Kinetic Theory of Gases

In around 1661, Boyle, Newton, and some other scientists eagerly tried to find the theory behind gases and their behavior even gave a theory named after Boyle. However, years later, around 150 years later, the actual kinetic theory of gases was discovered and it was done by Maxwell, Boltzmann, and others. The theory was based on the fact that gases have atoms and molecules that move very rapidly and the intermolecular force that acts upon solids and liquids, does not act upon gases. Let’s learn in-depth regarding the kinetic theory of gases.

Kinetic Theory of Gases

The kinetic theory of gases was introduced to explain the structure and composition of molecules with respect to submicroscopic particles. The 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 to the walls of the container, in such a case, it is difficult as well important to learn the dynamics of the gases.

Assumptions of Kinetic theory of gases

There are several assumptions that were taken into account in order to develop the kinetic theory of gas. Let’s first take a look at those assumptions and then dig deep into the theory,

• Every gas consists of molecules that are microscopic particles. All the molecules of a single gas are the same and identical but are different in properties to those of other gases’ molecules.
• 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, are rigid, and are elastic masses.
• 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.

Kinetic Theory and Gas Pressure

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

P = 1/3ρc^-2

Where c = mean square speed of a gas molecule.

ρ = Density of the gas.

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

P = 

Where V = Volume of the 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.

V ∝ 1/P

PV = Constant

For a given gas, P₁V₁ = P₂V₂

• Charles’s Law: According to Charles’s law, at constant pressure, the volume of a gas is directly proportional to the absolute temperature of the gas.

V ∝ T

V/T = Constant

Therefore, V₁/T₁ = V₂/T₂

V₁T₂ = V₂T₁

• Pressure Law (Gay lussac’s law): According to this law, at constant volume, the pressure of a given gas is directly proportional to its absolute temperature.

P ∝ T

P/T = Constant

P₁/T₁ = P₂/T₂

P₁T₂ = P₂T₁

• Avogadro’s Law: According to this law, an equal amount of volume of all gases under S.T.P. (Standard temperature and pressure) contain the same number of molecules equalling 6.023 × 10²³
• 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 ∝ 

• 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. There is a formula obtained after derivation for the mean speed of a gas molecule,

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. There is a formula derived for this term as well,

v_rms = 

The formula for the root mean speed, v_rms = 

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,

Formula for most probable speed, v_mp = 

• 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,

1. 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.
2. 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.
3. 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.

Sample Problems

Question 1: Explain the three main components of the Kinetic theory of gases?

Answer:

The three main components are,

1. The molecules of the gas have linear motion
2. No gain or loss in energy during a collision
3. The particles do not have mass and occupy negligible space in a container.

Question 2: A gas occupies 10 liters at a pressure of 30 mmHg. What will be the volume when the pressure is increased to 50mmHg?

Solution:

Applying Boyle’s law, 

P₁V₁ = P₂V₂

Now, P₁ = 30mmHg, V₁ = 10 liters, P₂ = 50mmHg

30 × 10 = 50 × V₂

V₂ = 6 liters

Question 3: A gas occupies a volume of 300 cm³. Upon heating it to 200° celsius, the volume increases to 1500cm³. Find the initial temperature of the gas.

Solution:

According to Charles’s law,

V₁T₂ = V₂T₁

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

300 × 200 = T₁ × 1500

T₁ = 40° C

Question 4: 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 = 55mmHg, V₁ = 15.5 liters, P₂ = 75mmHg

55 × 15.5 = 75 × V₂

V₂ = 11.36 liters

Question 5: 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:

The 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