Study Guide - Class XII Physics CHAPTER III: Kinetic ...physics-eclass.weebly.com/uploads/1/6/2/2/16227420/chapter_3_.pdf · Study Guide - Class XII Physics Prepared by Amit Dahal,

Study Guide - Class XII Physics

Prepared by Amit Dahal, YHSS Page 1 of 16

CHAPTER III: Kinetic Theory of Gases [5%]

Introduction The kinetic theory of gases (also known as kinetic-molecular theory) is a law that explains the behavior of a

hypothetical ideal gas. According to this theory, gases are made up of tiny particles in random, straight line

motion. They move rapidly and continuously and make collisions with each other and the walls. This describes

the gas pressure in terms of collisions with the walls of the container, rather than from static forces that push

the molecules apart. Kinetic theory also explains how the different sizes of the particles in a gas can give them

different, individual speeds.

Kinetic theory makes the following assumptions.

1. Gases consist of particles in constant, random motion. They continue in a straight line until they collide

with something—usually each other or the walls of their container.

2. Particles are point masses with no volume. The particles are so small compared to the space between

them, that we do not consider their size in ideal gases.

3. No molecular forces are at work. This means that there is no attraction or repulsion between the

particles.

4. Gas pressure is due to the molecules colliding with the walls of the container. All of these collisions are

perfectly elastic, meaning that there is no change in energy of either the particles or the wall upon

collision. No energy is lost or gained from collisions.

5. The time it takes to collide is negligible compared with the time between collisions.

6. The kinetic energy of a gas is a measure of its Kelvin temperature. Individual gas molecules have

different speeds, but the temperature and kinetic energy of the gas refer to the average of these speeds.

7. The average kinetic energy of a gas particle is directly proportional to the temperature. An increase in

temperature increases the speed in which the gas molecules move.

8. All gases at a given temperature have the same average kinetic energy.

9. Lighter gas molecules move faster than heavier molecules.

The mole and the Avogadro constant

The unit for the amount of a substance is the mole. It is defined as the amount of a substance that contains the

same number of particles (atoms or molecules) as the number of atoms in 12 g of the isotope carbon-12.

1mole = 6.02214179×1023 atoms

This number is known as the Avogadro constant (NA) i.e. and is 6.02 x 1023 particles per mole.

NA = 6.02 x 1023 per mole

The ratio of the mass of one mole of the substance to one twelfth of the mass of one mole of carbon-12 is

called the relative molecular mass of the substance - it is 32 for oxygen, 2 for hydrogen and so on.

A knowledge of the Avogadro constant enables us to calculate the number of molecules in any mass of a

substance and therefore to get an idea of the size of one molecule.

For example, a drop of water of volume 1.0 cm3 has a mass of 1g. The relative molecular mass of water is 18,

and therefore this drop of water must contain 6.02 x 1023 ÷ 18 = 3.34 x 1022 molecules.

Avogadro's constant is the number of particles in a mole of the substance.

This number is always the same. So in:

1 mole of hydrogen (2 g) there are 6.02 x 1023 molecules (hydrogen exists as H2)

1 mole of oxygen (32 g) there are 6.02 x 1023 molecules (oxygen exists as O2)

1 mole of copper (63 g) there are 6.02 x 1023 atoms

1 mole of uranium 235 (235 g) there are 6.02 x 1023 atoms



For example if we have 2 kg of uranium in a fuel rod we have 2000 ÷ 235 = 8.51 moles and this contains

8.51 x 6.02 x 1023 = 5.12 x 1023 atoms and so 5.12 x 1023 uranium nuclei.

The average volume of a water molecule must therefore be 2.99 x 10- 23 cm3.

Molar mass and molecular mass

Molar mass of a substance is the mass of one mole of the substance i.e. 6.02x1023 molecules. The molecular

mass is the mass of one molecule of the substance.

