Revision: Kinetic Theory of Gases and Radiation Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

The average distance travelled by the molecule between collisions is called mean free path (λ).

λ = `"kT"/(sqrt(2)π"d"^2"p")`

The specific heat capacity of a substance is the amount of heat energy required to raise the temperature of unit mass of that substance through 1°C (or 1 K).

OR

Heat capacity of a body when expressed for the unit mass is called the specific heat capacity of the substance of that body.

OR

The amount of heat energy required to raise the temperature of a unit mass of an object by 1 °C is called the specific heat of that object.

The heat capacity of a body is the quantity of heat required to raise its temperature by 1°C. It depends upon the mass and the nature of the body.

Diathermanous substances that allow transmission of infrared radiation through them are called diathermanous substances.

For example - rock salt, pure air, glass, etc.

Athermanous substances that don't allow transmission of infrared radiation through them are called athermanous substances.

For example - wood, metal, CO₂, water, benzene, etc.

It is the molar gas constant (R) to Avogadro constant (Avogadro number) ratio N_A.

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.

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

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

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 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.

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

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

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

Specific heat capacity c = \[\frac{\text{Heat capacity of body } C'}{\text{Mass of the body } m}\]

or

Specific heat capacity c = \[\frac{Q}{m\times\Delta t}\]

P ∝ T (at constant V)

V ∝ T (at constant P)

V ∝ \[\frac {1}{P}\] (at constant T)

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

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 = \[\frac {C_p}{C_v}\]

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

C_p - C_v = R

An ideal or perfect gas is a gas which obeys the gas laws (Boyle’s law, Charles’ law, and Gay-Lussac’s law) at all pressures and temperatures. An ideal gas cannot be liquefied by the application of pressure or by lowering the temperature.

Kirchhoff’s law of thermal radiation deals with wavelength specific radiative emission and absorption by a body in thermal equilibrium. 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.

Since we can describe the emissive power of an ordinary body in comparison to a perfect blackbody through its emissivity, Kirchhoff’s law can also be stated as follows: for a body emitting and absorbing thermal radiation in thermal equilibrium, the emissivity is equal to its absorptivity.

Symbolically, a = e or more specifically a(λ) = e(λ).

Thus, if a body has high emissive power, it also has high absorptive power and if a body has low emissive power, it also has low absorptive power.

Kirchhoff’s law can be theoretically proved by the following thought experiment. Consider an ordinary body A and a perfect blackbody B of identical geometric shapes placed in an enclosure. In thermal equilibrium, both bodies will be at same temperature as that of the enclosure.

Let R be the emissive power of body A, R_B be the emissive power of blackbody B and a be the coefficient of absorption of body A. If Q is the quantity of radiant heat incident on each body in unit time and Q_a is the quantity of radiant heat absorbed by the body A, then Q_a = a Q. As the temperatures of the body A and blackbody B remain the same, both must emit the same amount as they absorb in unit time. Since emissive power is the quantity of heat radiated from unit area in unit time, we can write

Quantity of radiant heat absorbed by body A = Quantity of heat emitted by body A

or a Q = R    ...(1)

For the perfect blackbody B,

Q = R_B   ...(2)

From Eqs. (1) and (2), we get,

a = `R/Q = R/R_b`   ...(3)

From Eq. (3), we get, `R/a = R_b`

By definition of coefficient of emission,

`R/R_b`   ...(4)

From Eqs. (3) and (4), we get, a = e.

Hence, the proof of Kirchhoff‘s law of radiation.

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.

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

• Heat energy absorbed (Q) depends on: mass (m), rise in temperature (Δt), and specific heat capacity (c), i.e., Q ∝ m × Δt × c.
• Heat capacity (C') and specific heat capacity (c) are related by: C′ = m × c.

• 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.

• 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.

• 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.

• 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.