Derivation from Ideal Gas Law

Section 1: Understanding the Ideal Gas Law

The Ideal Gas Law is a principle that combines several gas laws that were historically stated separately.
It is expressed as PV = nRT, where P is the pressure, V the volume, n the number of moles of gas, R the gas constant, and T the temperature.
It assumes gases exhibit ideal behaviour, meaning they follow this law at all temperature and pressure conditions.
This law indicates that volume is directly proportional to the amount of gas and temperature, and inversely proportional to the pressure.

Section 2: Deviations from the Ideal Gas Law

Real gases deviate from the Ideal Gas Law due to the interactions between molecules and the finite size of gas particles.
At low temperatures or high pressures, these interactions and volumes become significant, causing the Real gas to deviate from ideal behaviour.
For instance, a real gas condenses to a liquid at low temperatures, unlike an ideal gas which according to the ideal gas law, should not liquify irrespective of how cold or pressurised it is.
At high-pressure conditions, the volume of a Real gas is more than that predicted by the Ideal Gas Law because of repulsive interactions amongst gas molecules.

Section 3: The van der Waals Equation

The van der Waals equation accounts for these deviations by incorporating the size of the gas particles and interparticle attractions into the Ideal Gas Law.
It is expressed as [P + a(n/v)²] (V-nb) = nRT, where a and b are specific constants for each gas, and other symbols carry their usual meanings.
In this equation, a accounts for the attractions between particles, while b is the volume occupied by one mole of the gas molecules.
By considering the finite size of particles and their mutual attractions, the van der Waals equation provides a more accurate description of the behaviour of real gases.

Section 4: Importance of Ideal Gases and Real Gases Understanding

Understanding the concepts of Ideal Gases and Real Gases helps in comprehending the principles of various natural phenomena and man-made systems such as weather patterns, tyre pressure, and aerosol sprays.
It also aids in understanding how deviations from ideal gas behaviour occur and how the properties of a gas change with different conditions.