Gases: -

The state of matter which has no fixed shape and volume is called gas.
Gas is a state of matter that has no fixed shape and no fixed volume. Gases have lower density than other
states of matter, such as solids and liquids. There is a great deal of empty space between particles, which
have a lot of kinetic energy

General Characteristics of Gases

1. Expansibility: -
Gases have limitless expansibility. They expand to fill the entire vessel they are placed in.
2. Compressibility: -
Gases are easily compressed by application of pressure to movable piston fitted in the container.
3. Infusibility: -
Gases can diffuse rapidly through each other to form a homogeneous mixture.
4. Pressure: -
Gases exert pressure on the walls of the container in all directions.
5. Effect of Heat
When a gas, confined in a vessel is heated, its pressure increases. Upon heating in a vessel fitted with
a piston, volume of the gas increases.
6. Fluidity: -
They have high fluidity due to free motions of particles.
7. Density: -
They have low densities.
8. Inter-atomic Spaces: -
In gases, the particles are part about 300 times than their sizes.
9. No Definite shape and Volume: -
They have no definite shape and volume due to the negligible attractive forces. They acquire the
volume of that vessel in which they are trapped.
10. Inter Molecular Attractive Forces: -
They have negligible attractive force.
The above properties of gases can be easily explained by the Kinetic Molecular Theory which will be
considered later.
Parameters of Gas: -
A gas sample can be described in terms of four parameters (measurable properties):
1. The volume, V of the gas.
2. Its pressure, P.
3. Its temperature, T.
4. The number of moles, n, of gas in the container.

Volume, V: -
The volume of the container is the volume of the gas sample. It is given in liter (l or L) or
milliliters (ml or mL).
1 𝑙𝑖𝑡𝑟𝑒 (𝑙) = 1000𝑚𝑙 𝑎𝑛𝑑 1𝑚𝑙 = 10−3 𝑙
One milliliter is practically equal to one cubic centimeter (cc). Actually
1 𝑙𝑖𝑡𝑟𝑒 (𝑙) = 1000.028𝑐𝑐
The SI unit for volume is cubic meter (m3) and the smaller unit is decimeter3 (dm3).
1 𝑙𝑖𝑡𝑟𝑒 (𝑙) = 1 𝑑𝑚3
Pressure P: -
The pressure of gas is defined as the force exerted by the impacts of its molecules per unit surface area in
constant. The pressure of a gas sample can be measured with the help of a mercury manometer.
 Manometer

Similarly, the atmospheric pressure can be determined with a mercury barometer.

The pressure of air that can support 760mmHg column at sea level is called one atmosphere (1atm). The
unit of pressure, millimeter of mercury is also called torr.

Thus
1𝑎𝑡𝑚 = 760𝑚𝑚𝐻𝑔 = 760 𝑡𝑜𝑟𝑟
The SI unit of pressure is the Pascal (Pa). The relation between atmosphere, torr and pascal is:
1𝑎𝑡𝑚 = 760 𝑡𝑜𝑟𝑟 = 1.013 × 105 𝑃𝑎

The unit of pressure “𝑃𝑎𝑠𝑐𝑎𝑙“is not in common use.

Temperature, T: -
The temperature of a gas may be measured in Centigrade degree (℃) or Celsius degree. The SI unit of
temperature in Kelvin (K) or Absolute degree. The centigrade degree can be converted to Kelvins by using
the equation.
𝐾 = ℃ + 273
The kelvin temperature (or absolute temperature) is always used in calculations of other parameters of
gases. Remember that the degree sign (o) is not used with K.
The moles of a Gas Sample, n: -
The number of moles, n, of a sample of gas in a container can be found by diving the mass, m of the sample
by the molar mass M (molecular mass).
Mass of gas sample (m)
Moles of gas (n) =
molecular mass of gas (M)

The Gas Laws: -

The volume of a given sample of gas depends on the temperature and pressure applied to it. Any change
in temperature or pressure will affect the volume of the gas. As results of experimental studies from 17th
to 19th century, scientists derived the relationships among the pressure, temperature and volume of a
given mass of gas. These relationships, which describe the general behavior of gases, are called the gas
laws.
Boyle’s Law: -
In 1660 Robert Boyle found out experimentally the change in volume of a given sample of gas with
pressure at room temperature. From his observations he formulated a generalization known as Boyle’s
Law. It states that: at constant temperature, the volume of a fixed mass of gas is inversely proportional to
its pressure. If the pressure is doubled, the volume is halved.
The Boyle’s Law may be expressed mathematically as
1
𝑉 ∝ (𝑇, 𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑃
Or
1
𝑉 = 𝑘×
𝑃
Where 𝑘 is proportionality constant.
𝑃𝑉 = 𝑘
If 𝑃1 , 𝑉1 are initial pressure and volume of a given sample of gas and 𝑃2 , 𝑉2 the changed pressure and
volume, we can write,
𝑃1 𝑉1 = 𝑘 = 𝑃2 𝑉2
Or
𝑃1 𝑉1 = 𝑃2 𝑉2
This relationship is useful for the determination of the volume of a gas any pressure, if its volume at any
other pressure is known.

The Boyle’s Law can be demonstrated by adding liquid mercury to the open end of a𝐽 − 𝑡𝑢𝑏𝑒. As the
pressure is increased by addition of mercury the volume of the sample of trapped gas decreases. Gas
pressure and volume are inversely related: one increases when the other decreases.
 Boyle’s J-tube experiment
Charles’s Law: -
In 1787 Jacques Charles investigated the effect of change of temperature on the volume of a fixed amount
of gas at constant pressure. He established a generalization which is called the Charles’ Law. It states that:
at constant pressure, the volume of a fixed mass of gas is directly proportional to the Kelvin temperature
of absolute temperature. If the absolute temperature is doubled, the volume is doubled.

