high pressures.

The volume occupied by a real gas is greater than the volume available to its particles, which is the volume it would occupy in an ideal case, by nb, where b is an experimentally determined constant that depends on the gas and has units of L/mol.

Deducting nb adjusts the volume of a real gas downward to that available to its particles, which is equivalent to the ideal volume.

Another assumption of the kinetic molecular theory — that intermolecular forces between gas molecules are negligible — is valid only under high-temperature and low-pressure conditions.

Typically, gases exert very weak attractive forces. Under low-pressure conditions, the gas particles are separated by large distances and, therefore, do not perceive the attractive forces of other particles.

Similarly, under high-temperature conditions, the particles have high kinetic energies relative to the attractive forces and move very quickly. When particles collide, they bounce off each other because the high kinetic energy overcomes the small attractive forces.

However, when the gas is at higher pressures, the particle density is greater. The particles are, therefore, separated by shorter distances, and the likelihood that they will interact thereby increases. The attractive forces between the particles accordingly become more significant at high pressures.

This is more evident as the temperature is lowered. The kinetic energy of the particles decreases, and they move more slowly. When intermolecular attraction becomes significant, particles are increasingly likely to ‘stick’ to each other upon colliding.

As gas particles spend more time interacting with neighboring particles, the frequency of collisions with the container surface decreases.

Consequently, the pressure exerted by a real gas is lower than that of an ideal gas by an²/V². Here, a is an experimentally determined constant that depends on the gas and has units of L²·atm/mol², and V is the real volume.

Adding this term adjusts the real pressure upward to that exerted by an ideal gas. The modified equation with the pressure and volume correction factors is called the van der Waals equation for non-ideal or real gases.