JEE Physics Kinetic Theory Notes Set 02

Ideal Gas Laws

An ideal gas obeys \( PV=nRT=NkT \) with \( R=8.314\ \text{J/mol K} \) and \( k=1.38\times10^{-23}\ \text{J/K} \). Special cases: Boyle's law \( PV= \) const at constant \( T \); Charles's law \( V\propto T \) at constant \( P \); Gay-Lussac's law \( P\propto T \) at constant \( V \). For a fixed amount of gas \( \dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2} \), and the density satisfies \( \rho=\dfrac{PM}{RT} \). Temperature must always be in kelvin. A real gas behaves ideally at low pressure and high temperature, when molecular volume and attractions are negligible.

Assumptions and Pressure of a Gas

Kinetic theory assumes a very large number of identical point-like molecules in random motion, no attraction or repulsion except during collisions, perfectly elastic collisions with each other and with the walls, and negligible molecular size compared with the container. Pressure arises from the momentum transferred to the walls by colliding molecules:

\( P=\dfrac13\dfrac NVm\overline{v^{2}}=\dfrac13\rho\overline{v^{2}}=\dfrac23\,\dfrac{E}{V} \)

where \( E \) is the total translational kinetic energy. If only one component of velocity is given, \( P=\dfrac NVm\overline{v_x^{2}} \).

Temperature and Molecular Speeds

The mean translational kinetic energy per molecule is \( \tfrac32kT \), depending on temperature alone. Hence at the same temperature all gases have the same mean kinetic energy per molecule, and lighter molecules move faster.

The three speeds are in the ratio \( v_p:\bar v:v_{rms}=1:1.13:1.22 \) and all scale as \( \sqrt{T/M} \). In a mixture every component has the same temperature, so each keeps its own speed distribution. Average speed is the mean of the magnitudes, which is not zero even when the mean velocity is zero.

Degrees of Freedom and Internal Energy

By the law of equipartition each degree of freedom carries \( \tfrac12kT \) per molecule (or \( \tfrac12RT \) per mole). With \( f \) degrees of freedom the internal energy of \( n \) moles is \( U=\dfrac f2nRT=\dfrac f2PV \).

The specific heat relation is \( C_p-C_v=R \) per mole (or \( R/M \) per unit mass), and \( C_v=\dfrac{R}{\gamma-1} \). For a mixture, add the \( nC_v \) of the components and divide by the total number of moles; \( \gamma \) is then found from the mixture \( C_p/C_v \).

Mean Free Path and Other Results

1. Mean free path \( \lambda=\dfrac{1}{\sqrt2\,\pi d^{2}n}=\dfrac{kT}{\sqrt2\,\pi d^{2}P} \), where \( d \) is the molecular diameter and \( n \) the number density. The mean time between collisions is \( \lambda/\bar v \).
2. Number density \( n=\dfrac{P}{kT} \). At fixed pressure and volume, the number of molecules in an open vessel varies as \( 1/T \).
3. Equal volumes of different gases at the same temperature and pressure hold equal numbers of molecules (Avogadro's law).
4. Dalton's law: the pressure of a mixture is the sum of the partial pressures of its components.
5. In a free expansion into vacuum, an ideal gas does no work and its temperature stays the same.