Charles Law may be expressed mathematically as,

𝑉 ∝ 𝑇 (𝑃, 𝑛 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
Or
𝑉 = 𝑘𝑇
Where k is a constant
𝑉
=𝑘
𝑇
If V1 , T1 are the initial volume and temperature of a given mass of gas at constant pressure and V2 ,T2
be the new values, we can write
V1 V2
=𝑘=
T1 T2
Or
 V1 V2
=
T1 T2
Using this expression, the new volume V2 , can be found from the experimental values of V1 , T1 𝑎𝑛𝑑 T2 .

Avogadro’s Law: -
Let us take balloon containing certain mass of gas. If we add to it more mass of gas, holding the
temperature (𝑇) and Pressure (𝑃) constant, the volume of gas (𝑉) will increases. It was found
experimentally that the amount of gas in moles is proportional to the volume. That is,
𝑉 ∝ 𝑛 (𝑇 𝑎𝑛𝑑 𝑃 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
Or
𝑉=𝑘𝑛
Where k is constant of proportionality.
𝑉
=𝑘
𝑛
For any two gases with volumes V1, V2 and n1, n2 at constant T and P,
V1 V2
=𝑘=
n1 n2
If
𝑉1 = 𝑉2 , 𝑛1 = 𝑛2

Thus, for equal volumes of the two gases at fixed 𝑇and 𝑃, number of moles is also equal. This is the basis
of Avogadro’s Law which may be stated as: equal volumes of gases at the same temperature and pressure
contain equal number of moles or molecules. If the molar amount is doubled, the volume is doubled.
The Molar Gas Volume. If follows as a corollary of Avogadro’s Law that one moles of any gas at a
given temperature (𝑇) and pressure (𝑃) has the same fixed volume. It is called the molar gas volume or
molar volume. To compare the molar volumes of gases, chemists use a fixed reference temperature and
pressure. This is called standard temperature and pressure (abbreviated 𝑆𝑇𝑃). The standard temperature
used is 273𝐾(0o𝐶)and the standard pressure is 1𝑎𝑡𝑚 (760𝑚𝑚𝐻𝑔). At 𝑆𝑇𝑃 we find experimentally that
one mole of any gas occupies a volume of 22.4 liters. To put in it in the form of an equation, we have
1𝑚𝑜𝑙𝑒 𝑜𝑓 𝑔𝑎𝑠 𝑎𝑡 𝑆𝑇𝑃 = 22.4𝑙𝑖𝑡𝑟𝑒𝑠

inetic and molecular Model of gases: -

In 1738 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 was the first proposed the theoretical justification for the behaviours
of gases.𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖proposed that the pressure of the gas is due to the collisions of the gas molecules at
the walls of the vessel. So, therefore Bernoulli was the first scientist who gave the Idea of kinetic molecular
theory of gases and this theory was subsequently developed by
𝐶𝑙𝑎𝑢𝑠𝑖𝑢𝑠, 𝐶𝑙𝑎𝑟𝑘 𝑀𝑎𝑥𝑤𝑒𝑙𝑙, 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛, 𝑣𝑎𝑛 𝑑𝑒𝑟 𝑤𝑎𝑙𝑙𝑠 𝑎𝑛𝑑 𝑜𝑡ℎ𝑒𝑟𝑠to provide a theoretical explanation
for the properties of Ideal gases.
𝑬𝒔𝒔𝒆𝒏𝒕𝒊𝒂𝒍 𝑷𝒐𝒔𝒕𝒖𝒍𝒂𝒕𝒆𝒔 𝒐𝒇 𝑲𝒊𝒏𝒆𝒕𝒊𝒄 𝒎𝒐𝒍𝒆𝒄𝒖𝒍𝒂𝒓 𝒕𝒉𝒆𝒐𝒓𝒚
𝒐𝒇 𝒈𝒂𝒔𝒆𝒔 (𝑲𝑴𝑻𝑮)
1. 𝐸𝑣𝑒𝑟𝑦 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑠 𝑜𝑓 𝑎 𝑣𝑒𝑟𝑦 𝑙𝑎𝑟𝑔𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑛𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑐𝑎𝑙𝑙𝑒𝑑
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠.
2. 𝑀𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑔𝑎𝑠 𝑎𝑟𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑖𝑛 𝑎𝑙𝑙 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑠. 𝑖. 𝑒. 𝑚𝑎𝑠𝑠 𝑎𝑛𝑑
𝑠𝑖𝑧𝑒.
3. 𝑇ℎ𝑒𝑠𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑎𝑟𝑒 𝑖𝑛 𝑎 𝑠𝑡𝑎𝑡𝑒 𝑜𝑓 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑟𝑎𝑛𝑑𝑜𝑚 𝑚𝑜𝑡𝑖𝑜𝑛 𝑖𝑛 𝑎𝑙𝑙
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠.
4. 𝑇ℎ𝑒𝑦 𝑐𝑜𝑙𝑙𝑖𝑑𝑒 𝑤𝑖𝑡ℎ 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝑎𝑛𝑑 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙𝑠 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟.
5. 𝑇ℎ𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑖𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟.
6. 𝑇ℎ𝑒 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑤𝑖𝑡ℎ 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝑎𝑛𝑑 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒
𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 𝑎𝑟𝑒 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑙𝑦 𝑒𝑙𝑎𝑠𝑡𝑖𝑐.
7. 𝑇ℎ𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑎𝑟𝑒 𝑤𝑖𝑑𝑒𝑙𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑑 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠
𝑛𝑜 𝑓𝑜𝑟𝑐𝑒 𝑜𝑓 𝑎𝑡𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑟 𝑟𝑒𝑝𝑢𝑙𝑠𝑖𝑜𝑛 𝑎𝑚𝑜𝑛𝑔 𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠.
8. 𝐷𝑢𝑒 𝑡𝑜 𝑙𝑎𝑟𝑔𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠, 𝑡ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒
𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠𝑖𝑠 𝑛𝑒𝑔𝑙𝑖𝑔𝑖𝑏𝑙𝑒 𝑖𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠.
9. 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑒𝑓𝑓𝑒𝑐𝑡 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑜𝑛 𝑡ℎ𝑒 𝑚𝑜𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒
𝑔𝑎𝑠.
10. 𝑇ℎ𝑒 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑜𝑓 𝑎 𝑔𝑎𝑠 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠.
11. 𝑇ℎ𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡. 𝐴𝑙𝑙 𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑚𝑜𝑣𝑒 𝑤𝑖𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒𝑟𝑒 𝑓𝑜𝑟 ℎ𝑎𝑣𝑖𝑛𝑔 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠.
Equation of State for an Ideal gas/Universal or Ideal or Perfect gas law:
Equation of states is obtained by the combination of Boyle’s Law, Charles Law and
Avogadro’s Law.
According to Boyle’s Law,
1
𝑉 ∝ 𝑖𝑓(𝑇, 𝑛)𝑎𝑟𝑒 𝑘𝑒𝑝𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … … … … (1)
𝑃
According to Charles Law,
𝑉 ∝ 𝑇 𝑖𝑓 (𝑃, 𝑛)𝑎𝑟𝑒 𝑘𝑒𝑝𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … … … … (2)
According to Avogadro’s Law
𝑉∝ 𝑛 𝑖𝑓 (𝑃, 𝑇)𝑎𝑟𝑒 𝑘𝑒𝑝𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … … … … (3)
Combining equations 1,2 and 3 we get
 1 𝑇𝑛
𝑉 ∝ × 𝑇 × 𝑛 (𝑜𝑟) 𝑉 ∝ … … … … . (4)
𝑃 𝑃
1
𝑉 = 𝑅 × × 𝑇 × 𝑛
𝑃
𝑃𝑉 = 𝑛𝑅𝑇 … … … … … . (5)
Where 𝑅 is called gas constant.

