Deduction of Gas Laws From kinetic Theory

Boyle’s law

At constant temperature, the average kinetic energy and hence the average speed of the molecules is constant. The number of molecules present in a given mass of a gas is also constant.

Let the volume of a given mass of a gas be reduced to one half of its original volume. The same number of molecules with their same average speed will now have half the original space to move about. As a result, the number of molecules striking the unit area of the walls of the container in a given time will be doubled and consequently the pressure is also doubled.

If the volume of a given mass of a gas is doubled at constant temperature the same number of molecules with their same average speed will now have double the space to move about. The number of molecules striking the unit area of the walls of the container in a given time will now become one half of the original value. As a result, the pressure of the gas will be reduced to one half of its original volume.

Deduction from Kinetic Gas Equation

PV = 1/3 mnc² 

PV = 2/3 × 1/2 (Mc²)

But 1/2 Mc² = Kinetic energy of the gas

PV = 2 K.E/ 3

K.E. ∝ T

K.E. = kT

PV = 2kT/3

As 2/3 and k are constant

Hence PV = constant

 

Charles law

According to kinetic theory of gases, the average kinetic energy and hence the average speed of the gas molecules is directly proportional to its absolute temperature.

When the temperature of a gas is increased at constant volume the average kinetic energy of its molecules increases and hence the molecules would move faster. As a result, the molecules of a gas will strike the unit area of the walls of the container more frequently and vigorously. The pressure of the gas will increase accordingly. Thus, at constant volume the pressure of a gas increases with rise in temperature.

If the pressure of the gas is to be maintained constant, the force per unit area on the walls of the container in a given time must be kept the same. This can be achieved by increasing the volume proportionately. Thus at constant pressure, the volume of a given mass of a gas increases with increase in temperature. This explains Charles law.

Deduction from Kinetic Gas Equation

PV = 2kT / 3

V/T = 2k / 3P

2/3 is constant, k is also constant, hence P is kept constant, V/T =constant. which is charles law.

Dalton’s law of partial pressure

According to the kinetic theory of gases, the attractive forces between the molecules of the same or different gases are very weak under ordinary conditions of temperature and pressure. Therefore the molecules of a gaseous mixture move completely independent of one another. As a result, each molecule of the gaseous mixture would strike the unit area of the walls of the container the same number of times per second as if no other molecules were present.

 

Therefore the pressure due to a particular gas is not changed by the presence of other gases in the container. The total pressure exerted by a gaseous mixture must be kept equal to the sum of partial pressure of each gas when present alone in that space. Hence kinetic theory explains Dalton’s law of partial pressure.

Deduction from Kinetic Gas Equation

PV = 1/3 (mnc²)

P= mnc² / 3V

If only the first gas is enclosed in the vessel of volume V, the pressure exerted would be,

P₁ = m₁n₁c₁² / 3V

If second gas is enclosed in the same vessel, then the pressure exerted would be

P₂ = m₁n₂ c₂² /3V

P= m₁n₁c₁² / 3V  + m₁n₂ c₂² /3V + ………

P =P₁ + P₂