Table of contents  
Basics of Kinetic Theory  
Kinetic Theory  
Gas Laws  
Standard or Perfect Gas Equation  
Kinetic Energy of a Gas 
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.
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
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.
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.
Assumptions of Kinetic Theory of Gases:
Assuming permanent gases to be ideal, through experiments, it was established that gases irrespective of their nature obey the following laws.
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, p_{1}V_{1} = p_{2}V_{2}
Fig: Boyle's 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, V_{1}/T_{1} = V_{2}/T_{2}
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} = V_{0} (1 + t/273.15)
where V_{0} is the volume of gas at 0°C.
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,
p_{1}/T_{1} = p_{2}/T_{2}
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} = p_{0} (1 + t/273.15)
where P_{0} is the pressure of gas at 0°C.
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 10^{23} 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 10^{23}molecules.
(iii) One mole of any gas at STP occupies 22.4 L of volume.
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^{1}K^{1}.
Real gases deviate slightly from ideal gas laws because:
(p + a/V^{2}) (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). c^{2} = 1/3 ρ c^{2}
For one mole of an ideal gas
P = (1/3).(M/V).c^{2}
where, m = mass of one molecule, n = number of molecules, V = volume of gas, c = (c_{1}^{2}+ c_{2}^{2} + … + c_{n}^{2}) / 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
(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 ⇒ E_{1}/E_{2} = T_{1}/T_{2}
(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.
102 videos411 docs121 tests

1. What is the kinetic theory of an ideal gas? 
2. How does the kinetic theory explain the pressure of a gas? 
3. What is the relationship between temperature and the average kinetic energy of gas particles? 
4. Does the kinetic theory apply to real gases? 
5. How does the kinetic theory explain the expansion and contraction of gases with changing temperature? 

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