Equation number (5) is called equation of state because it contains the entire variable
(𝑃, 𝑉, 𝑇 𝑎𝑛𝑑 𝑛) which describe completely the condition or state of any gas. This equation is also called
universal or ideal or perfect or general gas equation.
According to equation (4)
𝑛𝑇
𝑉 ∝
𝑃
“The volume of a given amount of a gas is directly proportional to the number of moles and temperature
and inversely proportional to the pressure” is called universal gas law or general gas law or ideal or perfect
gas law.
Nature and Numerical value of “R”: -
According to equation of state.
𝑃𝑉 = 𝑛𝑅𝑇

𝑃𝑉
𝑅 = … … … … (1)
𝑛𝑇
We know
𝐹𝑜𝑟𝑐𝑒
𝑃 = … … … … (2)
𝐴𝑟𝑒𝑎

𝑉 = 𝐴𝑟𝑒𝑎 × 𝑙𝑒𝑛𝑔𝑡ℎ … … … … (3)

Putting the values of equation 2 and 3 in equation 1.
𝐹𝑜𝑟𝑐𝑒
× (Area × length)
𝑅 = 𝐴𝑟𝑒𝑎
𝑛 × T

𝐹𝑜𝑟𝑐𝑒 × length
𝑅=
𝑛 ×T

𝑤𝑜𝑟𝑘 = 𝐹𝑜𝑟𝑐𝑒 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑟 𝑙𝑒𝑛𝑔𝑡ℎ

𝑤𝑜𝑟𝑘
𝑅=
𝑛 × T
Hence “𝑅” by nature is work or energy per degree per mole.
Numerical Value of “𝑹” from the Ideal gas equation:
We can write
𝑃𝑉
𝑅= … … … . . (1)
𝑛𝑇

At𝑆𝑇𝑃 𝑃 = 1 𝑎𝑡𝑚

𝑉 = 22.4 𝑙𝑖𝑡𝑒𝑟𝑠𝑜𝑟𝑑𝑚3

𝑛 = 1 𝑚𝑜𝑙

𝑇 = 0℃𝑜𝑟 273 𝐾
Putting the value in equation (1) we get
1 𝑎𝑡𝑚 × 22.4 liters
𝑅=
1 𝑚𝑜𝑙 × 273 K

𝑅 = 0.0821 𝑎𝑡𝑚. 𝐿𝑖𝑡𝑒𝑟 𝑚𝑜𝑙-1𝐾 -1 (or)