𝑴𝒐𝒍𝒆𝒄𝒖𝒍𝒂𝒓 𝒎𝒂𝒔𝒔 = 𝑴𝒐𝒍𝒂𝒓 𝒎𝒂𝒔𝒔 ÷ 𝑨𝒗𝒐𝒈𝒂𝒅𝒓𝒐 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

For example, the molecular mass of hydrogen is [2/6.02x1023] gm = 3.32 x 10-24 gm = 3.32 x 10-27 kg, and for

oxygen [16/6.02 x 1023] gm = 2.66 x 10-24 gm = 2.66 x 10-26 kg.

STP

Standard temperature and pressure, refers to nominal conditions in the atmosphere at sea level.

Standard temperature is defined as zero degrees Celsius (00C), which translates to 32 degrees Fahrenheit (320F)

or 273.15 K. This is essentially the freezing point of pure water at sea level, in air at standard pressure.

NTP

Normal Temperature and Pressure - is defined as air at 200C (293.15 K, 680F) and 1 atm (101.325 kN/m2,

101.325 kPa, 760 torr).

Gas Laws

Boyle's Law

Boyle’s law is a quantitative relationship between volume and pressure of a gas at constant temperature.

It states that “the volume of a given mass of a gas is inversely proportional to pressure if temperature

remains constant.”

At constant temperature, the product of pressure and volume of a gas

remains constant

According to Boyle’s law

𝑉 ∝1

𝑃

𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ×1

𝑃

𝑃𝑉 = 𝐾 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)

𝐿𝑒𝑡 𝑎𝑡 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖. 𝑃1, 𝑣𝑜𝑙𝑢𝑚𝑒 𝑏𝑒 𝑉1

∴ 𝑃1𝑉1 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … … … . (𝑖)

𝑖𝑖. 𝑃2, 𝑣𝑜𝑙𝑢𝑚𝑒 𝑏𝑒 𝑉2

∴ 𝑃2𝑉2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … … … . . (𝑖𝑖) Comparing (i) and (ii)

𝑷𝟏𝑽𝟏 = 𝑷𝟐𝑽𝟐

Graph between V & 1/P at constant temperature is a straight line.

The value of k always stays the same so that P and V vary appropriately.

For example, if pressure increases, k must remain constant and thus

volume will decrease. This is consistent with the predictions of Boyle's

law.



Charles' Law It is quantitative relation between volume and absolute temperature of a gas at constant pressure.

It states that “the volume of a given mass of a gas at constant pressure is directly proportional to absolute

temperature.

The volume of a given mass of a gas increases or decreases

by 1/273 times of its original volume at 0 0C for every degree fall or

rise of temperature at given pressure.

The ratio of volume to absolute temperature of a gas at given

pressure is always constant.

Let the volume of a gas at T Kelvin be V

Then according to Charles’s law

𝑉 ∝ 𝑇

𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝑇 𝑉

𝑇= 𝐾 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)

𝐿𝑒𝑡 𝑎𝑡 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑖. 𝑇1(𝑘𝑒𝑙𝑣𝑖𝑛) , 𝑣𝑜𝑙𝑢𝑚𝑒 𝑏𝑒 𝑉1

∴𝑉1

𝑇1= 𝐾 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) … … … . (𝑖)

𝑖𝑖. 𝑇2 (𝑘𝑒𝑙𝑣𝑖𝑛), 𝑣𝑜𝑙𝑢𝑚𝑒 𝑏𝑒 𝑉2

∴𝑉2

𝑇2= 𝐾 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) … … … . (𝑖𝑖)

Comparing (i) and (ii) 𝑽𝟏

𝑻𝟏=

𝑽𝟐

𝑻𝟐

According to Charles' law, gases will expand when

heated. The temperature of a gas is a measure of the average

kinetic energy of the particles. As the kinetic energy

increases, the particles will move faster and makes more

collisions with the container. However, in order for the law

to apply, the pressure must remain constant. The only way to

do this is by increasing the volume.

Let V0 be the volume of the given mass of a gas at 00C, and

Vt be the volume at t0C.

