Gas laws were based on simple experimental facts hence, a theoretical proof or derivation for the same was proposed, on the basis of Kinetic Molecular Theory of Gases (KMT) was given by Maxwell and Boltzmann in the 19th century.
This theory lays the bridges the gap between macroscopic characteristics like pressure, temperature, and volume and microscopic particle motion by explaining the behavior of gases at the molecular level, and provided assistance for the study of statistical mechanics and thermodynamics.
Postulates of the Kinetic Molecular Theory
Kinetic molecular theory is built on the following key assumptions:
1.Gases Consist of a Large Number of Particles:A large number of tiny particles, e.g atoms or molecules, formed a gas. Their individual volumes are irrelevant because these particles are so tiny in relation to the distances between them.
2. Random Motion:Particles of gas move randomly and continuously, until they hit the container’s walls or one another, they move in straight lines. Macroscopic pressure that the gas exerts is caused by these encounters.
3.Elastic Collisions: Perfectly elastic collisions occur between the gas particles and the container walls. This indicates that energy may be transferred between the colliding particles or there is no net loss of kinetic energy during these collisions.
4. Negligible Intermolecular Forces: There are no strong forces of attraction or repulsion between gas particles. They only interact when they collide. In ideal gases, this assumption is accurate; but, in real gases, it is not, mainly at low temperatures and high pressures.
5. Proportionality to Temperature: Absolute temperature of the gas has a direct relationship with the average kinetic energy of its particles. This suggests that particles travel more quickly at higher temperatures and more slowly at lower ones.
Derivation of Gas Laws from Kinetic Molecular Theory of Gases (KMT): Kinetic molecular theory provides a molecular interpretation of the basic gas laws:
Boyle’s Law (Pressure-Volume Relationship): A gas’s pressure and volume are inversely proportional when its temperature remains constant. According to KMT, pressure rises as the container’s volume shrinks because gas particles hit with the walls more frequently.
Charles’s Law (Volume-Temperature Relationship): A gas’s volume and temperature are directly proportional while the pressure is constant. Gas expands and takes up more space as the temperature rises because the particles have more kinetic energy and travel more quickly.
Avogadro’s Law (Volume-Mole Relationship): Number of moles of a gas is directly proportional to its volume at constant temperature and pressure. As additional particles are added, collisions occur more frequently, the container’s expansion to keep the pressure constant.
Root Mean Square Speed and Kinetic Energy: Root mean square speed, or rms speed, is the square root of the average of the squares of the individual particle speeds. The velocity of gas particles varies. In terms of mathematics, it is stated as: (Vrms)2 = 3 kBT / m
Where:
kB is the Boltzmann constant,
T is the absolute temperature, and
m is the mass of a single particle.
The average kinetic energy of gas particles is given by:
KEavg = 3 / 2kBT
This formula emphasises that temperature is a measure of the average kinetic energy of gas molecules, indicating the direct correlation between temperature and kinetic energy.
Applications and Significance of KMT
1.Pressure: When gas particles collide with the container walls, pressure is created. The frequency of collisions and the force applied on them determine how much pressure is applied.
2. Diffusion and Effusion: The process by which gas particles disperse to uniformly fill a container is known as diffusion. According to KMT, this is the result of particles moving randomly. According to Graham’s Law, effusion occurs when gas particles flow through a small hole without colliding. Because of their greater rms speeds, lighter gases effuse more quickly than heavier gases.
Limitations of Kinetic Molecular Theory
On successful explanation of ideal gas behavior, the theory has limitations:
1.Real Gas Behavior: Because of strong intermolecular interactions and limited particle volumes, gases behave differently than they should under high pressures and low temperatures. The van der Waals equation explains these variations.
2. Non-Elastic Collisions: In real gases, collisions are not perfectly elastic, especially at very high densities.
3. Neglect of Quantum Effects: For extremely small or lightweight particles, quantum mechanical effects become important, which are not addressed by KMT.
Main points:-
Kinetic molecular theory of gases lays the basis for the gas laws by contributing a difficult and rational explanation of the characteristics of gases.
A vast number of microscopic particles made up gases whose size is negligible as compared to the distance of separation between them.
Molecules move randomly in different directions with different speeds in continuous manner.
Particle and container wall collisions are completely elastic.
Intermolecular forces of attraction and repulsion are almost negligible.
The absolute temperature of the gas is related to the average kinetic energy of its constituent particles.
Collisions are called elastic collisions because no energy changes occurs when two molecules collide i.e the total kinetic energy remain constant.
The average kinetic energy of a molecule is proportional to its absolute temperature.
The behavior of gases at the molecular level is explained by the kinetic molecular theory of gases. It gives gas laws like Boyle’s law and Charles’s law a microscopic foundation by describing the motion, interaction, and pressure exerted by gas particles.
The key assumptions are:
A vast number of microscopic particles make up gases.
• Particles move randomly and continuously.
• Particle and container wall collisions are completely elastic.
• There are very few intermolecular forces.
• The absolute temperature of the gas is related to the average kinetic energy of its constituent particles.
When gas particles collide with their container walls, pressure is created. The pressure of gas exerts is determined by the frequency and force of these impacts.
Absolute temperature of the gas has a direct relationship with the average kinetic energy of its particles. This means that the particles move more quickly at higher temperatures, which results in more kinetic energy.
According to Boyle’s law, a gas’s pressure and volume are inversely proportional at constant temperature. Kinetic molecular theory claims that when the volume drops, more gas particles smash with the walls, raising the pressure.
Real gases deviate because:
Intermolecular forces become important, particularly at low temperatures;
• Gas particles have finite volumes, which makes their individual volumes significant at high pressures.
Applications include:
Providing an explanation of gas laws such as Avogadro’s, Charles’, and Boyle’s laws.
Determining particle kinetic energy and root mean square speeds.
Giving information about how actual gases behave in harsh environments.
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