𝑅 = 0.0821 𝑎𝑡𝑚, 𝑑𝑚3𝑚𝑜𝑙 -1𝐾 -1
𝑺𝑰 𝒖𝒏𝒊𝒕𝒔 𝒐𝒇 “𝑹”.
We
𝑃 = 1 𝑎𝑡𝑚 = 101325 𝑝𝑎
𝑉 = 22.4 𝑑𝑚3 = 0.0224𝑚3
𝑛 = 1 𝑚𝑜𝑙
𝑇 = 273 𝐾
101325 𝑝𝑎 × 0.0224 𝑚3
𝑅=
1 𝑚𝑜𝑙 × 273 K
𝑅 = 8.314 𝑃𝑎 𝑚3 𝑚𝑜𝑙 −1 𝐾 −1 … … … … (2)
As
1 𝑃𝑎 = 1𝑁𝑚−2
Then,
𝑅 = 8.314𝑁𝑚−2 𝑚3 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 8.314𝑁𝑚 𝑚𝑜𝑙 −1 𝐾 −1
As,
1𝑁𝑚 = 1 𝐽𝑜𝑢𝑙𝑒
𝑅 = 8.314 𝐽𝑚𝑜𝑙 −1 𝐾 −1
Similarly
1𝐶𝑎𝑙 = 4.18𝐽
𝑅 = 1.987𝑐𝑎𝑙 𝑚𝑜𝑙 −1 𝐾 −1
Value of R in different units
𝑅 = 0.0821 𝑎𝑡𝑚 𝑙𝑖𝑡𝑒𝑟 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 82.1 𝑎𝑡𝑚 𝑚𝑙 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 62.3 𝑚𝑚𝐻𝑔 𝑙𝑖𝑡𝑒𝑟 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 62.3 𝑡𝑜𝑟𝑟 𝑙𝑖𝑡𝑒𝑟𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 8.314 × 107 𝑒𝑟𝑔 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 8.314 𝐽 𝑚𝑜𝑙 −1 𝐾 −1
𝑅 = 1.987 𝑐𝑎𝑙 𝑚𝑜𝑙 −1 𝐾 −1
The actual value of 𝑅 depends on the units of 𝑃 and 𝑉 used in calculating it.
deal gas or perfect gas:
 A gas which obeys Boyle’s law, Charles law, etc. under all conditions of temperature and pressure is
known as an ideal gas.
OR
A gas which obeys ideal gas law (𝑃𝑉 = 𝑛𝑅𝑇) under all conditions of temperature and pressure is called
ideal gas it is observed that there is no gas which obeys the gas laws or ideal gas law under all set of
conditions of temperature and pressure.
The natural occurring gases are found to obey the gas laws well if the pressure is low and the temperature
is high.
Real or non-Ideal gas: -
A gas which obeys the gas laws well under low pressure or high temperature. All gases are real. They
show more and more deviation from the gas laws, as pressure is increased, and the temperature is
decreased.
Critical Phenomena: - (𝑪𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒗𝒂𝒍𝒖𝒆𝒔 𝒐𝒇 𝑻, 𝑷, 𝒂𝒏𝒅 𝑽): -
𝑇ℎ𝑒 𝑠𝑚𝑜𝑜𝑡ℎ 𝑚𝑒𝑟𝑔𝑖𝑛𝑔 (𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛)𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑤𝑖𝑡ℎ 𝑖𝑡𝑠 𝑙𝑖𝑞𝑢𝑖𝑑 𝑠𝑡𝑎𝑡𝑒 𝑎𝑡 𝑎
𝑓𝑖𝑥𝑒𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑓𝑜𝑟 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑔𝑎𝑠𝑒𝑠 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠
𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝ℎ𝑒𝑛𝑜𝑚𝑒𝑛𝑜𝑛.
OR
𝑇ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑛𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑛𝑒𝑐𝑒𝑠𝑠𝑎𝑟𝑦
𝑑𝑢𝑟𝑖𝑛𝑔 𝑡ℎ𝑒
𝑙𝑖𝑞𝑢𝑒𝑓𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑎𝑠 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑡𝑎𝑛𝑡𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝ℎ𝑒𝑛𝑜𝑚𝑒𝑛𝑎 𝑖𝑠
𝑐𝑎𝑙𝑙𝑒𝑑 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑝ℎ𝑒𝑛𝑜𝑚𝑒𝑛𝑎 .
The critical phenomenon was first studied by 𝐴𝑛𝑑𝑟𝑒𝑤𝑠 in 1869. He studies the pressure temperature
conditions for the liquefaction of several gases. Every gas can be converted into liquid state by decrease
of temperature and increase of pressure. However, the effect of temperature is more prominent than
that of pressure. Andrews concluded that for each gas there is temperature above which the gas cannot
be liquefied, however high pressure we may apply. This temperature is called critical temperature (𝑇𝑐 )
which is different for different gases.
Thus, critical temperature (𝑇𝑐 )) is that temperature above which the gas cannot be liquefied, how so ever
high pressure we may apply on the gas. Similarly, the minimum pressure required to liquefy the gas at
critical temperature (𝑇𝑐 )) is called critical pressure (𝑃𝑐 ), and the volume occupied by one mole of the gas
at critical temperature (𝑇𝑐 ) and critical pressure (𝑃𝑐 )) is called critical volume (𝑉𝑐 ).
The state of the gas at critical temperature and pressure is said to be critical state at critical state the gas
becomes identical with its liquid.
The critical temperature, critical pressure and critical volume of a gas are collectively called as the critical
constant. The isotherm of the gas at critical temperature, pressure is known as critical isotherm and the
point representing the critical state of the gas at critical isotherm is called critical point.
𝑇ℎ𝑒 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑖𝑛𝑡𝑜 𝑙𝑖𝑞𝑢𝑖𝑑 𝑎𝑡 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑
𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝ℎ𝑒𝑛𝑜𝑚𝑒𝑛𝑎.
The critical constants of the carbon dioxide (𝐶𝑂2) are
𝑇𝑐 = 31.1℃ 𝑜𝑟 304.2 𝐾
𝑃𝑐 = 72.9 𝑎𝑡𝑚
𝑉𝑐 = 95.65 𝑐𝑚3 𝑚𝑜𝑙−1
Critical phenomena and Van der Waals equation: -
Van der Waals equation for real gases can be used to determine the values of𝑇𝑐 , 𝑉𝑐 and𝑃𝑐 .
We know Van der Waals equation
 𝑎
(𝑃 + ) (𝑉 − 𝑏) = 𝑅𝑇
𝑉2
Simplify
𝑎
𝑃(𝑉 − 𝑏) + (𝑉 − 𝑏) = 𝑅𝑇
𝑉2
Or
𝑎 𝑎𝑏
𝑃𝑉 − 𝑃𝑏 + − = 𝑅𝑇
𝑉 𝑉2
Multiply by 𝑉 2
𝑎 𝑎𝑏 2
(𝑃𝑉 − 𝑃𝑏 + − )𝑉 = 𝑅𝑇𝑉 2
𝑉 𝑉2
𝑃𝑉 3 − 𝑃𝑏𝑉 2 + 𝑎𝑉 − 𝑎𝑏 = 𝑉 2 𝑅𝑇
Or
𝑃𝑉 3 − 𝑃𝑏𝑉 2 − 𝑉 2 𝑅𝑇 + 𝑎𝑉 − 𝑎𝑏 = 0
𝑃𝑉 3 − 𝑃𝑏𝑉 2 − 𝑉 2 𝑅𝑇 + 𝑎𝑉 − 𝑎𝑏 = 0
Divide the equation by 𝑃
𝑃𝑉 3 𝑃𝑏𝑉 2 𝑉 2 𝑅𝑇 𝑎𝑉 𝑎𝑏
− − + − =0
𝑃 𝑃 𝑃 𝑃 𝑃
2
𝑉 𝑅𝑇 𝑎𝑉 𝑎𝑏
𝑉 3 − 𝑏𝑉 2 − + − =0
𝑃 𝑃 𝑃
Taking 𝑉 2 common we get
𝑅𝑇 𝑎𝑉 𝑎𝑏
𝑉 3 − (𝑏 + )𝑉 2 + − =0
𝑃 𝑃 𝑃
𝑅𝑇 𝑎𝑉 𝑎𝑏
𝑉 3 − ( + 𝑏) 𝑉 2 + − = 0 … … … (1)
𝑃 𝑃 𝑃
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (1) 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠 𝑉𝑎𝑛 𝑑𝑒𝑟 𝑊𝑎𝑎𝑙𝑠 𝑐𝑢𝑏𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛. In this equation 𝑉 gives either three real
values of “𝑉” or one real and two imaginary values on solving. At critical conditions the distinction
between liquid and gas disappears and three values of “𝑉” are identical. It means that for a given value of
𝑃 and 𝑇 “𝑉” has only real value. Now “𝑉” represents critical volume (𝑉𝑐 ).
𝑉 = 𝑉𝑐
𝑉 − 𝑉𝑐 = 0 … … … . . (2)
Taking cubic form
(𝑉 − 𝑉𝑐 )3 = 0
On expanding
𝑉 3 − 3𝑉 2 𝑉𝑐 + 3𝑉𝑐 2 𝑉 − 𝑉𝑐 3 = 0 … … … . (3)
When equation (1) has three roots. It must be identically equal to equation (3). Equating the co-efficient
of 𝑉 2 in equation (3) and (1) when (𝑉 = 𝑉𝑐 , 𝑃 = 𝑃𝑐 , 𝑇 = 𝑇𝑐 we get
𝑅𝑇
3𝑉 2 𝑉𝑐 = ( 𝑃 + 𝑏) 𝑉 2
𝑅𝑇
3𝑉𝑐 = ( + 𝑏) … … … … . . (4)
𝑃
Equating the coefficient of 𝑉
𝑎𝑉
3𝑉𝑐 2 𝑉 =
𝑃
2 𝑎
3𝑉𝑐 = … … … . . (5)
𝑃

