Maharashtra HSC · Class XII · Physics

Chapter 3: Kinetic Theory of Gases & Radiation

Full revision notes for Chapter 3 - Kinetic Theory of Gases & Radiation.

3.1

Ideal Gas and Gas Laws

PV = nRT = \frac{N}{N_A}RT = N k_B T

$P$ pressure (Pa), $V$ volume (m³), $n$ number of moles, $T$ absolute temperature (K), $R$ universal gas constant (8.314 J/mol·K), $N$ number of molecules, $N_A$ Avogadro number and $k_B$ the Boltzmann constant ( $R/N_A$ ).

3.2

Kinetic Theoretic Pressure and Speeds

Pressure from kinetic theory

P = \frac13 \rho v_{rms}^2 = \frac13\, n\, m\, v_{rms}^2

$\rho$ gas density, $n$ molecule number density, $m$ mass of one molecule and $v_{rms} = \sqrt{\overline{v^2}}$ the root-mean-square speed.

rms, average and most-probable speeds

v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}

v_{avg} = \sqrt{\frac{8RT}{\pi M}} \qquad v_{mp} = \sqrt{\frac{2RT}{M}}

$M$ is the molar mass (kg/mol). The ratio is $v_{rms} : v_{avg} : v_{mp} = \sqrt{3} : \sqrt{8/\pi} : \sqrt{2}$ .

Translational kinetic energy

\tfrac12 m v_{rms}^2 = \tfrac32 k_B T \qquad \text{per molecule}

\overline{KE}_{avg} = \tfrac32 R T \qquad \text{per mole}

The mean translational kinetic energy depends on temperature alone, connecting microscopic motion to temperature.

3.3

Blackbody Radiation

Blackbody and Féry's blackbody

A perfect blackbody absorbs all radiation incident on it and, in thermal equilibrium, radiates the maximum possible energy for its temperature. Féry's blackbody is a hollow box with a small hole: any ray entering the cavity is effectively trapped (multiple reflections), so the hole acts as a near-ideal blackbody.

Stefan-Boltzmann law

E = \sigma_{SB}\, T^4

The power radiated per unit area of a blackbody at absolute temperature $T$ . $\sigma_{SB} = 5.67\times 10^{-8}$ W/m²K⁴. Net power emitted into surroundings at $T_0$ : $P = \sigma_{SB} A\,(T^4 - T_0^4)$ .

Wien's displacement law

\lambda_{max}\,T = b

$\lambda_{max}$ is the wavelength of peak emission, $T$ temperature and $b = 2.898\times 10^{-3}$ m·K. Hotter bodies radiate most energy at shorter wavelengths.

Kirchhoff's law of radiation

At thermal equilibrium, for matched wavelength and direction, the spectral emissive power $e_\lambda$ equals the spectral absorptivity $a_\lambda$ . A good absorber is a good emitter: bodies radiate and absorb in proportion to their absorptivity.