Let T0 an T are the temperatures in Kelvin, then T0 = 273+0 = 273 K and T = (273 + t) K

According to Charle’s law

𝑉1

𝑇1=

𝑉2

𝑇2

𝑉0

𝑇0=

𝑉𝑡

𝑇⇒

𝑉0

273=

𝑉𝑡

(273 + 𝑡)⇒ 𝑉𝑡 = 𝑉0 (

(273 + 𝑡)

273) ⇒ 𝑽𝒕 = 𝑽𝟎 (𝟏 +

𝒕

𝟐𝟕𝟑)



What is an ideal gas?

Since it's hard to exactly describe a real gas, the concept of an ideal gas allows us to model and predict the

behavior of real gases. The term ideal gas refers to a hypothetical gas composed of molecules which follow a

few rules:

1. Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas

molecules would be an elastic collision upon impact with each other or an elastic collision with the

walls of the container.

2. Ideal gas molecules themselves take up no volume. The gas takes up volume since the molecules

expand into a large region of space, but the Ideal gas molecules are approximated as point particles that

have no volume of themselves.

3. Ideal gas follows all the gas laws perfectly.

The pressure, P, volume V, and temperature T of an ideal gas are related by a simple formula called

the ideal gas law.

PV = nRT

Where

P is the pressure of the gas,

V is the volume taken up by the gas,

T is the temperature of the gas,

R is the gas constant, and

n is the number of moles of the gas.

According to Boyle’s law, for a given mass of a gas at constant temperature, volume is related to pressure as

𝑉 ∝1

𝑃… … … (𝑖)

According to Charle’s law, for a given mass of a gas at constant pressure, volume is related to temperature as

𝑉 ∝ 𝑇 … … . (𝑖𝑖) Combining (i) and (ii)

𝑉 ∝𝑇

𝑃⟹ 𝑽 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 ×

𝑻

𝑷⇒

𝑷𝑽

𝑻= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

For 1 mol of a gas, the constant has the same value for all gases and is called Universal Gas Constant denoted

by R. So 𝑃𝑉

𝑇= 𝑅 ⇒ 𝑃𝑉 = 𝑅𝑇

For n moles of gas,

𝑷𝑽 = 𝒏𝑹𝑻 … 𝑖𝑑𝑒𝑎𝑙 𝑜𝑟 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑔𝑎𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛

Boltzmann’s constant (k) is a gas constant per molecule. It is the ratio of Universal constant to Avogadro’s

number. So it is represented as

𝑘 =𝑅

𝑁𝐴 ⇒ 𝑹 = 𝒌𝑵𝑨

The numerical value of Boltzmann’s factor “k” is 1.38 x 10-23 Jk-1

Number of moles is given by

𝑛 =𝑛𝑜. 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠

𝐴𝑣𝑜𝑔𝑎𝑑𝑟𝑜′𝑠𝑛𝑢𝑚𝑏𝑒𝑟=

𝑁

𝑁𝐴 ⟹ 𝑃𝑉 =

𝑁

𝑁𝐴. 𝑘. 𝑁𝐴. 𝑇 ⟹ 𝑷𝑽 = 𝑵𝒌𝑻

Universal Gas Constant (R)

𝑹 =𝑷𝑽

𝒏𝑻=

𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆 × 𝒗𝒐𝒍𝒖𝒎𝒆

𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒎𝒐𝒍𝒆𝒔 × 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆=

𝒘𝒐𝒓𝒌 𝒅𝒐𝒏𝒆

𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒎𝒐𝒍𝒆𝒔 × 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆

So the universal gas constant represents the work done by a gas per mole per Kelvin.

S.I unit of R is J mole-1 K-1 and its numerical value is 8.31 J mole-1 K-1.