Equating numerical terms.

 𝑎𝑏
𝑉𝑐 3 = … … … . (6)
𝑃
Dividing equation (6) by equation (5) we get
𝑎𝑏
𝑉𝑐 3 𝑃
2 = 𝑎
3𝑉𝑐
𝑃
𝑉𝑐 3 𝑎𝑏 𝑃
= ×
3𝑉𝑐 2 𝑃 𝑎
𝑉𝑐
=𝑏
3
𝑉𝑐 = 3𝑏 … … … (7)
When all roots are equal than 𝑉 = 𝑉𝑐 , 𝑃 = 𝑃𝑐 , 𝑇 = 𝑇𝑐
We know equation (6)
𝑎𝑏
𝑉𝑐 3 = 𝑃
𝑐
Putting the values of 𝑉𝑐 from equation (7) we get
𝑎𝑏
(3𝑏)3 =
𝑃𝑐
𝑎𝑏
27𝑏 3 =
𝑃𝑐
𝑎𝑏
𝑃𝑐 =
27𝑏 3
𝑎
𝑃𝑐 = … … … (8)
27𝑏 2
According to equation (4)
𝑅𝑇
3𝑉𝑐 = +𝑏
𝑃𝑐
Or
𝑅𝑇
3𝑉𝑐 = 𝑃 𝑐 + 𝑏 … … … . (9)
𝑐
Putting the values of 𝑉𝑐 and 𝑃𝑐 in equation (9)
𝑅𝑇𝑐
3(3𝑏) = 𝑎
27𝑏 2
𝑅𝑇𝑐
3(3𝑏) = 𝑎 +𝑏
27𝑏2
𝑅𝑇𝑐
9𝑏 = 𝑎 +𝑏
2
27𝑏
𝑅𝑇𝑐
8𝑏 = 𝑎
2
27𝑏
𝑅𝑇𝑐
8𝑏 = 𝑎 × 27𝑏 2
8𝑏 𝑅𝑇𝑐
2
=
27𝑏 𝑎
8 𝑅𝑇𝑐
=
27𝑏 𝑎
8𝑎
𝑇𝑐 = … … … (10)
27𝑅𝑏
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕𝒔 “𝒂”, “𝒃” 𝒂𝒏𝒅 “𝑹”
We know from equation (7)
𝑉𝑐 = 3𝑏
𝑉𝑐
𝑏 = 3 … … …. (1)
We know from equation (9)
𝑅𝑇𝑐
3𝑉𝑐 = +𝑏
𝑃𝑐
Putting the value of 𝑉𝑐
𝑅𝑇𝑐
3(3𝑏) = +𝑏
𝑃𝑐

