Chapter–13: Kinetic Theory of gases

Kinetic theory of gases – assumptions, concept of pressure. Kinetic interpretation of temperature; rms speed of gas molecules; degrees of freedom, law of equi-partition of energy (statement only) and application to specific heat capacities of gases; concept of mean free path, Avogadro’s number. Equation of state of a perfect gas, work done in compressing a gas.

Unit-9 Kinetic Theory of Gases

Boyle’s Law:-

According to Boyle’s law, at constant temperature, the volume of a gas is inversely proportional to its pressure. i.e.

Here are constant of proportionality depending upon mass, temperature and units of

If is initial pressure and volume of the gas and are final pressure and volume of the gas then

As graph shows that when pressure decreases then volume increases or vice versa.

Charle’s Law:

According to Charle’s law at constant pressure, the volume of gas is directly proportional to the temperature of the gas. i.e.

Or Or

If is the initial volume and temperature of the gas and is the final volume and temperature of the gas then

The graph between T/V is as shown in fig. The graph is a straight line graph.

The other form of Charle’s law states that if pressure remains constant then volume of gas increases or decreases by of its volume at for each rise or fall of temperature. I.e. volume at one degree is which can be given as

Example: 1.The volume of a given mass of a gas at 27˚C, 1 atm is 100 cc. What will be its volume at 327˚C ?

Ans. Given that

If P is constant then or

Gay Lussac’s Law:

According to this law at constant volume, the pressure of a gas is directly proportional to the temperature of the gas. i.e.

If are the initial pressure and temperature of the gas and are the final pressure and temperature of the gas then at constant volume

The Gay Lussac’s law may also be given as

Where is pressure at and is pressure at of the gas

Perfect Gas and Perfect gas equation:

Perfect Gas or Ideal Gas:

A gas which strictly obeys the Boyle’s law Charle’s law and Gay Lussac’s law is called perfect gas. In reality there is no any gas is perfect gas.

Derivation:

According to Boyle’s Law, at constant temperature

And according to Charle’s law, at constant pressure

Combing the above eq. we get

Now we may write for one mole of a gas as

For mole of gas this eq. is called perfect gas equation.

Universal Gas Constant:-

As we know for ideal gas

Hence universal gas constant may be defined as the work done by a gas per unit mole per unit Kelvin.

Example: 2 calculate the number of molecules in each cubic metre of a gas at 1 atm and 27 °C

Solution: We have

Example:3. Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and Pressure (STP: 1 atm pressure, 0°C). Show that it is 22.4 liters.

Ans. Here n =1 mol, T = 273 K,

Using the relation

Or,

Kinetic Theory of An Ideal Gas:

The Kinetic theories given by Claussius and Maxwell have the following assumptions.

All the gases are consisting of molecules. The molecules of a gas are identical but are different of different gases.
The size of the molecules is negligible compared with the distance between the molecules
The molecules of the gas are in a stating continuous random motion, moving in all directions with all possible velocities.
During the random motion, the molecules collide with one another and with the wall of container.
The collision of gas molecules are perfect elastic.
There is no any force of attraction or repulsion between the molecules.
Between two collisions, the gas molecule moves in a straight line.
The average distance travelled by particle between two successive collision is called mean free path.
The collision are almost instantaneous, the time during collision is very negligible small.
The density remains uniform in the container containing gas.

How does a gas exert pressure:

As gases collides the container containing gas and impart momentum on the wall of the container, now according to Newton’s second law of motion, the rate of momentum exerted on the wall is equal to the force.

This force per unit on the wall of the container gives the pressure.

Express for pressure exerted by a gas:

Suppose numbers of gas molecules each of mass are moving with velocities in a cubical vessel of each side

Now suppose one molecule move toward axis and collide with the container and impart momentum = and regain its original path with back momentum .

So total change in momentum

By conversion of total change in momentum

Total time taken by the gas molecules to move distance with velocity is

So force exerted by gas molecules on the container

If there are number of molecules per unit volume then total number of molecules

So total force

Similarly &

As we may let

Where is mean square speed of the gas molecules.

Hence

( For unit volume )

This is the required expression for pressure exerted by gas molecules.

Relation between Pressure and K.E. of Gas Molecules:

As we know

Mass of the unit volume of gas

Dividing eq. (i) by (ii) we get

Hence pressure exerted by a gas is equal to two-third of kinetic energy of the gas.

Kinetic Interpretation of temperature:

As we know

And from kinetic theory of gases

For unit volume of gas molecules

Or average K.E. of gas molecules

Hence K.E. of gas molecules is directly proportional to the temperature of gas.

Maxwell’s Speed Distribution:

James Clerk Maxwell was the first to derive a mathematically relation for the most probable distribution of speed among the molecules of gas.