Pressure of ideal gas in terms of density “𝝆” and molecular mass “M” of a gas

From Ideal Gas equation

𝑷𝑽 = 𝒏𝑹𝑻

Density is

𝝆 =𝒎

𝑽… … … . (𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝑚 = 𝑚𝑎𝑠𝑠; 𝑉 = 𝑉𝑜𝑙𝑢𝑚𝑒)

Mass m is

𝒎 = 𝑴 × 𝒏 … … (𝑀 = 𝑚𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠; 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑠)

Therefore density can be expressed as

∴ 𝜌 =𝑚

𝑉=

𝑴 × 𝒏

𝑽

⇒𝝆

𝑴=

𝒏

𝑽… … … . (𝒂)

From Ideal gas equation 𝑃𝑉 = 𝑛𝑅𝑇

𝒏

𝑽=

𝑷

𝑹𝑻… … … … (𝒃)

Comparing equations (a) and (b)

∴𝝆

𝑴=

𝑷

𝑹𝑻

𝑷 =𝝆𝑹𝑻

𝑴

Real Gas and deviation of the behavior of real gas from ideal gas

A gas which obeys the gas laws and the gas equation PV = nRT strictly at all temperatures and pressures is

said to be an ideal gas. The molecules of ideal gases are assumed to be volume less points with no attractive

forces between one another. But no real gas strictly obeys the gas equation at all temperatures and

pressures. Deviations from ideal behavior are observed particularly at high pressures or low temperatures.

Causes of Deviation from Ideal Behaviour The causes of deviations from ideal behaviour may be due to the following two assumptions of kinetic theory

of gases.

i. The volume occupied by gas molecules is negligibly small as compared to the volume occupied by the

gas.

ii. The forces of attraction between gas molecules are negligible.

Consider a gas molecule (black square

at the bottom left).

Volume of the gas molecule is

insignificant/negligible in first figure.

Volume becomes significant in the

second figure.



The first assumption is valid only at low pressures and high temperature, when the volume occupied by the gas

molecules is negligible as compared to the total volume of the gas. But at low temperature or at high pressure,

the volumes of molecules are no more negligible as compared to the total volume of the gas.

The second assumption is not valid when the pressure is high and temperature is low. But at high pressure

or low temperature when the total volume of gas is small, the forces of attraction become appreciable and

cannot be ignored.

The deviations from ideal gas behavior can also be illustrated as follows:

The isotherms obtained by plotting pressure, P against volume, V for real gases do not coincide with that of

ideal gas, as shown below.

It is clear from above graphs that the volume of real gas is more than or less than expected in certain cases. The

deviation from ideal gas behaviour can also be expressed by compressibility factor, Z.

https://files.askiitians.com/cdn1/images/2014916-145217457-5993-realgas.gif



Compressibility factor (Z):

The deviation from ideal behaviour is expressed by introducing a factor Z known as compressibility factor in

the ideal gas equation.

The ratio of PV to nRT is known as compressibility factor. The ratio of volume of real gas, Vreal to the ideal

volume of that gas, Vperfect calculated by ideal gas equation is known as compressibility factor.

𝑍 =𝑃𝑉𝑟𝑒𝑎𝑙

𝑛𝑅𝑇

But from ideal gas equation:

PVperfect = nRT

𝑉𝑝𝑒𝑟𝑓𝑒𝑐𝑡/𝑖𝑑𝑒𝑎𝑙 =𝑛𝑅𝑇

𝑃

Therefore

𝑍 =𝑃𝑉𝑟𝑒𝑎𝑙

𝑛𝑅𝑇=

𝑉𝑟𝑒𝑎𝑙

𝑉𝑝𝑒𝑟𝑓𝑒𝑐𝑡/𝑖𝑑𝑒𝑎𝑙

For ideal or perfect gases, the compressibility factor, Z = 1.

But for real gases, Z ≠1.

Case-I: If Z>1 (Positive deviation) Vreal > Videal

The repulsion forces become more significant than the attractive forces.

The gas cannot be compressed easily.

Usually the Z > 1 for so called permanent gases like He, H2 etc.

Case-II: If Z < 1 (Negative deviation)

Vreal < Videal

The attractive forces are more significant than the repulsive forces.

The gas can be liquefied easily.

Usually the Z < 1 for gases like NH3, CO2, SO2 etc.

The isotherms for one mole of different gases, plotted against the Z value and pressure, P at 0 oC are shown

below:

i. For gases like He, H2 the Z value increases with increase in pressure (positive deviation).

It is because, the repulsive forces become more significant and the attractive forces become less dominant.

