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11P13

Kinetic Theory of Gases

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Learning Objectives

Equipartition law of Energy and Energy of Gases

Behaviour of Gases and Their Kinetics

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11P13.1

Behaviour of Gases and Their Kinetics

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Learning Objectives

Pressure Calculation Applied by Gas

Postulates of Kinetic Theory of Gases

Kinetic Interpretation of Temperature

Maxwell’s Speed Distribution Law

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11P13.1

CV 1

Postulates of Kinetic Theory and Maxwell’s Speed Distribution Law

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Postulates of Kinetic Theory of Gases :

  • All gases are made up of molecules moving randomly in all directions.

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Postulates of Kinetic Theory of Gases :

  • The size of a molecule is much smaller than the average separation between the molecules.
  • The molecules exert no force on each other or on the walls of the container except during collision.

 

 

 

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Postulates of Kinetic Theory of Gases :

  • All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also, the time spent during a collision is negligibly small.
  • The molecules obey Newton’s laws of motion.

 

 

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Postulates of Kinetic Theory of Gases :

  • When a gas is left for sufficient time, it comes to a steady state. The density and distribution of molecules with different velocities are independent of position, direction and time.

 

 

 

 

 

 

 

 

Number of Particles

 

 

 

 

Speed of Particles

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Postulates of Kinetic Theory of Gases :

  • When a gas is left for sufficient time, it comes to a steady state. The density and distribution of molecules with different velocities are independent of position, direction and time.

 

 

 

 

 

 

 

 

Number of Particles

 

 

 

 

Speed of Particles

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Maxwell’s Speed Distribution Law :

In a given mass of gas, the velocities of all molecules are not the same, even when the bulk parameters like pressure, volume and temperature are fixed.

Collisions changes the direction and the speed of molecules.

Distributions are very important when we are dealing with system containing large number of objects.

 

Equation of Maxwell’s Speed Distribution :

 

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Maxwell’s Speed Distribution Law :

 

 

 

 

 

It is the speed attained by maximum number of gas molecules.

 

 

By solving given equation

 

Most Probable Speed

 

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Maxwell’s Speed Distribution Law :

 

 

 

 

 

 

For Average Speed

 

Substituting values and solving,

 

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ConcepTest

Ready for a Challenge

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Sol.

 

 

 

 

 

 

 

 

 

We also know

and

We know

Pause the Video

(Time Duration : 2 Minutes)

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11P13.1

CV 2

Pressure Calculation and Kinetic Interpretation of Temperature

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Some Definitions :

Root Mean Square Speed :

The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values.

 

Translational Kinetic Energy of a Gas :

The total translational kinetic energy of all the molecules of the gas will be equal to the summation of kinetic energy of all the molecules.

 

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Calculation of the pressure of an ideal gas :

 

 

 

Ideal Gas

 

 

We know,

 

 

Pressure by gas on the wall

 

Change in momentum of one particle in one collision

 

Time between two collision with wall

 

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Calculation of the pressure of an ideal gas :

 

 

 

Ideal Gas

 

 

 

 

Total Force by all molecules

 

 

 

 

 

Due to symmetry,

 

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Calculation of the pressure of an ideal gas :

For velocity of any particle

 

 

 

 

 

 

 

 

 

 

Ideal Gas

 

 

 

 

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Calculation of the pressure of an ideal gas :

 

 

 

 

 

 

 

Ideal Gas

 

 

 

 

 

 

 

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Calculation of the pressure of an ideal gas :

 

 

 

Ideal Gas

 

 

 

 

 

 

 

 

 

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Kinetic Interpretation of Temperature :

According to KTG

 

We know

 

 

 

For 1 mole ideal gas

 

 

 

 

 

Constant for a gas

 

 

 

 

 

i.e.

 

 

 

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Translational Kinetic Energy of a Gas :

The total translational kinetic energy of all the molecules of the gas will be equal to the summation of kinetic energy of all the molecules.

 

 

 

 

So

 

The average K.E of a molecule

 

 

 

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Translational Kinetic Energy of a Gas :

 

 

 

We know,

 

 

 

 

For ideal gas

 

 

 

 

 

 

 

 

 

 

 

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Root Mean Square Speed (RMS Speed) :

The square root of mean square speed is called root-mean-square speed or rms speed.

 

 

We know

 

 

 

For ideal gas

 

 

 

 

 

 

 

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Root Mean Square Speed (RMS Speed) :

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We know

 

 

 

 

Root Mean Square Speed (RMS Speed) :

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or

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Boyle’s Law on the basis of Kinetic theory :

It states that the pressure of a given mass of an ideal gas is inversely proportional to its volume at a constant temperature.

