Molecular nature of matter
* Kinetic theory was developed successfully by Maxwell, Boltzmann Brown and others.
* Kinetic theory explains the behavior of gases based on the idea that a gas consists of atoms or molecules which are in random motion.
* Dalton’s atomic theory proposes that every element is made up of tiny particles called atoms.
* The atoms of different elements differ from each other and are identical for the same element.
* Different molecules combine to form a compound whereas a molecule is formed by the combination of a small number of atoms.
Behavior of gases
* The value of KB is constant for all gases, which is called the Boltzmann constant. Its value is 1.38 X 10-23 JK-1.
* PV=ṅRT is called the ideal gas equation. Where R = N_{A} K_{B} is a universal gas constant. Its value is 8.314 J mol-1 K-1. This value is constant for a given system of units.
* A gas that satisfies the ideal gas equation at all temperatures and pressures is defined as an ideal gas.
* At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of the gas, and V/n is constant. This is called Avagadro’s Law.
* One mole of any gas can occupy a volume of 22.4 litres at S.T.P and it contains Avagadro’s number of molecules, that is, 6.02 x 1023 molecules.
* For a given mass of a gas at constant temperature, pressure is inversely proportional to its volume. This is called Boyle’s law.
Volume of a gas is directly proportional to its absolute temperature. This is called Charle’s law.
* The total pressure of a mixture of gases is equal to the sum of individual partial pressures exerted by all the gases. This is called Dalton’s law of partial pressures.
Kinetic theory – pressure of an ideal gas
* An ideal gas is defined by making the following assumptions based on the kinetic theory of gases. The postulates are:
• A gas consists of a large number of small particles called molecules
• The number of molecules is of the order of Avagadro’s number.
• Molecules are in constant random motion and obey Newton’s laws of motion.
• The rapidly moving particles constantly collide with one another and with the walls of the container.
• All these conditions are elastic and of negligible duration.
• The interaction between the molecules is negligible except during a collision.
• The volume of the molecules is negligible compared with the volume of the container.
• The molecules are perfectly spherical and elastic.
* Pressure is defined as force per unit area.
* From the kinetic theory of gases, pressure exerted by a gas is: P=1/3 mNv2
Kinetic theory – Temperature
* RMS speed is root mean square speed. It is defined as the square root of the mean of the squares of velocities of all the molecules in a gas. It is denoted by Vrms.
* The RMS speed is denoted by Vrms =V2.
* The total translational kinetic energy of all the molecules is: K= ∑ ½ mv2.
* The total kinetic energy of all the molecules of the gas is K=3/2nKBT
* As the RMS speed increases, the diffusion of the molecules of the gas increases.
* The diffusion is also inversely proportional to the mass of the molecule at given temperature because at the given temperature the average kinetic energy of the gas is fixed.
Law of equipartition of energy
The total energy of a molecule is translational energy plus rotational energy plus vibrational energy. That is,
E=1/2mvx2+1/2mvy2+1/2 mvz2+1/2Ixwx2+1/2 Iywy2+1/2µ(dr/dt)2 +1/2kr2
* The law of Equipartition of Energy states that in thermal equilibrium the average energy of a molecule in a gas associated with each degree of freedom is ½ kBT.
* According to the law of Equipartition of Energy, the average energy of a molecule of a monoatomic gas is equal to 3/2KBT.
* In the case of a diatomic gas, if the molecule does not vibrate, then the average energy of a molecule in a diatomic gas is equal to 5/2kBT.
* In the case of diatomic gas, if the molecule does not vibrate, then the average energy of a molecule In a diatomic gas is equal to 7/2kBT.
* The total average energy of a gas is called the internal energy of the gas.
* The internal energy of a monoatomic gas, U, is equal to 3/2ṅRT.
* The internal energy of a diatomic gas, if it is not vibrating, U, is equal to 5/2ṅRT.
* The internal energy of a diatomic gas, if it is not vibrating, U, is equal to 7/2ṅRT.
* Finally in the case of polyatomic gases, the number of independent terms in the energy, degree of freedom and internal energy of the gas depends on the arrangement of the molecules.
Kinetic theory – specific heat capacity
* The specific heat capacity of a substance is defined as the amount of heat energy required per unit mass to change the temperature by one unit.
* The amount of heat energy required for one mole of substance to change the temperature by one unit is called the molar specific heat capacity.
* The molar specific heat capacity of a monoatomic gas at a constant volume Cv is 3/2R.
* The Cp of a monoatomic gas is equal to 5/2R.
* For a monoatomic gases, the ratio of specific heat capacities is equal to 1.67.
* The molar specific heat capacity of a diatomic gas at a constant volume Cv is 5/2R.
* The Cp of a diatomic gas, where the molecules do not have vibrational motion, is equal to 7/2R.
* The ratio of specific heat capacities for a diatomic gas, when the molecule is not vibrating is 1.4.
* The molar specific heat capacity of a diatomic gas at a constant volume Cv is 7/2R.
* The Cp of a diatomic gas, when the molecules have vibrational motion, is equal to 9/2R.
* The ratio of specific heat capacities of a diatomic gas, when the molecule is vibrating is 1.29.
* The molar specific heat capacity of a polyatomic gas at a constant volume Cv is (3+f)R.
* The Cp of a polyatomic gas is equal to (4+f)R.
* The ratio of specific heat capacity of a polyatomic gas is (3+f)/(4+f).
Mean free path
* Molecules of a gas move with an average velocity of the order of the velocity of sound.
* The average distance travelled by a molecule between collisions is called the mean free path.
* The number of collisions =( Total number of collisions/Time) = nvπd2ṽ in one second. This is called collision frequency.
* Mean free path, ɭ = 1/nvπd2.
* The magnitude of the average relative velocity vrel = √ vrel .vrel
* Mean free path is inversely proportional to the number of molecules per unit volume and it is also inversely proportional to square of the diameter of the molecule.
* The resultant mean free path can be written as λ = 1/√2nvπd2.