Kinetic Theory of Gases

Kinetic Theory of Gases is a theoretical model which helps us understand the behavior of gases and their constituent particles. This theory suggests that gas is made up of a larger number of tiny particles which collide with each other and their surroundings and exchange kinetic energy between them. The kinetic theory of gases has various applications throughout physics, chemistry, and engineering and it is essential to understand many phenomena like diffusion, effusion, and Brownian motion.

In this article, we will learn about the assumptions of kinetic theory, its limitations, and others in detail.

What is Kinetic Theory of Gases?

The kinetic theory of gases was introduced to explain the structure and composition of molecules with respect to submicroscopic particles which make up the gaseous matter around us. This theory talks about the increase in pressure due to the constant movement and collision of the submicroscopic particles. It also discusses other properties of a gas such as temperature, pressure, volume, viscosity, diffusion, thermal conductivity, etc. The theory develops a relationship between the microscopic particles and the macroscopic properties. The molecule of gas is always in constant motion and keeps colliding with each other and the walls of the container, in such a case, it is difficult as well important to learn the dynamics of the gases.

Kinetic Theory of Gases Assumptions

There are several assumptions that were taken into account in order to develop the kinetic theory of gas which are stated as follows:

• Every gas consists of molecules that are microscopic particles. 
• There are uncountably large numbers of particles making up any Gas.
• The size of the molecule also known as the molecular size is negligible as compared to the molecular distance between two molecules (which is approximately 10^-9 m).
• The speed of the molecules of a gas is very high generally and it can lie anywhere between 0 and infinity.
• The molecule shape of gas is spherical, rigid, and elastic masses.
• The mean free path is known as the mean of all free paths. The free path is defined as the distance covered by the molecules between their two successive collisions.
• The number of collisions per unit volume always remains the same in gas and is a constant.
• There is no force of attraction or repulsion acting between the gas molecules.
• The force of gravitation is also negligible due to the fact that the molecules have a very very small mass and they travel at a very high speed.

Postulates of Kinetic Theory of Gases

Based on the assumptions, the following Postulates of the Kinetic Theory of Gases are given:

• All the molecules of a single gas are the same and identical but are different from the other gas molecules which have different properties as well.
• Every gas consists of molecules that are microscopic particles such that the volume of all the molecules combined is negligible when compared with the total volume of the container i.e., in other words, the size of the molecule is negligible as compared to the molecular distance between two molecules (which is approximately 10^-9 m).
• The time in which particles collide with the container’s wall is negligible compared to the time taken by molecules in two successive collisions.
• The number of particles in the system is a very large number, so we can use statistics instead of considering individual particles. This assumption is known as the Thermodynamic limit.
• The collision between two particles of gases is perfectly elastic i.e., molecules of a gas are hard round spheres. 
• There is no exchange of energy between gas particles except the collision. 

Kinetic Theory and Gas Pressure

The continuous bombardment of the gas molecules against the walls of the container results in an increase in gas pressure. According to the Kinetic theory of gases, the pressure at that point exerted by a gas molecule can be represented as,

P = 1/3ρ×c^-2

where 

• c is Mean Square Speed of a Gas Molecules
• ρ is Density of Gas

Suppose the container has n number of molecules of gas, with each of mass m, then the pressure can be represented as,

P = 1/3(nm/v)×c^-2

where, 

• V is Volume of Gas

Gas Laws for Ideal Gas

If the gases are assumed to be ideal in nature, the following gas laws are applicable to them. The laws are defined to understand the ideal gases and their parameters like volume, pressure, etc. Let’s take a look at the laws,

Boyle’s Law

According to Boyle’s law, the volume of a given gas is inversely proportional to its pressure at a constant temperature. As this law is given by Robert A. Boyle in 1662, hence the name Boyle’s Law.

V ∝ 1/P

PV = Constant

For a given ideal gas,

P₁V₁ = P₂V₂

Charles’s Law

Charles’s law (named after Jacques Charles) states that at constant pressure, the volume of a gas is directly proportional to the absolute temperature of the gas. 

V ∝ T

V/T = Constant

Thus, for any given ideal gas,

V₁T₂ = V₂T₁

Pressure Law (Gay-Lussac’s Law)

Pressure Law or Gay-Lussac’s laws state that at constant volume, the pressure of a given gas is directly proportional to its absolute temperature. This law is named after Joseph-Louis Gay-Lussac who published this law in 1809.

P ∝ T

P/T = Constant

For any ideal gas,

P₁T₂ = P₂T₁

Avogadro’s Law

Avogadro’s Law or Avogadro-Ampere’s hypothesis states that an equal amount of volume of all gases under S.T.P. (Standard temperature and pressure) contain the same number of molecules i.e., one mole of any ideal gas at STP always has a volume of 22.4 liters.

V ∝  n

V/n = Constant

For any ideal gas,

V₁n₂ = V₂n₁

Graham’s Law of Diffusion of Gases

According to Graham’s law of diffusion of gases, the rate of diffusion of a gas is inversely proportional to the square root of the density of the gas. Therefore, the more the density of the gases slower will be its rate of diffusion.

r ∝ √(1/p)

Dalton’s Law of Partial Gases

According to this law, the net pressure applied by a mix of non-interacting gas is equivalent to the sum of the individual pressures.