Hence these gases are difficult to be condensed.



ii. For gases like CH4, CO2, NH3 etc., the Z value decreases initially (negative deviation) but increases at

higher pressures.

It is because: at low pressures, the attraction forces are more dominant over the repulsion forces, whereas

at higher pressures the repulsion forces become significant as the molecules approach closer to each

other.

iii. But for all the gases, the Z value approaches one at very low pressures, indicating the ideal behavior.

Also consider the following graphs of Z vs P for a particular gas, N2 at different temperatures.

In above graphs, the curves are approaching the horizontal line with increase in the temperature i.e., the

gases approach ideal behaviour at higher temperatures.

Dalton’s Law of Partial Pressure

Partial Pressure

In a mixture of different gases which do not react chemically, each gas behaves independently of the other

gases and exerts its own pressure. This individual pressure that a gas exerts in a mixture of gases is called its

partial pressure.

Dalton’s Law of Partial Pressure

Based on this behavior of gases, JOHN DALTON formulated a basic law which is known as "The Dalton's law

of partial pressure".

"If two or more gases (which do not react with each other) are enclosed in a vessel,

the total pressure exerted by them is equal to the sum of their partial pressure".

Consider a mixture of three non-reacting gases a, b and c. Partial pressures of these gases are Pa, Pb and Pc .

According to Dalton's law of partial pressure, their total pressure is given by

Ptotal = Pa + Pb + Pc

According to kinetic molecular theory of gases there is no force of attraction or repulsion among the gas

molecules. Thus each gas behaves independently in a mixture and exerts its own pressure.

In terms of Kinetic molecular theory, Dalton's law of partial pressure can be explained as:

"In a non-reacting mixture of gases, each gas exerts separate pressure on the container in which it is

confined due to collision of its molecules with the walls of container. The total pressure exerted by the

gaseous mixture is equal to the sum of collisions of the molecules of individual gas."



Expression for Partial Pressure

Consider a gaseous mixture of three different gases a , b and c enclosed in a container of

volume Vdm3 at T Kelvin. Let the partial pressures of these gases are Pa, Pb and Pc respectively and total

pressure of mixture is Pt. Let there are na, nb and nc moles of each gas respectively and the total number

of moles are nt.

Three gases confined in a cylinder under similar conditions:

Using equation of state of gas:

𝑃𝑉 = 𝑛𝑅𝑇

OR

𝑃 =𝑛𝑅𝑇

𝑉

For gas a

𝑃𝑎 =𝑛𝑎𝑅𝑇

𝑉… … … … … … … (𝑖)

For gas b

𝑃𝑏 =𝑛𝑏𝑅𝑇

𝑉… … … … … … … (𝑖𝑖)

For gas c

𝑃𝑐 =𝑛𝑐𝑅𝑇

𝑉… … … … … … … (𝑖𝑖𝑖)

For any gas

𝑃𝑔𝑎𝑠 =𝑛𝑔𝑎𝑠𝑅𝑇

𝑉

OR 𝑃𝑔𝑎𝑠

𝑛𝑔𝑎𝑠=

𝑅𝑇

𝑉… … … . . (𝑎)

Adding equation (i) , (ii) and (iii), we get,

𝑃𝑡 =𝑛𝑎𝑅𝑇

𝑉+

𝑛𝑏𝑅𝑇

𝑉+

𝑛𝑐𝑅𝑇

𝑉= (𝑛𝑎 + 𝑛𝑏 + 𝑛𝑐)

𝑅𝑇

𝑉

But

𝑛𝑡 = 𝑛𝑎 + 𝑛𝑏 + 𝑛𝑐

𝑃𝑡 = 𝑛𝑡

𝑅𝑇

𝑉

𝑃𝑡

𝑛𝑡=

𝑅𝑇

𝑉… … … … … (𝑏)

Comparing equation (a) and (b), we get,

𝑃𝑡

𝑛𝑡=

𝑃𝑔𝑎𝑠

𝑛𝑔𝑎𝑠

𝑷𝒈𝒂𝒔

𝑷𝒕𝒐𝒕𝒂𝒍=

𝒏𝒈𝒂𝒔

𝒏𝒕𝒐𝒕𝒂𝒍

This expression indicates that the pressure of a gas is proportional to number of moles if confined under

similar conditions.