𝑅𝑇𝑐
9𝑏 = +𝑏
𝑃𝑐
𝑅𝑇𝑐
9𝑏 − 𝑏 =
𝑃𝑐
𝑅𝑇𝑐
8𝑏 =
𝑃𝑐
𝑅𝑇𝑐
𝑏= … … … . (2)
8𝑃𝑐
Equation (1) and (2) show the values of “b”
We know from equation (5)
𝑎
3𝑉𝑐 2 =
𝑃
𝑎 = 𝑃𝑐 3𝑉𝑐 2 … … … . (3)

Putting the values of 𝑉𝑐

𝑎 = 3(3𝑏)2 𝑃𝑐
𝑎 = 27𝑏 2 𝑃𝑐
Putting the values of 𝑏 from equation (2)
𝑅𝑇𝑐
𝑎 = 27 ( 8𝑃 )2 𝑃𝑐
𝑐
27 𝑅 2 𝑇𝑐 2
𝑎= 𝑃
64 𝑃𝑐 22 𝑐
2
27 𝑅 𝑇
𝑎 = 64 𝑃 𝑐 … … … . . (4)
𝑐
From equation (1) and (2)
𝑉𝑐
𝑏=
3
𝑅𝑇𝑐
𝑏=
8𝑃𝑐

Compare
𝑅𝑇𝑐 𝑉𝑐
=
8𝑃𝑐 3
 8𝑃𝑐 𝑉𝑐
𝑅= … … … … (5)
3𝑇𝑐
We know for one mole of ideal gas.
𝑃𝑉 = 𝑅𝑇
𝑃𝑉
𝑅 = 𝑇
At critical conditions
𝑃𝑐 𝑉𝑐
𝑅 = … … … (6)
𝑇𝑐
Putting the value of 𝑃𝑐 , 𝑉𝑐 and 𝑇𝑐 in equation (6)
𝑃𝑐 𝑉𝑐
𝑅 =
𝑇𝑐
𝑎
2 3𝑏
𝑅 = 27𝑏
8𝑎
27𝑅𝑏
𝑃𝑐 𝑉𝑐 𝑎 27𝑅𝑏
= 2
3𝑏 ×
𝑇𝑐 27𝑏 8𝑎
𝑃𝑐 𝑉𝑐 3
𝑇
= 8 𝑅 … … … . (7)
𝑐
Experimental determination of critical constants: -
Measurement of Tc and Pc: -
𝐶𝑎𝑔𝑛𝑖𝑎𝑟𝑑 𝑑𝑒 𝑙𝑎 𝑡𝑜𝑢𝑟’𝑠 𝑎𝑝𝑝𝑎𝑟𝑎𝑡𝑢𝑠is used to determine the critical temperature (Tc) and critical pressure
(Pc). This apparatus determines both.
Tc and Pc simultaneously.
It consists of a glass bulb “𝐵” containing only the liquid and its vapour sealed over mercury. The bulb is
connected to manometer “𝑀” containing air above the mercury. The bulb “𝐵” is surrounded by a heating
Jacket as shown.

Procedure: -
The bulb is heated gently by circulating hot water in the heating jacket and the temperature is noted
at which the meniscus disappears. The bulb is then allowed to cool till the meniscus reappears or the
cloudiness appears on the surface of the bulb due to condensation of vapour. The means of the two
temperatures gives the critical temperature (𝑇𝑐 ) of the liquid/gas enclosed in the bulb “𝐵”.
Similarly, the corresponding pressure from the manometer at the two stages is noted and taking their
means, gives the critical pressure (𝑃𝑐 ).
Measurement of 𝑽𝒄 :-
The critical volume (𝑉𝑐 ) determination is based upon a rule which states that,” the means of the densities
of any substance in the state of liquid and saturated vapour at the same temperature is a linear function
of the temperature”.

Procedure: -
To determine the critical volume of any substance, the densities of the substance in the liquid
state and those of the vapour in equilibrium with it are determined at several temperatures.
If these densities are plotted against temperature the curve obtained is shown.

The densities of the vapour lie along the curve 𝐴𝐶 while those of the liquid along curve 𝐵𝐶. The two curves
𝑑𝑙+𝑑𝑣
meet at the point “𝐶” which corresponds to Tc. If the mean values of the densities ( ) are plotted
2
against temperature a straight line 𝐶𝐷 is obtained. The line on extrapolation meets the density axis at “𝐷”
which is the value of critical density. Dividing the molecular weight of the substance by critical density
gives critical volume𝑉𝑐 .
We know
𝑀𝑎𝑠𝑠
𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑉𝑜𝑙𝑢𝑚𝑒
𝑀
𝑑=
𝑉
𝑀
𝑑𝑐 =
𝑉𝑐
𝑀
𝑉𝑐 =
𝑑𝑐

Andrews experiments on critical Phenomenon (𝑪𝑶𝟐 isotherms)

