Derivation of Ideal Gas Equation

An ideal gas is an imaginary entity that does not exist in reality. Almost all gases are real, and they only approach perfect gas behaviour under particular circumstances. Here The concept of an ideal gas, the ideal gas law, and the ideal gas equation are explained briefly. It also describes the limitations of ideal gas law.

The Ideal Gas Law is an equation in thermodynamics describing the relationship between temperature, pressure, and volume of gases. The Ideal Gas Law equation is PV = nRT, where P is the pressure, V is the volume, ‘n’ is the number of moles of gas molecules, R is the universal gas constant, and T is the absolute temperature. 

Ideal Gas

An ideal gas is a hypothetical gas developed to simplify calculations of a gas molecule’s temperature and pressure. The concept of an ideal gas is that it is made up of molecules that can travel in all directions at random. The collision between the particles is thought to be fully elastic, meaning that there is no loss of kinetic energy of the particles due to the collision.

In actuality, Ideal gas does not exist. It is a theoretical concept. When the density is low, the gas molecules are too far apart to interact with each other, that is why all gasses are real and tend to behave like ideal gases. An ideal gas should follow a few rules which are-

• The molecules of an ideal gas neither attract nor repel each other.
• Elastic collisions between perfect gas molecules cause them to interact.
• These molecules have no volume on their own.
• The molecules in an ideal gas are moving point particles with no volume of their own.

 As a result, the ideal gas notion helps us in studying real gases.The ideal gas notion is advantageous because it adheres to the ideal gas law. In this section, we will look at the Ideal Gas Law. 

The Ideal Gas Equation

The purpose of creating Ideal gas law was to show the relationship between pressure, volume, moles of gas, and temperature. It is a hypothetical or speculative equation for an ideal gas. Pressure and volume have an inverse relationship; however, temperature has a direct relationship. The equation for the Ideal Gas Law is:

P × V = n × R × T

Where,

‘P’ denotes the ideal gas pressure. ‘V’ denotes the volume of the ideal gas. ‘n’ represents the quantity of ideal gas measured in moles. ‘R’ is the proportionality constant or the gas constant and ‘T’ stands for the ideal gas temperature.

Derivation Of Ideal Gas Equation:

Let us consider,

The pressure exerted by the gas: P

The volume of the gas: V

Temperature: T

The number of moles of gas: n

Universal gas constant: R

According to Boyle’s Law, At constant n & T, the volume makes an inverse relation with the pressure exerted by a gas.

i.e. 

      v ∝ 1/P ————————-(i)

According to Charles’ Law, When P & n are constant, the volume of gas directly relates to the temperature.

i.e. 

       v ∝ T     ————————-(ii)

According to Avogadro’s Law, When P & T are constant, then the volume of gas makes a direct relation with the number of moles of gas.

i.e. 

       v ∝ n     ———————–(iii)

Combining all the three equations, we have-

V ∝ nT/P 

or 

PV = nRT

Where R is the Universal gas constant, which has a value of 8.314 J/mol-K

Ideal gas behaviour conditions

Many of the properties or ideal behaviour of gases are very similar to those of real gases. The following are some of the properties that are unique to Ideal gases:

• Molecule collisions are elastic, and their motion is frictionless, which means that the molecules do not lose energy
• The overall volume of individual molecules is orders of magnitude smaller than the volume occupied by the gas.
• Between the molecules and their surroundings, there are no intermolecular forces at work
• The molecules are constantly moving, and the distance between them is much greater than the size of an individual molecule
• According to Newton’s Laws of Motion, the gas particles move at random
• Molecules are only exposed to external forces when they collide. These collisions are elastomeric and last only a few seconds

At room temperature, an ideal gas would not form a liquid due to these assumptions for deriving real gases from ideal behaviour.

Conclusion:

The ideal gas is a hypothetical gas in which the molecules take up very little space and have no interactions, and thus obeys the gas laws perfectly. The study of the ideal behaviour of gases allows us to understand the matter at its most fundamental level: individual particles acting independently, almost completely free of interactions and interference.