P = P₁ + P₂ + P₃ +… P_n

The arithmetic mean of the speed of gas molecules is known as the average speed of molecules or the mean speed of the gas molecules. If there are  and are given by

Mean speed = v_mean = (v₁ + v₂ + v₃ +… v_n)/n

Formula for mean speed, v_mean = 

Similarly, there is another term known as the root mean square speed of gas molecules, it is defined as the root mean of the squares of speeds of gas molecules. The formula for the root mean speed is given as follows:

v_rms = √(3RT/M)

Similarly, there is a term known as the most probable speed of a gas molecule, which is defined as the speed obtained by the maximum number of gas molecules, and formula for the most probable speed is given as follows:

v_mp = √(2RT/M)

Kinetic Interpretation of Temperature

The overall average energy present in the molecules is directly proportional to the temperature. Therefore, average kinetic energy is formed by the measure of the average temperature of the gas. According to this, the average energy of the molecules is 0 when the temperature is 0. Therefore, the motion of the molecules stops at absolute 0. The formula for the average energy of the molecules is given as,

U = 3/2 RT

Non-Ideal Gas Behavior

Under low pressure and high temperature, it is presumed that all gases obey the ideal gas behavior and hence the gas laws. For the real gases, or during the study of real gases, the deviation from the ideal gas behavior is mostly pointed out. It involves talking about the wrong postulates defined for ideal gases that do not follow up in real gas behavior. Let’s take a look at them,

• Gas particles are point charges and have no volume. In such a case, it was possible for the particles to get compressed to 0 volume, but is it true? No. Gases cannot be compressed to 0 volume, not practically, hence, they do have volume and that cannot be neglected.
• Particles do not interact with each other and are independent. This postulate is false as the particles do interact with each other depending upon nature. It also affects some of the terms like the pressure of gas molecules.
• The collision of the particles is not elastic in nature. Again, the statement is false. The collision of the particles is indeed elastic in nature and they do exchange energy upon colliding. Hence, the distribution of energy is defined.

Read More,

• Ideal Gas Equation
• Newton’s Law of Cooling
• Stefan-Boltzman Constant

Sample Problems on Kinetic Theory of Gases

Problem 1: A gas occupies 10 liters at a pressure of 30 mmHg. What will be the volume when the pressure is increased to 50 mmHg?

Solution:

Applying Boyle’s law, 

P₁V₁ = P₂V₂

Now, 

• P₁ = 30 mmHg
• V₁ = 10 liters
• P₂ = 50mmHg

30 × 10 = 50 × V₂

V₂ = 6 liters

Problem 2: A gas occupies a volume of 300 cm³. Upon heating it to 200° Celsius, the volume increases to 1500 cm³. Find the initial temperature of the gas.

Solution:

According to Charles’s law,

V₁T₂ = V₂T₁

• V₁ = 300 cm³
• T₂ = 200° C
• V₂ = 1500 cm³

300 × 200 = T₁ × 1500

T₁ = 40° C

Problem 3: A gas occupies 15.5 liters at a pressure of 55 mmHg. What will be the volume when the pressure is increased to 75mmHg?

Solution:

Applying Boyle’s law, 

P₁V₁ = P₂V₂

Now, 

• P₁ = 55 mmHg
• V₁ = 15.5 liters
• P₂ = 75 mmHg

55 × 15.5 = 75 × V₂

V₂ = 11.36 liters

Problem 4: The root mean square speed of a gas molecule at 300K temperature and 2 bar pressure is 2 × 10⁴ cm/sec. If the temperature is increased two times, find the new root mean square speed of the gas molecule.

Solution:

Formula for the root mean speed, v_rms = 

Therefore, v ∝ √T

v₁/v₂ = √T₁/ √T₂

• V₁ = 2 × 10⁴ cm/sec
• T₁ = 300K
• T₂ = 2 × 300 = 600K

(2 × 10⁴)/v₂ = √300/√600

v₂ = 2√2 × 10⁴ cm/sec

FAQs on Kinetic Theory of Gases

Q1: What is Kinetic Theory of Gases?

Answer:

The Kinetic Theory of Gases is a model based on some assumptions, which help us explain the behavior of gases. In this model, gases are assumed to be made up of a large number of tiny molecules that are always in constant motion.

Q2: What are Assumptions of Kinetic Theory of Gases?

Answer:

There are several assumptions of the kinetic theory of gases, some of which are as follows:

• Gases are made up with a large number of tiny particles.
• These molecules are always in contant motion.
• The particles of gases are collide with each other and with the wall of the container.
• These collision of between particles are elastic in nature.

Q3: How is Temperature related to Kinetic Theory of Gases?

Answer:

In the kinetic theory of gases, temperature is directly proportional to the average kinetic energy of the gas particles.

Q4:  How does Kinetic Theory of Gases Explain Pressure in Gases?

Answer:

In the kinetic theroy of gases, pressure is the result of collision between particles and the wall of the container.

Q5: How does Kinetic Theory of Gases Explain Volume in Gases?

Answer:

The kinetic theory of gases explains volume as the result of the random motion of gas particles. As gas particles move around, they fill up the space available to them.

Q6: How does Kinetic Theory of Gases Explain Diffusion?

Answer:

The kinetic theory of gases explains diffusion as the result of the motion of the gas particles which is random in nature. Gas particles move randomly and collide with each other also with the other objects in their enviroment. Over time, this random motion leads the mixing of gases.

Q7: What is Relation Between Pressure, Volume, and Temperature in Kinetic Theory of Gases?

Answer:

The relationship between Pressure, Volume and Temerature in the kinetic theory of gases by the Ideal Gas equation, which is as follows:

PV = nRT

where, 

• P is pressure, 
• V is volume, 
• n is the number of moles of gas, 
• R is the gas constant, and 
• T is temperature.