JEE Physics Kinetic Theory Notes Set 01

Assumptions of the Kinetic Theory

A gas is taken as a very large number of identical molecules in random motion. The molecules are point-like compared with the distances between them, exert no forces on one another except during brief elastic collisions, and obey Newton's laws. Collisions with the walls are elastic and cause the pressure. In equilibrium the motion is the same in all directions.

Pressure and Temperature

The kinetic theory gives \( P=\dfrac13\dfrac NV m\overline{v^2}=\dfrac13\rho\overline{v^2} \). With the ideal gas law this shows that the average translational kinetic energy per molecule is \( \tfrac32kT \), so temperature is a measure of the mean kinetic energy of translation and has meaning only for a collection of molecules. The gas constant R equals \( N_Ak \).

Speeds of Molecules

The speeds of molecules are distributed according to the Maxwell distribution. Three characteristic speeds are the most probable speed \( v_p=\sqrt{2RT/M} \), the average speed \( v_{av}=\sqrt{8RT/\pi M} \) and the rms speed \( v_{rms}=\sqrt{3RT/M} \), in the ratio \( 1:1.13:1.22 \). All of them depend on \( \sqrt{T/M} \): at a given temperature lighter molecules are faster, and for the same speed a heavier gas must be hotter. Example: hydrogen has an rms speed of 1930 m/s at 300 K.

Equation of State and Gas Mixtures

For an ideal gas \( PV=nRT=NkT \). For a mixture, Dalton's law gives the total pressure as the sum of partial pressures, each proportional to the number of moles of that gas, and the temperature of an isolated mixture follows from energy conservation: \( T=\dfrac{\sum n_iC_{vi}T_i}{\sum n_iC_{vi}} \). A reaction that changes the number of gas moles changes the pressure in the same ratio at constant V and T.

Law of Equipartition and Internal Energy

Each quadratic degree of freedom of a molecule carries average energy \( \tfrac12kT \). A monatomic gas has 3 degrees of freedom, a diatomic gas has 5 (3 translational and 2 rotational at ordinary temperatures, with vibration appearing at high temperature), and a non-linear polyatomic gas has 6. The internal energy of n moles is \( U=\dfrac f2nRT=\dfrac f2PV \), which depends only on temperature for an ideal gas.

Worked Example

Four containers of monatomic gases have the same temperature order as their \( m\,v_{rms}^2 \) values, because \( \tfrac12mv_{rms}^2=\tfrac32kT \); the number of moles does not matter. A container with atoms of mass 2m and speed \( 2V \) is therefore hotter than one with atoms of mass 4m and speed V (8 against 4).