Kinetic Theory of Gases: Assumptions, Postulates, Gas Laws, Formulas

As the word kinetic itself justifies that this topic is related to the ‘motion’ or ‘movement’ of particles. Kinetic Theory of Gases elucidates the behaviour of gases that consist of rapidly moving atoms or molecules. According to the Kinetic Theory of gases, it is supposed that the molecules are very tiny particles relative to the distance between molecules. These molecules are in constant motion and frequently collide with each other and with the walls of any container as they move randomly. This randomness increases with the temperature. In this article, let’s learn everything about the kinetic theory of gases in detail.

Introduction to Kinetic Theory of Gases

James. C. Maxwell a British scientist, and Ludwig Boltzmann an Austrian physicist established the kinetic theory of gases in the \({\rm{1}}{{\rm{9}}^{{\rm{th}}}}\) century. Based on their assumptions, they formulated the Postulates of kinetic theory of gases. The macroscopic properties of gases, such as volume, pressure, and temperature, as well as the movement of gaseous particles, are explained by the kinetic theory of gases.

Kinetic Theory of Gas is the basis of the Ideal gas equation i.e. \({\rm{PV = nRT}}\). According to this, there is no attractive force between gaseous molecules and individual gas molecules that occupy almost negligible volume as compared to the total volume of gases in a container. 

But in practice, this assumption proves to be wrong and there is no gas molecule that behaves ideally. For example, the liquification of gases proves the presence of attractive force between the gaseous molecules. Thus, the concept of ‘Real Gases’ emerged. Kinetic theory of gases describes the behaviour of gases in constant motion.

What is Kinetic Theory of Gases?

Kinetic Theory of Gases states that gaseous particles are in constant motion and undergo perfectly elastic collisions. In a collection of gas particles, the average kinetic energy is directly proportional to absolute temperature.

According to this theory, the gas molecule is composed of a huge number of tiny molecules compared to the distances between them. The kinetic theory of gases is necessary for clarifying the process of trapping particles by the diffusion mechanism. Let’s learn about the relation between these macroscopic properties with the microscopic phenomenon.

Kinetic Theory of Gases Assumptions

The various assumptions of kinetic theory of gases are discussed as under:
1. It is assumed that gas molecules are constantly moving in random directions.
2. The gaseous molecules are very tiny particles relative to the distance between them.
3. Gas molecules have negligible volume and intermolecular forces.
4. While random motion, gas particles undergo perfectly elastic collisions.
5. They have an average kinetic energy proportional to the absolute temperature of the ideal gas.

Kinetic Theory of Gases Postulates

James Clark Maxwell, Clausius, and Rudolph studied and proposed the basic postulates of the kinetic theory of gases which are as follows:

1. Gases consist of a large number of extremely small particles i.e. atoms and molecules. As compared to the distance between the particles, the size of these particles is extremely small.

2. The volume of the gaseous particles is almost negligible.

3. These molecules are in constant motion that moves randomly and results in colliding with each other and with the walls of the container. Pressure is exerted by the gas molecules due to such collisions. 

4. The collisions between the molecules and the walls are completely elastic. This means that after the collision between the particles, the kinetic energy remains the same. 

5. The average kinetic energy of the gas particles fluctuates with temperature. As temperature increases, the (Eav) average kinetic energy of the gases increases.

6. The molecules do not employ any force of attraction or repulsion on one another except during collisions.

Kinetic Theory of Gases Formulas

1. Total translational kinetic energy of gas

\({\rm{K}}{\rm{.E}}{\rm{. = }}\frac{{\rm{3}}}{{\rm{2}}}{\rm{nRT}}\)
Where: \({\rm{n}}\) is the number of moles
\({\rm{R}}\) is the universal gas constant
\({\rm{T}}\) is the absolute temperature

2. RMS speed

The root-mean-square speed denoted by \({{\rm{V}}_{{\rm{rms}}}}\) is defined as the average velocity of the molecules in a gas.
\({{\rm{V}}_{{\rm{rms}}}}{\rm{ = }}\sqrt {\frac{{{\rm{3RT}}}}{{\rm{M}}}} \)
Where: \({\rm{R}}\) is the gas constant
\({\rm{T}}\) is the absolute temperature
\({\rm{M}}\) is molar mass

3. Boltzmann constant (k)

\({\rm{k = n}}\frac{{\rm{R}}}{{\rm{N}}}\)
Where: \({\rm{R}}\) is the gas constant
\({\rm{n}}\) is the number of moles
\({\rm{N}}\) is the number of particles in one mole (the Avogadro number)

4. Average velocity \(\left( {{{\rm{V}}_{{\rm{av}}}}} \right)\)

\({{\rm{V}}_{{\rm{av}}}}{\rm{ = }}\sqrt {\frac{{{\rm{8RT}}}}{{{\rm{\pi M}}}}}\)
Where: \({\rm{R}}\) is the gas constant
\({\rm{T}}\) is the absolute temperature
\({\rm{M}}\) is molar mass