In 1861 Andrews carried out his experiment on the effect of temperature and pressure on the volume of
𝐶𝑂2 . The 𝑃 − 𝑉 curves of a gas at constant temperature are called isotherms or isothermals.
Andrews plotted the isotherms of 𝐶𝑂2 for a series of temperature. There are three types of isotherms.
 (1) 𝐴𝑏𝑜𝑣𝑒 31℃
(2) 𝐵𝑒𝑙𝑜𝑤 31℃
(3) 𝐴𝑡 31.1℃
(1) Isotherms above 31℃:-
The isotherms above 31℃ are hyperbolic curves in accordance with Boyle’s law
𝑃𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. Above 31℃ no liquefaction of 𝐶𝑂2 is observed and 𝐶𝑂2 always exists in gaseous state
above 31℃.
(2) Isotherms below 31℃:-
The isotherms below 31℃ are discontinuous. The isotherm of 21℃ consists of three parts.
i. Curve 𝑨𝑩:-
It is a “𝑃𝑉” curve for gaseous 𝐶𝑂2 . Along “𝐴𝐵” the volume decreases gradually with increase in
pressure at“𝐵” the volume decreases suddenly due to the formation of liquid 𝐶𝑂2 having high
density.
ii. Curve 𝑩𝑪:-
Along BC the liquefaction continues while the pressure is constant at “𝐶” all the gas is converted to
liquid.
iii. Curve 𝑪𝑫:-
The curve “𝐶𝐷” is the PV curve of liquid 𝐶𝑂2 the vertical portion shows liquid is not very
compressible.
(3) Isotherm at 31.1℃:-
The isotherm at 31.1℃ shows that 𝐶𝑂2 has becomes liquid at the point “𝐹” at this point “F” it is not
possible to distinguish has from liquid. 𝐶𝑂2 In this state is called critical state. The point 𝐹 is called critical
point and temperature, pressure, and volume at point “𝐹” are called critical temperature (𝑇𝑐 ) critical
pressure 𝑃𝑐 and critical volume 𝑉𝑐 .
The dotted area encloses the region where both liquid and gas can exist together. While the point marks
the boundary between the gaseous 𝐶𝑂2 on the right and liquid 𝐶𝑂2 on the left.

Molecular collisions: -
The gas molecules are in constant random motion. “The striking of the gas molecules with each
other is called molecular collisions”.
According to kinetic molecular theory the gas molecules colliding among themselves and with the walls
of the vessel, during these collisions they suffer a change in their directions and change their velocities,
but the total energy remains the same. “The collisions in which there is no net loss or gain of energy are
called elastic collisions”.
Types of Collisions: -
There are three types of collisions.
1. Grazing or glancing collisions.
2. Head on collisions.
3. Right angle collisions.
1. Grazing or Glancing Collisions: -
When molecules are moving parallel to each other with average velocity 𝑢̅ or 𝑐̅ and their outer boundaries
touch each other is called Grazing or Glancing Collisions. The value of grazing collision is the same as
average velocity (𝑐̅).

2. Head on Collisions: -
When the molecules approach each other on a straight line and collide with each other such collisions are
called head on collisions.
The colliding molecules move in the same straight line with greater collision but in the reverse direction
the relative speed becomes 2𝑐.

3. Right angled Collisions: -

When two molecules approach each other, and their approaching lines are approximately 90o to each
other. Then the collision is right angled. The relative speed is√2𝑐̅.

Collision properties: -
As the molecules of a gas are constantly colliding with one another. The properties of gases such as
diffusion, viscosity and mean free path depend on the molecular collisions. Some collision properties are
given below.
Collision diameter: -
“The closest distance between the centers of two molecules during collision is known a collision
diameter”.
OR
“The distance between centers of the point of closed approach is called collision diameter”.
Is denoted by sigma (𝜎).
The molecules come close to do collision at the line of constant of the boundaries. There is a limit
beyond which they cannot come close to each other. This is called the distance of closest approach.
Collisions Number: -
“The number of molecules with which single molecule will collide per unit time per unit volume of a gas
is called collision number”. It is given by
ZI = √2𝜋𝜎 2 𝑐̅𝜌
𝑐 ̅ = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝜎 = 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
𝜌 = 𝑛𝑜 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠
Collision Frequency: -
“The number of molecular collisions taking place per unit volume (𝑐. 𝑐) of the gas”.
Total number of molecules colliding per unit volume per unit time is obtained by multiplying collision
number by number density𝜌. Thus, total number of colliding molecules=√2𝜋𝜎 2 𝑐̅𝜌2 . Since each collision
involves two molecules of the same types, number of collisions is given by one half of this.
1
Z11 = √2𝜋𝜎2 𝑐̅𝜌2
2
1
= 𝜋𝜎2 𝑐̅𝜌2
√2
It readily follows that number of collisions of molecules of types 1 with type 2 would be given by
Z12 = 𝜋𝜎 2 𝑐̅𝜌1 𝜌2
Where 𝜌1 𝑎𝑛𝑑 𝜌2 are the densities of type 1 and type 2 respectively?

Mean Free Path “𝝀”: -

The mean distance travelled by a molecule between two successive collisions is called mean free path.
Ti is denoted by “𝝀”.
It l1, l2, l3, ----------------- are the free paths for a molecule of a gas then
𝑙1+ 𝑙2 +𝑙3 + −−−−−−−−−−−−𝑙𝑛
Mean free path 𝝀 =
𝑛
Where “𝑛” is the number of molecules with which the molecule collides.
Velocity
𝑀𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ =
collision number
𝑐̅ 𝑐̅
𝝀= or 𝝀=𝑍
√2𝜋 𝜎2 𝑐̅𝜌2 𝐼

