Overview: Kinetic Theory of Gases and Radiation

V ∝ \[\frac {1}{P}\] (at constant T)

V ∝ T (at constant P)

P ∝ T (at constant V)

A gas that obeys the equation PV = nRT at all pressures and temperatures is called an ideal gas.

The equation relating pressure, volume, temperature and other state variables of a gas is called the equation of state.

A real gas is a gas whose molecules interact with each other and therefore does not obey the ideal gas equation under all conditions.

A real gas behaves like an ideal gas at low pressure and high temperature when intermolecular forces become negligible.

The mean free path is the average distance travelled by a gas molecule between two successive collisions.

\[\lambda=\frac{1}{\sqrt{2}\pi d^2\left(\frac{N}{V}\right)}\]

Where:

• λ = mean free path
• d = diameter of molecule
• N/V = number density of molecules

P =\[\frac{1}{3}\frac{N}{V}m\overline{\mathrm{v}^2}\]

V_s = \[\sqrt{\frac{\gamma RT}{M_{0}}}\]

where,
γ = \[\frac {C_p}{C_v}\]

\[\frac{1}{2}m\overline{v^2}=\frac{3}{2}k_BT\]

Degrees of freedom of a system are defined as the total number of coordinates or independent quantities required to describe the position and configuration of the system completely.

• A diatomic molecule has 3 translational and 2 rotational degrees of freedom.
• Each degree of freedom contributes \[\frac {1}{2}\]k_BTenergy (law of equipartition).
• At high temperatures, vibrational motion also adds extra energy to the molecule.

C_p - C_v = R

• Monatomic gas: Has 3 translational degrees of freedom, so
  C_v = \[\frac {3}{2}\]R, C_p = \[\frac {5}{2}\]R, and γ = \[\frac {5}{3}\]​.
• Diatomic gas (rigid): Has 3 translational + 2 rotational degrees of freedom, so
  C_v = \[\frac {5}{2}\]R, C_p = \[\frac {7}{2}\]R, and γ = \[\frac {7}{5}\]​.
• Polyatomic gas: Has translational, rotational, and vibrational degrees of freedom; more degrees of freedom → higher internal energy and specific heat.

Radiation is the mode of transfer of heat in the form of electromagnetic waves without requiring a material medium.

Thermal radiation is the electromagnetic radiation emitted by a body due to its temperature.

The ratio of amount of heat absorbed to total quantity of heat incident is called the coefficient of absorption.

The ratio of the amount of radiant energy reflected to the total energy incident is called the coefficient of reflection.

The ratio of amount of radiant energy transmitted to total energy incident is called the coefficient of transmission.

A body, which absorbs the entire radiant energy incident on it, is called an ideal or perfect blackbody. 

A blackbody is a body that absorbs all incident radiation of all wavelengths and does not reflect or transmit any radiation.

(Absorptivity a = 1, reflectivity r = 0, transmissivity t = 0)

Ferry’s blackbody is a practical model of a perfect blackbody consisting of a hollow cavity with a small aperture that absorbs almost all incident radiation.

Radiation emitted by a heated cavity is called cavity radiation and depends only on the temperature of the cavity walls.

• All bodies above 0 K emit and absorb heat radiation continuously.
• If a body emits more than it absorbs, its temperature falls; if it absorbs more, its temperature rises.
• Heat radiated depends on temperature, surface area, surface nature, and time.
• Emissive power is heat radiated per unit area per unit time.
• Emissivity (e) compares a body’s radiation to a blackbody; 0 < e ≤ 1.

It states that at a given temperature, the ratio of emissive power to coefficient of absorption of a body is equal to the emissive power of a perfect blackbody at the same temperature for all wavelengths.

• Blackbody radiation has many wavelengths, and its pattern depends only on temperature.
• For a given temperature, intensity rises, reaches a maximum (λₘₐₓ), then decreases.
• As the temperature increases, the peak shifts to shorter wavelengths.
• Higher temperature means more total energy is emitted.
• Classical theory failed to explain this, but Planck’s theory did.

It is observed that the wavelength, for which the emissive power of a blackbody is maximum, is inversely proportional to the absolute temperature of the blackbody. This is Wien's displacement law.

λ_max T = b
b = Wien's constant

According to this law, "The rate of emission of radiant energy per unit area or the power radiated per unit area of a perfect blackbody is directly proportional to the fourth power of its absolute temperature".

R = eσT⁴