5. Most probable velocity \(\left( {{{\rm{V}}_{{\rm{mp}}}}} \right)\)

\({{\rm{V}}_{{\rm{mp}}}}{\rm{ = }}\sqrt {\frac{{{\rm{2RT}}}}{{\rm{M}}}} \)
Where: \({\rm{R}}\) is the gas constant
\({\rm{T}}\) is the absolute temperature
\({\rm{M}}\) is molar mass

6. Pressure of gas

\({\rm{P = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{\rho V}}_{{\rm{rms}}}^{\rm{2}}\)
Where: \({\rm{\rho }}\) is density
\({{\rm{V}}_{{\rm{rms}}}}\) is RMS velocity

7. Equipartition of energy

\({\rm{K = }}\frac{{\rm{f}}}{{\rm{2}}}{\rm{kT}}\)
Where: \({\rm{f}}\) is degree ofdom
\({\rm{k}}\) is the Boltzmann’s constant
\({\rm{T}}\) is the temperature of the gas

8. Internal energy (U)

\({\rm{U = }}\frac{{\rm{f}}}{{\rm{2}}}{\rm{nRT}}\)
Where: \({\rm{f}}\) is degree ofdom
\({\rm{n}}\) is moles of an ideal gas.

Kinetic Theory of Gases Proof

Kinetic theory of gases proves all ideal gas laws: Boyles’s law, Charles’s law, and Avogadro’s Law. Let’s prove them one by one: 

Boyle’s Law

According to Boyle’s law at constant temperature pressure and volume of a gas are related as \({\rm{PV = }}\) Constant
According to the kinetic theory of gases, the pressure of the gas is:
\({\rm{P = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{\rho }}{{\rm{V}}^{\rm{2}}}\)
\({\rm{PV = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{mN}}{{\rm{V}}^{\rm{2}}}\)

Multiplying and dividing by \(2\) on R.H.S
\({\rm{PV = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{N}}\left( {\frac{{\rm{1}}}{{\rm{2}}}{\rm{m}}{{\rm{V}}^{\rm{2}}}} \right)\)
\({\rm{PV = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{NKE = }}\) Constant
Hence, Boyle’s law is proved.

Charles’s Law

We know that, from KTG
(Pressure) \({\rm{P = }}\frac{{{\rm{mn}}{{\rm{v}}^{\rm{2}}}}}{{\rm{2}}}\)
\({\rm{v}} \to\) R.M.S Speed
Also \({\rm{N = }}\frac{{\rm{n}}}{{\rm{V}}}\)
\(({\rm{N}} \to \) Total no. of gas molecules)
\(({\rm{V}} \to \)Volume)
\({\rm{PV = }}\frac{{{\rm{mN}}{{\rm{V}}^{\rm{2}}}}}{{\rm{3}}}\)

Now, \({\rm{E = }}\frac{{{\rm{m}}{{\rm{V}}^{\rm{2}}}}}{{\rm{2}}}\) (kinetic energy of a molecule)
Thus, \({\rm{PV = }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{NE}}\)
Now from the kinetic interpretation of temperature.
\({\rm{E = }}\frac{{\rm{3}}}{{\rm{2}}}{\rm{KT,K}} \to \) Boltzmann constant
Thus, \({\rm{PV = NKT}}\)

Since \({\rm{N}}\) and \({\rm{K}}\)are constants then for faxed \({\rm{P}}\)
\(\frac{{\rm{V}}}{{\rm{T}}}{\rm{ = }}\) Constant
So, Charles’s law is proved.

Avogadro’s Law

According to the kinetic theory of gases:
\({\rm{P = }}{\mkern 1mu} \frac{{\rm{1}}}{{\rm{3}}}\rho {{\rm{v}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{3}}}\frac{{\rm{M}}}{{\rm{V}}}{{\rm{v}}^{\rm{2}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{3}}}\frac{{{\rm{nm}}}}{{\rm{V}}}{{\rm{v}}^{\rm{2}}}\)

For gas having mass \({{\rm{m}}_{\rm{1}}}\)
\({\rm{P = }}{\mkern 1mu} \frac{{\rm{1}}}{{\rm{3}}}\frac{{{{\rm{n}}_{\rm{1}}}{{\rm{m}}_{\rm{1}}}}}{{\rm{v}}}{\rm{v}}_{\rm{1}}^{\rm{2}}\,\,\,\,…\left( 1 \right)\)