The graph between number of molecules and speed of the gas molecules is as shown in fig. the important feature of speed distribution curve are.

At any temperature the speed of molecules varies from zero or infinity.
The speed passed by the larger number of molecules is called most probable speed.

Average, Root means square and most probable speeds:

Average Speed:-

It is defined as the arithmetic mean of the speed of the molecules of a gas.

If are the speed of molecules, then average speed can be given as

From Maxwell’s speed distribution law, the average speed may be given as

Root Mean Square Speed:-

It is defined as the root of mean of square of speed of gas molecules.

And from Maxwell’s speed distribution law, the root mean square speed of gas is

Most Probable Speed:-

It is defined as the speed of the maximums number of molecules in gas and Maxwell’s distribution law

Relation between :-

Clearly

Example: 4 Find the rms speed of oxygen molecules in a gas at300 K.

Solution:

Example: 5. Two molecules of a gas have speed of and respectively. What is the root mean source speed of these molecules?

Ans. Root mean square speed

For two molecules

Degree of Freedom:-

The total number of independent co-ordinate required to describe completely the position and configuration of the system is called degree of freedom. e.g:-

A particle moving along a straight line can change only one axis so have only one degree freedom?
A particle moving in plane needs two co-ordinates to change so have two degree of freedom
A particle moving in space can change three co-ordinates so have three degree of freedom.
There are N number of particle, each having three degree of freedom and K number of constrains or independent relations, then total number of degree of freedom
A rigid body has six degree of freedom, 3 Translatory and three rotatory.
Degree of freedom of monatomic Gas:

As monatomic gas like are consist of single molecules having only translatory motion. So have three degree of freedom.

Degree of freedom of diatomic Gas:

The molecules like are consist of two atoms, connected by one bond so degree of freedom

Degree of freedom of Tri-atomic Gas:
Non linear gas like In this case the degree of freedom
Linear gas In this case there are only two constrains so

Law of Equipartion of Energy:

According to law of equipartition of energy, the energy is equally distributed among the various degrees of freedom and the average energy associated with each degree of freedom is

Proof: – as we know from kinetic theory of gases.

Here

Thus average kinetic energy per degree of freedom this is obtained by Boltzmann, so also known as Boltzman’s equip partition of energy.

Specific heat of Monatomic, Diatomic and Poly atomic Gases:

(a) Specific heat of monatomic gas:-

As monatomic gas have three degree of freedom. So energy associated with monatomic gas is

U= Where

So total energy associated with one mole of gas

Now Molar specific heat of gas at constant volume or

And Molar specific heat at constant pressure

So specific heat ratio

(b) Specific heat of Diatomic Gas:

The total degree of freedom of diatomic gas is

So energy associated with one mole of diatomic gas

Now And

(c) Specific Heat of Tri-atomic Gas:

(i) For non linear tri-atomic gas:

And And

For linear tri-atomic gas:

As linear tri-atomic gas has seven degree of freedom so

So And

(d) Specific heat of poly-atomic gas:

Suppose one mole of poly atomic gas having degree of freedom,

Then average energy associated with gas Or

So And

So specific heat coefficient

Specific heats of solids: Dulong and Petit’s Law:-

In solids a particle may have K.E as well as P.E each having three degree of freedom.

Now from equilibrium of energy

And

So the average Vibrational energy per atom

Or So

Hence according to Dulong and Petits law the specific heat of solids at constant volume is equal to 3R.

With rise in temperature the molar specific heat of solids increases with increases in temperature and becomes constant at temperature equal to 3R. This temperature is called Debye temperature.

Specific heat of water:-

As we know average Vibrational energy per water molecule

So energy associated with water molecule

This is in close agreement with the predicted value of water.

Unit of specific heat:- 1 Calorie=4.186J

One calorie is defined as the amount of heat required to raise the temperature of of water from

Mean free path:

The average distance travelled by the molecule between two successive collisions is called mean free path.

Suppose molecule covers paths between successive collisions. Then its mean free path is given by

Let’s consider our single-molecule to have a diameter of d moves through the gas; it sweeps out a short cylinder of cross-section area between successive collisions,

For a small-time t it will move a distance of vt where v is the velocity of the molecule,

Now if we sweep this cylinder we will get a volume of

Since N/V is the number of molecules per unit volume, so the number of molecule in the cylinder will be ,

The mean free path can be derived as follows,

we have assumed that all the particles are stationary with respect to the particle we are studying, in fact, all the molecules are moving relative to each other, actually the v in the numerator is the average velocity and v In the denominator is relative velocity hence they both differ from each other with a factor therefore the final equation would be,

Mean free path at sea level is 0.1 micrometers.

Factor on which mean free path depends.