 

 

We know

 

 

 

We know for a given gas

i.e. for a given temperature

 

 

 

 

 

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ConcepTest

Ready for a Challenge

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Sol.

Pause the Video

(Time Duration : 2 Minutes)

Q. What will be the change in rms speed of gas on increasing pressure ?

A) Increases

B) Decreases

D) Can’t say

C) Remains constant

 

We know

 

 

We can not comment on speed when pressure is changed.

Answer : C

Explanation :

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11P12.3

PSV 1

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Sol.

At STP,

 

 

We know,

 

 

 

 

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ConcepTest

Ready for a Challenge

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Q. Which of the following parameters is the same for molecules of all gases at a given temperature ?

Sol.

Pause the Video

(Time Duration : 2 Minutes)

Answer : D

A) Mass

B) Speed

C) Momentum

D) Kinetic Energy

Explanation :

 

We know

If temperature is fixed

 

 

 

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11P12.3

PSV 2

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Sol.

We know,

 

For same temperature,

 

For two different gases ,

 

 

 

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ConcepTest

Ready for a Challenge

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Sol.

Pause the Video

(Time Duration : 2 Minutes)

 

 

 

 

Answer : C

Explanation : If there is one molecule in the container then all the velocities will be equal.

 

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Summary :

  • Pressure of an ideal gas is given by,

 

  • Root mean square speed is given by,

 

  • Translation kinetic energy of the gas is given by,

 

 

 

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Summary :

  • Most probable speed of the gas is given by,

 

  • Average speed of the gas is given by,

 

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Reference Questions

NCERT : 13.1, 13.2, 13.3, 13.7, 13.8, 13.9.

Workbook : 1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 18, 19.

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11P13.2

Degree of Freedom and Energy of Gases

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Learning Objectives

Degree of Freedom

Law of Equipartition of Energy

Mean Free Path

Specific Heat Capacity of Gases

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11P13.2

CV 1

Degree of Freedom and Equipartition Law

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Degree of Freedom :

For Translatory Motion :

A particle moving in a straight line along any one of the axes has one degree of freedom.

A particle moving in a plane (X and Y axes) has two degrees of freedom.

A particle moving in space (X, Y and Z axes) has three degrees of freedom.

Simple pendulum

It is defined as the total number of independent variables required to describe the position and configuration of the system.

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A point mass can not undergo rotation, but only translatory motion.

A rigid body with finite mass has both rotatory and translatory motion.

The rotatory motion also can have three co-ordinates in space, like translatory motion.

For Rotatory Motion :

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Law of Equipartition of Energy :

It states that for a dynamical system in thermal equilibrium, the total energy is equally distributed in all possible energy modes (degree of freedom).

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According to kinetic theory of gases,

​

The mean kinetic energy of a molecule of monoatomic gas

Since molecules move at random,

Law of Equipartition of Energy :

 

 

 

Energy associated with each degree of freedom per molecule

 

 

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Examples : Molecules of rare gases like helium, argon, etc.

Monoatomic molecule :

Since a monoatomic molecule consists of only a single atom of point mass.

It has three degrees of freedom of translatory motion along the three co-ordinate axes .

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Monoatomic molecule :

 

 

 

 

 

 

 

 

 

Internal energy of a monoatomic gas

 

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Hence, a diatomic molecule has five degrees of freedom.

Diatomic molecule :

The diatomic molecule can rotate about any axis at right angles to its own axis.

Hence it has two degrees of freedom of rotational motion in addition to three degrees of freedom of translational motion along the three axes.

 

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Diatomic molecule :

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Potential energy of two molecules

Kinetic energy

of vibration

 

 

 

 

If Diatomic gas molecule have considerable vibration. Then,

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Diatomic molecule :

 

For non-vibrating:

For vibrating:

 

 

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Triatomic molecule (Linear Type) :

In the case of triatomic molecule of linear type, the center of mass lies at the central atom.

It therefore behaves like a diatomic molecule with three degrees of freedom of translation and two degrees of freedom of rotation, totally it has five degrees of freedom.

 

 

 

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Triatomic molecule (Nonlinear Type) :

A triatomic non-linear molecule may rotate, about the three mutually perpendicular axes.  Therefore,  it possesses three degrees of freedom of rotation in addition to three degrees of freedom of translation along the three co-ordinate axes.