For gas having mass \({{\rm{m}}_{\rm{2}}}\)
\({\rm{P = }}\frac{{\rm{1}}}{{\rm{3}}}\frac{{{{\rm{n}}_{\rm{2}}}{{\rm{m}}_2}}}{{\rm{v}}}{\rm{v}}_2^{\rm{2}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} …\left( 2 \right)\)

From the first and second equation
\({{\rm{m}}_{\rm{1}}}{{\rm{n}}_{\rm{1}}}{\rm{v}}_{\rm{1}}^{\rm{2}}{\rm{ = }}\,{{\rm{m}}_{\rm{2}}}{{\rm{n}}_{\rm{2}}}{\rm{v}}_{\rm{2}}^{\rm{2}}\,\,\,\,…\left( 3 \right)\)

Since the temperature of both the gases are same, So
\(\frac{{\rm{1}}}{{\rm{2}}}{{\rm{m}}_{\rm{1}}}{\rm{v}}_{\rm{1}}^{\rm{2}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{{\rm{m}}_{\rm{2}}}{\rm{v}}_{\rm{2}}^{\rm{2}}\,\,\,\,…\left( 4 \right)\)

From equation \(3\) and \(4\)
We can conclude that
\({{\rm{n}}_{\rm{1}}}{\rm{ = }}{{\rm{n}}_{\rm{2}}}\)
The number of molecules of both gases is the same.
Hence, Avogadro’s law is proved.

Equation of Kinetic Theory of Gases Derivation

The kinetic theory of gases says that the macroscopic properties of a gas i.e. pressure, volume, and temperature are just a result of the microscopic properties of gas molecules such as position, velocity, etc. So when particles in a box collide, due to kinetic energy, it again bounces back to its original position. So, We need to establish a relation between the pressure exerted by the collided particles and the wall of the box.

According to Newton’s second law of motion force exerted is equal to change in momentum.
Thus, \({\rm{F = }}\frac{{\Delta {\rm{p}}}}{{\Delta {\rm{t}}}}{\rm{ = }}\frac{{{\rm{m}}\Delta {\rm{v}}}}{{\Delta {\rm{t}}}}\)
Where: \({\rm{p}}\) denotes momentum
\({\rm{M}}\) denotes the mass of the particle
\({\rm{T}}\) denotes the time

Since the collided particle is moving twice in the same distance. Thus change in velocity will be twice. We can write:
\({\rm{F = }}\frac{{{\rm{m2Vx}}}}{{{\rm{\Delta t}}}}\) ………\((1)\)
Now, we can assume the length of the sides of the box as ‘L’. When the collision occurs, the particles travel twice the distance. So we can write:
\({{\rm{V}}_{\rm{x}}}{\rm{ = }}\frac{{{\rm{2L}}}}{{{\rm{\Delta t}}}}\)
Thus, \({\rm{\Delta t = 2L/Vx}}\)

Hence, we can replace \({\rm{\Delta t}}\) from equation \((1)\) and get:
\({\rm{F = }}\frac{{{\rm{mV}}_{\rm{x}}^{\rm{2}}}}{{\rm{L}}}\) …….\((2)\)
This particle when moves in various x directions (\({{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}\) and so on till \({{\rm{x}}_{\rm{n}}}\)), the equation \((2)\) become:
\({\rm{F = }}\frac{{{\rm{m}}\left( {{\rm{V}}_{{\rm{x1}}}^{\rm{2}}{\rm{ + V}}_{{\rm{x2}}}^{\rm{2}}{\rm{ +  \ldots V}}_{{\rm{xn}}}^{\rm{2}}} \right)}}{{\rm{L}}}\) ………\((3)\)

Dividing L.H.S. and R.H.S by N (number of particles):
\(\frac{{\rm{F}}}{{\rm{N}}}{\rm{ = }}\frac{{{\rm{m}}\left( {{\rm{V}}_{{\rm{x1}}}^{\rm{2}}{\rm{ + V}}_{{\rm{x2}}}^{\rm{2}}{\rm{ +  \ldots V}}_{{\rm{xn}}}^{\rm{2}}} \right)}}{{{\rm{LN}}}}\)
Therefore, \(\frac{{\rm{F}}}{{\rm{N}}}{\rm{ = }}\frac{{\rm{m}}}{{\rm{L}}}\overline {{\rm{V}}_{\rm{x}}^{\rm{2}}} \)
\({\rm{F = N}}\frac{{\rm{m}}}{{\rm{L}}}\overline {{\rm{V}}_{\rm{x}}^{\rm{2}}} \) …….\((4)\)

