Kinetic Theory of Gas, Gas Laws & Gas Equations | Physics Class 11 - NEET PDF Download

Basics of Kinetic Theory

It says that the molecules of gas are in random motion and are continuously colliding with each other and with the walls of the container. All the collisions involved are elastic in nature due to which the total kinetic energy and the total momentum both are conserved. No energy is lost or gained from collisions.

Ideal Gas Equation

The ideal gas equation is as follows
PV = nRT
the ideal gas law relates the pressure, temperature, volume, and number of moles of ideal gas. Here R is a constant known as the universal gas constant.

Assumptions

• The gas consists of a large number of molecules, which are in random motion and obey Newton’s laws of motion.
• The volume of the molecules is negligibly small compared to the volume occupied by the gas.
• No forces act on the molecules except during elastic collisions of negligible duration.

At the ordinary temperature and pressure, the molecular size is very very small as compared to the intermolecular distance. In case of gas, the molecules are very far from each other. So when the molecules are far apart and the size of the molecules is very small when compared to the distance between them. Therefore the interactions between the molecules are negligible.
In case there is no interaction between the molecules than there will be no force acting on the molecule. This is because it is not interacting with anything.  Newton’s first law states that an object at rest will be at rest and an object will be in motion unless an external force acts upon it.
So in this case, if the molecule is not interacting with any other molecule then there is nothing that can stop it. But sometimes when these molecules come close they experience an intermolecular force. So this basically something we call as a collision.

Solved Question

Q.1. The number of collisions of molecules of an ideal gas with the walls of the container is increasing per unit time. Which of the following quantities must also be increasing?
I. pressure 
II. temperature 
III. the number of moles of gas.
(a) I only
(b) I and II only
(c) II  only
(d) II and III only
Ans: (a)
Solution: If there are more collisions between the molecules and the walls of the container, there must be more pressure against the wall. If there are more collisions than the molecules must have high average kinetic energy. Since kinetic energy is proportional to temperature, the temperature is also increasing.

Q.2. When the volume of a gas is decreased at constant temperature the pressure increases because of the molecules
(a) Strike unit area of the walls of the container more often
(b) Strike unit area of the walls of the container with higher speed
(c) Move with more kinetic energy
(d) Strike unit area of the walls of the container with less speed
Ans: (a)
Solution: The kinetic theory of the molecules depends on the temperature and since here the temperature remains constant, the pressure cannot increase due to the other options mentioned. So option A is correct as more pressure is generated here and hence pressure increases.

Kinetic Theory

Assumptions of Kinetic Theory of Gases:

1. Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different from those of another gas.
2. The molecules of a gas are identical spherical, rigid and perfectly elastic point masses.
3. Their molecular size is negligible in comparison to intermolecular distance (10^-9 m).
4. The speed of gas molecules lies between zero and infinity (very high speed).
5. The distance covered by the molecules between two successive collisions is known as free path and mean of all free path is known as mean free path.
6. The number of collision per unit volume in a gas remains constant.
7. No attractive or repulsive force acts between gas molecules.
8. Gravitational to extremely attraction among the molecules is ineffective due small masses and very high speed of molecules.

Gas Laws

Assuming permanent gases to be ideal, through experiments, it was established that gases irrespective of their nature obey the following laws.

Boyle’s Law

At constant temperature the volume (V) of given mass of a gas is inversely proportional to its pressure (p), i.e.,

V ∝ 1/p ⇒ pV = constant

For a given gas, p₁V₁ = p₂V₂

Fig: Boyle's law

Charles’ Law

At constant pressure the volume (V) of a given mass of gas is directly proportional to its absolute temperature (T), i.e.,

V ∝ T ⇒ V / T = constant

For a given gas, V₁/T₁ = V₂/T₂

At constant pressure the volume (V) of a given mass of a gas increases or decreases by 1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature.

Fig: Charles' law

Volume of the gas at t°Celsius:

V_t = V₀ (1 + t/273.15)

where V₀ is the volume of gas at 0°C.

Gay Lussacs’ or Regnault’s Law

At constant volume, the pressure p of a given mass of gas is directly proportional to its absolute temperature T, i.e. ,

p ∝ T ⇒ V/T = constant

For a given gas,
p₁/T₁ = p₂/T₂

At constant volume (V) the pressure p of a given mass of a gas increases or decreases by 1/273.15 of its pressure at 0°C for each l°C rise or fall in temperature.

Fig: Gay Lussacs' law

Volume of the gas at t°C, p_t = p₀ (1 + t/273.15)

where P₀ is the pressure of gas at 0°C.

Avogadro’s Law

Avogadro stated that equal volume of all the gases under similar conditions of temperature and pressure contain equal number molecules. This statement is called Avogadro’s hypothesis. According to Avogadro’s law,

(i) Avogadro’s number: The number of molecules present in 1g mole of a gas is defined as Avogadro’s number.

N_A = 6.023 X 10²³ per gram mole

(ii) At STP or NTP (T = 273 K and p = 1 atm 22.4 L of each gas has 6.023 x 10²³molecules.

(iii) One mole of any gas at STP occupies 22.4 L of volume.

Standard or Perfect Gas Equation

Gases which obey all gas laws in all conditions of pressure and temperature are called perfect gases.

Equation of perfect gas pV=nRT

where p = pressure, V = volume, T = absolute temperature, R = universal gas constant and n = number of moles of a gas.

Universal gas constant R = 8.31 J mol^-1K^-1.

Real Gases

Real gases deviate slightly from ideal gas laws because:

• Real gas molecules attract one another.
• Real gas molecules occupy a finite volume.

Real or Van der Waal’s Gas Equation

(p + a/V²) (V – b) = RT

where a and b are called Van der Waal's constants.

Pressure due to an ideal gas is given by
p = (1/3).(mn/V). c² = 1/3 ρ c²

For one mole of an ideal gas
P = (1/3).(M/V).c²

where, m = mass of one molecule, n = number of molecules, V = volume of gas, c = (c₁²+ c₂² + … + c_n²) / n all the root mean square (rms) velocity of the gas molecules and M = molecular weight of the gas. If p is the pressure of the gas and E is the kinetic energy per unit volume is E, then

p = (2/3).E

Kinetic Energy of a Gas

(i) Average kinetic energy of translation per molecule of a gas is given by:

E = (3/2) kt

where k = Boltzmann’s constant.

(ii) Average kinetic energy of translation per mole of a gas is given by:

E = (3/2) Rt

where R = universal gas constant.

(iii) For a given gas kinetic energy

E ∝ T ⇒ E₁/E₂ = T₁/T₂

(iv) Root mean square (rms) velocity of the gas molecules is given by:

(v) For a given gas c ∝ √T

(vi) For different gases c ∝1/√M

(vii) Boltzmann’s constant k = R/N

where R is ideal gas constant and N = Avogadro number.

Value of Boltzmann’s constant is 1.38 x 10^-28 J/K.

(viii) The average speed of molecules of a gas is given by

(ix) The most probable speed of molecules of a gas is given by

⇒  

Important Points:

(i) With rise in temperature rms speed of gas molecules increases as

(ii) With the increase in molecular weight rms speed of gas molecule decrease as

(iii) Rms speed of gas molecules is of the order of knn/s, eg., at NTP for
hydrogen gas

(iv) Rms speed of gas molecules does not depend on the pressure of gas !if temperature remains constant) because p ∝ p (Boyle s law). If pressure is increased n times, then density will also increase by n times but v_rms remains constant.