Derivation of Ideal Gas Equation

Understanding the behaviour of gases is important in the field of chemistry, physics, and engineering. The ideal gas equation is one of the fundamental concepts in thermodynamics that links pressure, volume, temperature, & the number of moles of a gas. Let’s delve deep into the derivation of the ideal gas equation, especially from the perspective of kinetic theory. We will also answer some of the foundational questions, like “what is ideal gas?” and “what is an ideal gas equation?” and explore all the terms in the ideal gas equation in detail. 

1.0What is an Ideal Gas?

Before diving into all the equations, let’s answer, what is ideal gas?

An ideal gas is a theoretical gas that is composed of many moving particles that have negligible volume and no intermolecular forces. The ideal gas equation is a combination of empirical laws like Charles’ law, Boyle’s law, Gay-Lussac’s law, and Avogadro’s law into a single expression. 

Key Characteristics of an Ideal Gas

• No intermolecular forces between gas molecules.
• Collisions between molecules are perfectly elastic.
• The volume occupied by the molecules of any gas is negligible when compared to its container’s volume.
• All gas molecules are identical and in constant, random motion. 

2.0What is an Ideal Gas Equation?

It is important to know the answer to “What is an ideal gas equation?” to understand the concepts in a much clearer manner. The ideal gas equation describes the behaviour of an ideal gas under various conditions. It connects the macroscopic properties of a gas, such as pressure (P), volume (V), temperature (T), and the number of moles (n) in one equation. 

The equation is: 

PV=nRT

Where,

• P represents the pressure.
• n represents the no. of moles of gas
• V is the volume.
• R represents the universal gas constant.
• T is the absolute temperature in Kelvin.

This equation provides a powerful tool for calculating the behaviour of gases under different conditions. 

3.0Terms in Ideal Gas Equation

Here’s a breakdown of all the terms in ideal gas equation and what they represent:

4.0Derivation of Ideal Gas Equation from Kinetic Energy

Let’s explore the derivation of ideal gas equation from kinetic energy using the principles of kinetic molecular theory. 

Assumptions of the Kinetic Theory of Gases

• Gases are composed of countless tiny particles called molecules. 
• These gas molecules are always moving in constant, random motions.
• The volume of individual gas molecules is negligible compared to the total volume.
• Intermolecular forces of attraction or repulsion between gas molecules are considered negligible (except when the molecules collide).
• Collisions between gas molecules & the container walls are assumed to be perfectly elastic.
• The motion of gas molecules follows Newton's laws of motion.

Step-by-Step Derivation

Let a gas molecule of mass “m” move in the “x” direction with the velocity u. When it collides elastically with the wall and rebounds, the change in momentum is:

$\Delta P=-mu-(-mu)=2mu$

Time between collisions for the same molecule:

t = 2Lu

Force due to one molecule:

$F=\frac{\Delta p}{\Delta t}=\frac{2mu}{\frac{2L}{u}}=\frac{m{u}^2}{L}$

Since pressure ​​

$P=\frac{F}{A}andA=L^2$, we get

$P=\frac{m{u}^2}{L^3}=\frac{m{u}^2}{V}$

For 𝑁 molecules moving in all directions, the average velocity squared is:

$u^2+v^2+w^2=3c^2$

Thus, total pressure,

$P=\frac{1}{3}\frac{Nm{c}^2}{V}$

The average kinetic energy of a gas molecule is:

$KE=\frac{1}{2}m{c}^2$

From the pressure equation:

$PV=\frac{1}{3}m{c}^2\Rightarrow PV=\frac{2}{3}N\left(\frac{1}{2}m{c}^2\right)\Rightarrow PV=\frac{2}{3}N\cdot KE$

Therefore:

$KE=\frac{3}{2}KT$

Where k is Boltzmann’s constant.

This connects microscopic molecular motion to macroscopic gas properties, showing the derivation of ideal gas equation from kinetic energy.

Recall, 

$N=n\cdot N_a\text{ and }R=k\cdot N_a$

We get: 

$\begin{array}{rl}& PV=Nkt=n\cdot N_{a}kT=nRT\\ & PV=nRT\end{array}$

5.0Applications of Ideal Gas Equation

• It is widely used in chemical reactions to calculate reactant/product quantities in the gas form.
• It is used in thermodynamics to study energy exchange and internal energy.
• It has essential applications in engineering, such as HVAC systems and engine modelling.
• It is used to study the behaviour of gas in the atmosphere to help with climate science and pollution monitoring. 

6.0Limitations of Ideal Gas Equation

Although the ideal gas equation is greatly useful, it has some limitations:

• Real gases deviate from ideal behaviour at low temperatures and high pressures.
• The model ignores intermolecular forces and finite molecular size.
• It fails to explain the behaviour of gases near liquefaction.

7.0Conclusion

The derivation of ideal gas equation gives us deep insight into how gases behave at the molecular level. By connecting pressure, volume, temperature, and moles through kinetic energy and statistical physics, the ideal gas equation remains an important topic in science.