Mean free path depends directly on mass of gas molecules, temperature and inversely on density, molecular diameter, and pressure of the gas molecules.

Avogadro’s number:-

The number of particles present in one mole of the substance in called Avogadro number.

i.e

Brownian motion:

The Zigzag motion of gas molecules is called Brownian motion.

It depends upon Size of suspended particles, density of fluids, temperature and viscosity of the fluids.

SOME IMPORTANT MCQ:

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

(a) Mass (b) Speed (c) Momentum (d) Kinetic energy

A gas behaves more closely as an ideal gas at

(a) Low pressure and low temperature (b) low pressure and high temperature

The pressure of an ideal gas is written as ⋅ Here E refers to

(a) Translational kinetic energy (b) rotational kinetic energy

The energy of a given sample of an ideal gas depends only on its

(a) Volume (b) pressure (c) density (d) temperature.

Which of the following gases has maximum rms speed at a given temperature?

(a) Hydrogen (b) nitrogen (c) oxygen (d) carbon dioxide

Figure shows graphs of pressure vs density for an ideal gas at two temperatures and .

(a) (b) (c) (d) Any of the three is possible.

The mean square speed of the molecules of a gas at absolute temperature T is proportional to

(a)

Suppose a container is evacuated to leave just one molecule of a gas in it. Let and represent the average speed and the rms speed of the gas.

(a) (b) (c) (d)

The quantity pV=kT represents

(a) Mass of the gas (b) kinetic energy of the gas

There is some liquid in a closed bottle. The amount of liquid is continuously decreasing. The vapour in the remaining part

(a) Must be saturated (b) must be unsaturated (c) may be saturated (d) there will be no vapour.

There is some liquid in a closed bottle. The amount of liquid remains constant as time passes. The vapour in the remaining part

(a) Must be saturated (b) must be unsaturated (c) may be unsaturated (d) there will be no vapour.

Vapour is injected at a uniform rate in a closed vessel which was initially evacuated. The pressure in the vessel

(a) Increases continuously (b) decreases continuously

Which of the following quantities is zero on an average for the molecules of an ideal gas in equilibrium?

(a) Kinetic energy (b) Momentum (c) Density (d) Speed.

Keeping the number of moles, volume and temperature the same, which of the following are the same for all ideal gases?

(a) Rms speed of a molecule (b) Density (c) Pressure (d) Average magnitude of momentum

The average momentum of a molecule in a sample of an ideal gas depends on

(a) Temperature (b) number of moles (c) volume (d) none of these.

Which of the following quantities is the same for all ideal gases at the same temperature?

(a) The kinetic energy of 1 mole (b) The kinetic energy of 1 g

(c) The number of molecules in 1 mole (d) the number of molecules in 1 g

Consider the quantity of an ideal gas where M is the mass of the gas. It depends on the

(a) Temperature of the gas (b) volume of the gas (c) pressure of the gas (d) nature of the gas

according to kinetic theory of gases, at absolute zero temperature

(a) Water freezes (b) liquid helium freezes (c) molecular motion stops (d) liquid hydrogen freezes

At 0K which of the following of a gas will be zero?

(a) Kinetic energy (b) Potential energy (c) Vibrational energy (d) Density

Which of the following molecular properties is the same for all ideal gases at a given temperature?

(a) Momentum (b) rms velocity (c) mean kinetic energy (d) mean free path

If the pressure and the volume of certain quantity of ideal gas are halved, then its temperature

(a) Is doubled (b) becomes one-fourth (c) Remains constant (d) is halved

Boyle’ law is applicable for an

(a) Adiabatic process. (b) Isothermal process. (c) Isobaric process. (d) Isochoric process

A fly moving in a room has …X… degree of freedom. Here, X refers to

(a) One (b) two (c) three (d) four

The total number of degree of freedom of a CO2 gas molecule is

(a) 3 (b) 6 (c)5 (d) 4

The degree of freedom of a molecule of a tri-atomic gas is

(a) 2 (b) 4 (c) 6 (d) 8

Mean free path of a gas molecule is

(a) Inversely proportional to number of molecules per unit volume

(b) Inversely proportional to diameter of the molecule

(d) Directly proportional to the molecular mass

Maxwell’s laws of distribution of velocities shows that

(a) The number of molecules with most probable velocity is very large

(b) The number of molecules with most probable velocity is very small

(d) The number of molecules with most probable velocity is exactly equal to 1

Which of the following formula is wrong?

(a) (b) (c) (d) – = 2R

As per the law of equi-partition of energy each Vibrational mode gives how many degrees of freedom?

(a) 1 (b) 2 (c) 3 (d) 0

The mean kinetic energy of a perfect monatomic gas molecule at the temperature TºK is

(a) (b) k T (c) (d) 2 k T

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