Derivation the expression for collision frequency: -

Consider a molecule “𝐴” moving in an imaginary cylinder having a diameter equal to 2𝜎.
𝐿𝑒𝑡 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜎
𝐵𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝜎 2
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑖𝑛 𝑚𝑒𝑡𝑒𝑟𝑠 = 𝑐̅
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝜎 2 𝑐̅ (𝐴𝑟𝑒𝑎 × 𝐿𝑒𝑛𝑔𝑡ℎ)
𝐿𝑒𝑡 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑖𝑛 1. 𝑐𝑐 = 𝑛
𝑆𝑜, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 “𝐴” 𝑤𝑖𝑙𝑙 𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒 𝑖𝑛 𝑜𝑛𝑒
𝑠𝑒𝑐𝑜𝑛𝑑
𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 = 𝜋𝜎 2 𝑐̅.
𝐵𝑒 𝑐𝑎𝑟𝑒𝑓𝑢𝑙 𝑡ℎ𝑎𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑑 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 𝑖𝑠 𝑚𝑢𝑐ℎ 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛
𝑔𝑟𝑎𝑧𝑖𝑛𝑔 𝑎𝑛𝑑 ℎ𝑒𝑎𝑑 𝑜𝑛
𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 = √2𝜋𝜎 2 𝑐̅𝑛.
𝑆𝑜 𝑡ℎ𝑒 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏𝑦 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦
𝑍𝐼 = √2𝜋𝜎 2 𝑐̅𝑛
Since there are “n” molecules present in 1cm3 and each of these molecules’ experiences 𝑍𝐼 collision per
second the total number of collisions per second per cubic cm will be 𝑛𝑍𝐼 .
To obtain exact number of collision s-1cm-3 (𝑍𝐼𝐼 ) we divide n ZI by 2 becomes each collision involve two
molecules. This ensures that each collision will not be counted twice thus
1
𝑍𝐼𝐼 = 𝑛 𝑍𝐼
2
1
𝑍𝐼𝐼 = 2
√2 𝜋𝜎 2 𝑛2 𝑐̅
1
𝑍𝐼𝐼 = 𝜋𝜎 2 𝑛2 𝑐̅
√2
𝑍𝐼𝐼 Refers to the collision frequency.

Molecular Velocities: -
According to the molecular or kinetic gas equation that all the molecules are moving with some speed.
So, there for gas molecules possess velocities. We assumed in the derivation of kinetic gas equation that
all molecules are moving with same speed having roots mean square velocity (𝑈). But in actual practice
all the molecules cannot move with the same speed because they are frequently colliding with each other
as result the molecular velocities change from time to time.
Types of Velocities: -
There are three types of velocities.
1. Average Velocity 𝑈 ̅
2. Root Mean Square Velocity (𝑈𝑟𝑚𝑠 )
3. Most probable Velocity 𝑈(𝑚.𝑝)

1. Average Velocity: -
“The arithmetic mean of the velocities possessed by the different molecules of a gas at given
temperature”. OR
𝑈 + 𝑈2 + 𝑈3 + ⋯ … … … … … . . … … …. + 𝑈𝑛
𝑈̅= 1
𝑛
̅ can be obtained from Maxwell’s law of molecular velocities distribution.
The expression for 𝑈
8𝑅𝑇
̅=√
𝑈
𝜋𝑀

8 𝑅𝑇
√ √
𝜋 𝑀
 𝑅𝑇
1.59√ … … … … … (1)
𝑀

̅ = 1.59√𝑅𝑇
𝑈 𝑀

̅ ∝ √1
̅ ∝ √𝑇 and𝑈
The equation shows that the 𝑈 𝑀
2. Root Mean Square Velocities: -
It is defined as the square root of the mean of the squares of different velocities i.e.
𝑈21 + 𝑈22+ 𝑈13+⋯………………………………….. + 𝑈2𝑛
𝑈𝑟𝑚𝑠 = √ 𝑛
According to kinetic gas equation
1
𝑃𝑉 = 3 𝑚𝑛𝑢−2 𝑚𝑛 = 𝑀
1
𝑃𝑉 = 3 𝑀𝑢−2
3𝑃𝑉 3𝑅𝑇
𝑈2 = 𝑀
= 𝑀
∴ 𝑃𝑉 = 𝑅𝑇 𝑤ℎ𝑒𝑛 𝑛 =1
3𝑃𝑉 3𝑅𝑇
𝑈𝑟𝑚𝑠 = √ = √
𝑀 𝑀
𝑅𝑇
𝑈𝑟𝑚𝑠 = 1.73√ 𝑀 … … … … … . (2)

1
𝑈𝑟𝑚𝑠 ∝ √𝑇 𝑎𝑛𝑑 √
𝑀
3. The most probable velocities: -
“The velocity possessed by greatest fraction of the molecules in a gas at any temperature”.
According to Maxwell’s destruction law.
2𝑅𝑇
𝑈𝑚.𝑝 = √
𝑀

𝑅𝑇
𝑈𝑚.𝑝 = 1.414 √ … … … … . . (3)
𝑀
Relationship of Velocities: -
𝑈𝑟𝑚𝑝 ∶ ̅
𝑈 ∶ 𝑈𝑚.𝑝
𝑅𝑇 𝑅𝑇 𝑅𝑇
1.732√ ∶ 1.59√ ∶ 1.414√
𝑀 𝑀 𝑀
1.73 ∶ 1.59 ∶ 1.41
Dividing equation (1) by Equation (2)
𝑅𝑇
𝑈̅ 1.59 √ 𝑀
=
𝑈𝑟𝑚𝑠 𝑅𝑇
1.73 √ 𝑀
̅
𝑈 1.59
= = 0.91
𝑈𝑟𝑚𝑠 1.73
̅ = 0.91 × 𝑈𝑟𝑚𝑠
𝑈
Other relation can be drawn in the same manner.