Y

X

Z

L

vx

– vx

Pressure of an Ideal Gas

Consider an ideal gas (consisting of N molecules each of mass m) enclosed in a cubical box of side L.

It’s any molecule moves with velocity v

in any direction where

kvjvivv zyxˆˆˆ

This molecule collides with the shaded wall )( 1A with velocity xv

and rebounds with velocity xv .

The change in momentum of the molecule

xxx mvmvmvP 2)()(

As the momentum remains conserved in a collision, the change

in momentum of the wall A1 is xmvP 2

After rebound this molecule travel toward opposite wall A2 with velocity xv , collide to it and again rebound

with velocity xv towards wall A1.

(1) Time between two successive collision with the wall A1.

moleculeof Velocity

collision successive two betweenmolecule by travelledDistancet

xv

L2

Number of collision per second L

v

tn x

2

1

(2) The momentum imparted per unit time to the wall by this molecule 22

2xx

x vL

mmv

L

vPn

This is also equal to the force exerted on the wall 1A due to this molecule 2xv

L

mF

(3) The total force on the wall 1A due to all the molecules 2xx v

L

mF

(4) Now pressure is defined as force per unit area

22xx

xx v

V

mv

AL

m

A

FP Similarly 2

yy vV

mP and 2

zz vV

mP

So )( 222zyxzyx vvv

V

mPPP

23 vV

mP [As PPPP zyx and

2222zyx vvvv ]

......)(3 33

22

21 vvv

V

mP

or

or 23 rmsvV

NmP

As root mean square velocity of the gas molecule

N

vvvvvrms

.....24

23

22

21

or 2

3

1rmsv

V

NmP

Now the total mass of the gas M = mN, and since ρ = M÷V we can write

N

vvvv

V

mNP

....3

24

23

22

21



2

3

1rmsvP

Kinetic Interpretation of gases

According to the kinetic equation of pressure of a gas:

2

3

1rmsvP

But 𝝆 = density of gas

𝝆 = density of gas = mass of gas ÷ volume of gas

𝝆 = density of gas = (M) ÷ V

2

3

1rmsv

V

MP …………..M: mass; V: Volume

Multiplying and dividing by 2

2

2

1

3

2rmsMvPV

2

2

1rmsMv is the average kinetic energy of the gas

avgEKPV .3

2

For 1 mole of a gas

RTPV

RTEK avg .3

2

RTEK avg2

3.

RTMvrms2

3

2

1 2

M

RTVrms

3

Translational kinetic energy is the kinetic energy an object has due to its motion in a straight line from one

place to another place.

If NA is the Avagadro’s number, then the mean kinetic energy per molecule is given by

TN

R

N

EKEK

AA

rmsavg

2

3..

kTEK avg2

3. ……k: Boltzmann’s constant

TVrms

Faster the motion of the molecules, higher

will be the kinetic energy and hence higher

will be the temperature of the gas. Hence

the temperature of a gas is the measure of

the average kinetic energy of its molecules.

At T = 0, Vrms = 0

i.e. at absolute zero all the molecular motion

stops.



Types of Molecular Velocities

The three types of molecular velocities are

i. Most probable velocity

ii. Average velocity

iii. Root mean square velocity

Most probable velocity (Vp)

This is defined as the velocity possessed by maximum fraction of the total number of molecules of the gas at a

given temperature.

M

PV

M

kT

M

RTVp

222

Where

R: gas constant

T: temperature and

M: molecular weight

Average velocity (Vavg)

Average velocity is the mean of the velocities of all the molecules. It is calculated using the formula.

M

PV

M

kT

M

RTVavg

888

Root mean square velocity

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

M

RTVrms

3

M

PV

M

kT 33

Root mean square velocity = 1.085 x Average velocity

Boltzmann distribution

It is a distribution of velocities of gas particles in a particular temperature.