Hence, it has six degrees of freedom. 

 

 

 

 

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11P13.2

PSV 1

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Answer : D

Sol:

 

 

Since the gas is changed to monoatomic gas

 

 

 

 

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11P13.2

CV 2

Specific Heat Capacities of Gases and Mean Free Path

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Specific Heat Capacity of Gases :

Monoatomic Gases :

In monoatomic gas, molecules have three translational degrees of freedom.

 

 

For one mole of such gas at constant volume,

Internal Energy

 

 

 

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Specific Heat Capacity of Gases :

According to the first law of thermodynamics

Where

 

 

 

 

 

 

 

 

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Specific Heat Capacity of Gases :

 

If heat transfer takes place at constant volume

Since the volume is constant

 

 

 

 

 

 

For monoatomic gas

 

 

 

 

By the definition of molar specific heat

 

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Specific Heat Capacity of Gases :

We know from 1st law of thermodynamics

 

 

For ideal gas heat transfer at constant pressure

 

 

 

 

Substituting in 1st law of thermodynamics

 

 

 

 

 

 

 

 

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Specific Heat Capacity of Gases :

 

 

 

We know

and

 

 

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Specific Heat Capacity of Gases :

Diatomic Gases :

In case of diatomic gases, there are two possibilities :

i) Molecule is a Rigid Rotator :

In this scenario, the molecule will have five degrees of freedom.

 

3 Translational

5 Degrees of Freedom

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Specific Heat Capacity of Gases :

By the law of equipartition of energy

 

 

 

We know

We know

Internal energy of a molecule of diatomic gas

Internal energy at constant volume

 

 

 

By the definition of molar specific heat

 

 

 

 

 

 

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Specific Heat Capacity of Gases :

 

 

 

We know

and

 

 

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Specific Heat Capacity of Gases :

 

ii) Molecule is not a Rigid Rotator :

In this scenario, the molecule will have two additional vibrational degree of freedom.

So the internal energy can thus be calculated as

 

 

 

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Specific Heat Capacity of Gases :

Internal energy at constant volume

 

 

 

By the definition of molar specific heat

 

 

We know

 

 

 

 

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Specific Heat Capacity of Gases :

 

 

 

We know

and

 

 

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Specific Heat Capacity of Gases :

Polyatomic Gases :

The degrees of freedom of polyatomic gases are,

 

 

 

 

Deploying the Law of Equipartition of Energy for calculation of internal energy

we get,

 

 

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Specific Heat Capacity of Gases :

 

The molar specific heat capacities of polyatomic gases using same analysis done for monoatomic gases.

 

 

 

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Specific Heat Capacity of Solids :

 

Each atom is oscillating along its mean position

 

Potential energy of two molecules

Kinetic energy

of vibration

 

Hence, the average energy in three dimensions of the atom would be

 

 

 

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For one mole of solid

Specific Heat Capacity of Solids :

 

 

 

 

In case of solids, volume change is negligible.

 

From 1st law of thermodynamics

 

 

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Specific Heat Capacity of Water :

And, following a similar calculation like solids

For the purpose of calculation of specific heat capacity, water is treated as a solid.

A water molecule has three atoms.

 

Hence, its internal energy would be

 

 

 

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11P13.2

PSV 2

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A) 22 calorie 

B) 4 calorie 

C) 8 calorie 

D) 12 calorie

Sol:

We know

 

(For a monoatomic gas)

(For a diatomic gas)

 

Thus for the mixture

 

 

 

 

 

Answer : (B)

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Mean Free Path :

Hence, a molecule follows a chain of zigzag paths. Each of these paths is known as a free path which lies between two collisions.

The Kinetic theory of gases assumes that molecules are continuously colliding with each other and they move with constant speeds and in straight lines between two collisions.

Free path

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Mean Free Path :

The average of the distances travelled by the molecule between all the collisions is known as Mean Free Path. 

The number of collisions increases if the gas is denser or the molecules are large in size.

The mean free path

 

 

 

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Summary :

Degree of Freedom defined as the total number of independent variables required to describe the position and configuration of the system.

Law of equipartition of energy states that for a dynamical system in thermal equilibrium, the total energy is equally distributed in all possible energy modes (degree of freedom).

The average of the distances travelled by the molecule between all the collisions is known as Mean Free Path. 

 

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Reference Questions

NCERT – 13.4, 13.5 ,13.6, 13.10.

Workbook –2, 3, 8, 14, 16, 17, 20.