Kinetic Theory of Gases – JEE Mains Physics
1. Equation of State of a Perfect Gas
- The equation of state for a perfect gas is given by the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.
2. Work Done on Compressing a Gas
- Work done (W) on compressing a gas is given by:
W = ∫ P dV
where P is the pressure and V is the volume of the gas.
3. Kinetic Theory of Gases: Assumptions
- The gas consists of a large number of molecules in random motion.
- The volume of the gas molecules is negligible compared to the volume of the container.
- The molecules do not experience any intermolecular forces except during collisions.
- Collisions of gas molecules are perfectly elastic (no energy is lost in collisions).
- The time spent during collisions is negligible compared to the time between collisions.
4. Concept of Pressure
- Pressure (P) exerted by a gas is the force per unit area on the walls of its container, caused by the continuous collisions of the gas molecules with the walls.
P = (1/3) * (n * m * v^2)
where n is the number of molecules, m is the mass of a molecule, and v is the velocity of the molecules.
5. Kinetic Interpretation of Temperature
- The temperature of a gas is directly related to the average kinetic energy of the gas molecules:
K.E. = (3/2) kT
where k is the Boltzmann constant and T is the temperature in Kelvin.
6. RMS Speed of Gas Molecules
- The root mean square (RMS) speed of gas molecules is the square root of the average of the squares of the velocities of the molecules.
v_rms = √((3kT) / m)
where k is the Boltzmann constant, T is the temperature, and m is the mass of the molecule.
7. Degrees of Freedom
- Degrees of freedom refer to the number of independent ways in which a molecule can store energy. For a monatomic gas, there are 3 degrees of freedom (translational motion in x, y, and z directions). For a diatomic gas, there are 5 degrees of freedom (3 translational and 2 rotational).
8. Law of Equipartition of Energy
- The law of equipartition of energy states that energy is equally distributed among all degrees of freedom. Each degree of freedom contributes an energy of (1/2)kT to the system.
E = (f/2) kT
where f is the number of degrees of freedom.
9. Applications to Specific Heat Capacities of Gases
- Specific heat capacity is related to the degrees of freedom of the molecules. For a monoatomic ideal gas, the molar specific heat at constant volume is (3/2)R, and for a diatomic ideal gas, it is (5/2)R at constant volume.
10. Mean Free Path
- The mean free path is the average distance a molecule travels before colliding with another molecule. It is given by:
λ = 1 / (√2 * nσ)
where n is the number of molecules per unit volume, and σ is the effective collision cross-section.
11. Avogadro's Number
- Avogadro's number (Nₐ) is the number of molecules in one mole of a substance, approximately 6.022 × 10²³ molecules/mol.