It is obtained by plotting a graph between fraction (Δn/N) of molecules having different speeds against the

speeds of the molecules (along x-axis). This curve is

known as Maxwell's Boltzmann distribution curve.

Even if the air is at a single temperature, the air

molecules travels at different speed. Some of the air

molecules will be moving extremely fast, some will be

moving with moderate speeds, and some of the air

molecules will hardly be moving at all. So rather than

talking about the speed of an air molecule in a gas, we

discuss the distribution of speeds in a gas at a certain

temperature.

The Maxwell-Boltzmann distribution is often represented

with the graph as shown alongside.



The y-axis of the Maxwell-Boltzmann graph can be thought of as giving the number of molecules per unit

speed. So, if the graph is higher in a given region, it means that there are more gas molecules moving with

those speeds.

We notice that the graph is not symmetrical. There is a longer "tail" on the high speed right end of the graph.

The graph continues to the right to extremely large speeds, but to the left the graph must end at zero (since a

molecule can't have a speed less than zero).

The average speed Vavg of a molecule in the gas is

actually located a bit to the right of the peak. The

reason the average speed is located to the right of the

peak is due to the longer "tail" on the right side of the

Maxwell-Boltzmann distribution graph. This longer

tail pulls the average speed slightly to the right of the

peak of the graph.

Root-mean-square speed Vrms. is the square root of

the mean of the squares of the velocities

We can write the root-mean-square speed

mathematically as,

𝑉𝑟𝑚𝑠 = √(𝑉1

2 + 𝑉22 + 𝑉3

2 + ⋯ + 𝑉𝑛2)

𝑁

Area under a Maxwell-Boltzmann distribution

The y-axis of the Maxwell-Boltzmann distribution graph gives

the number of molecules per unit speed. The total area under the

entire curve is equal to the total number of molecules in the gas.

If we heat the gas to a higher temperature, the peak of the graph will

shift to the right (since the average molecular speed will increase).

As the graph shifts to the right, the height of the graph has to decrease

in order to maintain the same total area under the curve.

Similarly, as a gas cools to a

lower temperature, the peak of

the graph shifts to the left. As the graph shifts to the left, the height of

the graph has to increase in order to maintain the same area under the

curve. This can be seen in the curves below which represent a sample

of gas (with a constant amount of molecules) at different temperatures.

As the gas gets colder, the graph becomes taller and narrower.

Similarly, as the gas gets hotter the graph becomes shorter and wider.

This is required for the area under the curve (e.g. total number of

molecules) to stay constant.

If molecules enter the sample, the total area under the curve would increase. Similarly, if molecules were to

leave the sample, the total area under the curve would decrease.



Kinetic Energy Distribution

Maxwell and Boltzmann distribution helps us to examine the properties of very large numbers of molecules.

While we can't predict the behavior of any one molecule with any precision, we can quite exactly predict the

behavior of large quantities of molecules.

Kinetic energy of an atom or molecule is calculated by the formula:

KE = 1/2 mv2

where m is the mass of the particle and v is its

velocity.

When we have to describe the distribution of

kinetic energies in a large collection of atoms or

molecules, the distribution can be described by

plotting the graph as follows

Notice that the plot is not symmetrical so the most

probable speed is not the same as the average

speed.

Effect of temperature Temperature is directly proportional to the average

kinetic energy of molecules,

𝐾𝐸 =3

2𝑛𝑅𝑇

where n is the number of moles of gas, T is the temperature in Kelvins, and R is the Universal gas constant

If T is proportional to the average kinetic energy, then we'd expect for this change in the average kinetic

energy to be reflected in changes in the shape of the Boltzmann distribution. As temperature increases, the

curve will spread to the right and the value of the most probable kinetic energy will decrease. This is illustrated

below for several temperatures.

What happens to the fraction of molecules with kinetic energies equal to or greater than this value as we

change the temperature?

As T increases, the fraction of

molecules with energies greater than

the red line increase. This fact is of

fundamental importance to

properties of matter such as vapor

pressures.