Dividing L.H.S. and R.H.S by A (area):
\(\frac{{\rm{F}}}{{\rm{A}}}{\rm{ = N}}\frac{{\rm{m}}}{{{\rm{AL}}}}\overline {{\rm{V}}_{\rm{x}}^{\rm{2}}} \)
We know that Area is \({{\rm{L}}^{\rm{2}}}\) and F/A is pressure. Therefore:
\({\rm{P = }}\frac{{{\rm{NmV}}_{\rm{x}}^{\rm{2}}}}{{{{\rm{L}}^{\rm{3}}}}}\)
Now, we know that \({{\rm{L}}^{\rm{3}}}{\rm{ = V}}\) (volume) 
Thus, \({\rm{PV = Nm}}\overline {{\rm{v}}_{\rm{x}}^{\rm{2}}} \) ……….\((5)\)

If gas molecules move in all \({\rm{x,y}}\) and \({\rm{z}}\) directions:
\({{\rm{v}}_{{\rm{total }}}}{\rm{ = v}}_{\rm{x}}^{\rm{2}}{\rm{ + v}}_{\rm{y}}^{\rm{2}}{\rm{ + v}}_{\rm{z}}^{\rm{2}}\)
\(\left( {\overline {{\rm{v}}_{{\rm{total }}}^{\rm{2}}} } \right){\rm{ = }}\left( {\overline {{\rm{v}}_{\rm{x}}^{\rm{2}}} } \right){\rm{ + }}\left( {\overline {{\rm{v}}_{\rm{y}}^{\rm{2}}} } \right){\rm{ + }}\left( {\overline {{\rm{v}}_{\rm{z}}^{\rm{2}}} } \right)\)

Since particle can move randomly so we can generalize all the directions as \({{\rm{v}}_{\rm{x}}}{\rm{,}}\)_, Thus:
\(\left( {\overline {{\rm{v}}_{{\rm{total }}}^{\rm{2}}} } \right){\rm{ = 3}}\left( {\overline {{\rm{v}}_{\rm{x}}^{\rm{2}}} } \right)\)
Thus, \(\frac{{\left( {\overline {{\rm{v}}_{{\rm{total }}}^{\rm{2}}} } \right)}}{{\rm{3}}}{\rm{ = }}\left( {\overline {{\rm{v}}_{\rm{x}}^{\rm{2}}} } \right)\) ……\((6)\)

Substituting equation \((5)\) in equation \((6)\):
\({\rm{PV = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{Nm}}\overline {\left( {{\rm{v}}_{{\rm{total}}}^{\rm{2}}} \right)}\)
This is the equation of the kinetic theory of gases.

Summary

Thus we have learned that the model, called the kinetic theory of gases, assumes that the molecules are very small with respect to the distance between molecules. The molecules are in constant and random motion and an elastic collision occurs among them with the walls of any container.

Hence, we can conclude that the Ideal Gas Law holds true to some extent. The kinetic theory of gases is important for the diffusion mechanism. According to this theory, gas is composed of a large number of small-sized molecules or particles compared with the distances between them.

Frequently Asked Questions (FAQs) – Kinetic Theory of Gases

Q.1. What are the 5 assumptions of the kinetic theory of gases?
Ans: The 5 assumptions of kinetic theory of gases are:
a) It is assumed that gas molecules are constantly moving in random directions.
b) The gaseous molecules are very tiny particles relative to the distance between them.
c) Gas molecules have negligible volume and intermolecular forces.
d) While in random motion, gas particles undergo perfectly elastic collisions.
e) They have an average kinetic energy proportional to the absolute temperature of an ideal gas.

Q.2. What are the three main points of the kinetic theory of gases?
Ans: The three main points of the kinetic theory of gases are:
a) No energy is gained or lost when molecules collide with each other.
b) space and volume occupied by the gas molecules in a container are very negligible.
c) The gas molecules always move linearly.

Q.3. What are the two main ideas in the kinetic theory of matter?
Ans: The two main ideas in the kinetic theory of matter are that the solid particles have the least amount of energy, and gas particles have the highest amount of energy. The average kinetic energy of the particles is directly proportional to the temperature of a substance. With a change in energy, a change in the phase of matter may occur.

Q.4. Why is the kinetic theory of gases important?
Ans: The kinetic theory of gases is important for studying the nature of gas particles with respect to diffusion mechanism, i.e. deviation from ideal behaviour. According to this theory, gas is composed of a large number of tiny molecules compared with the distances between them.

Q.5. What are the limitations of the kinetic theory of gases?
Ans: The main limitations of the kinetic theory of matter come when the condensed state of matter is discussed: Kinetic theory is not satisfied in the liquid state due to the fact that in liquids, molecules are quite close together, but still have a certain degree of constant, random motion.

Q.6. Is the kinetic energy of gases high?
Ans: The kinetic energy of gases is higher than solids and liquids because gaseous molecules move randomly as they have the least inter-molecular force of attraction and when the temperature is increased then solid and liquid substances are converted into the gaseous state.

Q.7. How many postulates of the kinetic theory of gases are there?
Ans: There are five postulates of the kinetic theory of gases.

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