Boltzmann Factor Equation

Particles in a gas lose and gain energy at random due to collisions with each other. On average, over a large

number of particles, the proportion of particles which have at least a certain amount of energy ε is constant.

This is known as the Boltzmann factor. It is a value between 0 and 1. The Boltzmann factor is given by the

formula: 𝒏

𝒏𝟎= 𝒆

−𝜺𝒌𝑻

where n is the number of particles with kinetic energy above an energy level ε, n0 is the total number of

particles in the gas, T is the temperature of the gas (in kelvin) and k is the Boltzmann constant (1.38 x

10−23 JK−1).

This energy could be any sort of energy that a particle can have - it could be gravitational potential energy, or

kinetic energy, for example.

Derivation

In the atmosphere, particles are pulled downwards by gravity. They gain and lose gravitational potential energy

(mgh) due to collisions with each other.

Considering a small portion of the atmosphere.

Let

horizontal cross-sectional area be A,

height be dh,

molecular density be n

molecular mass be m.

number of particles in the portion be N.

𝑛 =𝑁

𝑉=

𝑁

𝐴 𝑑ℎ

By definition:

𝑁 = 𝑛𝑉 = 𝑛𝐴 𝑑ℎ

The total mass Σm is the mass of one molecule (m) multiplied by the number of molecules (N):

Σ𝑚 = 𝑚𝑁 = 𝑚𝑛𝐴 𝑑ℎ

Then the weight of the portion of atmosphere:

𝑊 = 𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐. 𝑑𝑢𝑒 𝑡𝑜 𝑓𝑟𝑎𝑣𝑖𝑡𝑦 = 𝑔Σ𝑚 = 𝑚𝑛𝑔𝐴 𝑑ℎ

The downwards pressure P is force per. unit area, so:

𝑃 =𝑊

𝐴=

𝑛𝑚𝑔𝐴 𝑑ℎ

𝐴= 𝑚𝑛𝑔𝑑ℎ

We know that, as we go up in the atmosphere, the pressure decreases. So, across the portion there is a

difference in pressure dP given by:

𝒅𝒑 = −𝒏𝒎𝒈𝒅𝒉 … … … (1)

In other words, the pressure is decreasing (-) and it is the result of the weight of this little chunk of atmosphere.

We also know that:

𝑃𝑉 = 𝑁𝑘𝑇

So:

𝑃 =𝑁𝑘𝑇

𝑉

But we know that

𝑛 =𝑁

𝑉



So, by substitution

𝑃 = 𝑛𝑘𝑇

So, for portion of atmosphere

𝒅𝒑 = 𝒌𝑻 𝒅𝒏 … … (2)

Equating (1) and (2)

𝑑𝑝 = −𝑛𝑚𝑔𝑑ℎ = 𝑘𝑇 𝑑𝑛

Rearranging, we get

𝑑𝑛

𝑑ℎ=

−𝑛𝑚𝑔

𝑘𝑇⇒

𝑑𝑛

𝑑ℎ=

−𝑘𝑇

𝑛𝑚𝑔

Integrating between the limits n0 and n:

ℎ =−𝑘𝑇

𝑚𝑔= ∫

1

𝑛

𝑛

𝑛0

𝑑𝑛 =−𝑘𝑇

𝑚𝑔[ln 𝑛]𝑛0

𝑛 =−𝑘𝑇

𝑚𝑔(ln 𝑛 − ln 𝑛0) =

−𝑘𝑇

𝑚𝑔ln

𝑛

𝑛0

ln𝑛

𝑛0=

−𝑚𝑔ℎ

𝑘𝑇⟹

𝑛

𝑛0= 𝑒

−𝑚𝑔ℎ𝑘𝑇

Since mgh is gravitational potential energy, ε = mgh, so 𝒏

𝒏𝟎= 𝒆

−𝜺𝒌𝑻

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Study Guide - Class XII Physics CHAPTER III: Kinetic ...physics-eclass.weebly.com/uploads/1/6/2/2/16227420/chapter_3_.pdf · Study Guide - Class XII Physics Prepared by Amit Dahal,