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

The distance travelled by a gas molecule between two successive collisions, during which it moves in a straight line with constant velocity, is called free path.

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

λ = `"kT"/(sqrt(2)π"d"^2"p")`

The average distance traversed by a molecule between two successive collisions, obtained by dividing the total distance travelled during nn collisions by the number of collisions nn, is called mean free path (λ).

The minimum number of independent coordinates needed to specify the position and configuration of a thermo-dynamical system in space is called the degree of freedom of the system.

The total number of coordinates or independent quantities required to describe the position and configuration of a system completely is called degrees of freedom (dof).

The state in which two objects are at the same temperature and there is no net flow of heat between them is called thermal equilibrium.

The thermodynamic state variables that do not depend on the size of the system (e.g., pressure, temperature) are called intensive variables.

The specific values of macroscopic variables that completely describe every equilibrium state of a thermodynamic system are called thermodynamic state variables.

The thermodynamic state variables that depend on the size of the system (e.g., internal energy, volume) are called extensive variables.

A device that transforms heat partly into work or mechanical energy (where T_H > T_C​, Q_H > 0, Q_C < 0) is called a heat engine.

Heat engine is a device which takes heat as input and converts this heat into work by undergoing a cyclic process.

"Temperature is a physical quantity that defines the thermodynamic state of a system."

"Heat is energy in transit. When two bodies at different temperatures are brought in contact, they exchange heat."

One mole of any substance is the amount of that substance which contains the Avogadro number (NA) of particles (such as atoms or molecules).

The heating-up of the earth’s atmosphere due to trapped infrared rays reflected from the earth's surface by atmospheric gases is called the greenhouse effect.

The law which states that for any system in thermal equilibrium, the total energy is equally distributed among all its degrees of freedom, with energy \[\frac {1}{2}\]kT associated with each degree of freedom per molecule, is called the Law of Equipartition of Energy.

Let λ₁, λ₂, λ₃,…λ_n​ be the distances travelled by a gas molecule during nn collisions respectively, then the mean free path is:

f = 3A − B

where:

• A = number of atoms in the molecule
• B = number of bonds between atoms

By substituting equation W = −p_ex . ΔV in the equation ΔU = q + W, we get

ΔU = q − p_ex . ΔV  ...(1)

If the reaction is carried out in a closed container so that the volume of the system is constant, then Δ = 0. In such a case, no work is involved.

The equation (1) becomes ΔU = q_v

Equation (1) suggests that the change in internal energy of the system is due to heat transfer. The subscript v indicates a constant volume process. As U is a state function, qv is also a state function. We see that an increase in the internal energy of a system is numerically equal to the heat absorbed by the system in a constant volume (isochoric) process.

\[E_k=\frac{3}{2}k_BT\]

Where:

• E_k = Average kinetic energy of the molecules (in joules)
• k_B = Boltzmann constant = 1.380649 × 10⁻²³ J/K
• T = Absolute temperature (in kelvin)

Q = mcΔT

Where:

• Q = Heat absorbed or released (in joules)
• m = Mass of the substance (in kg)
• c = Specific heat capacity (J/kg·K)
• ΔT = Change in temperature (T_final−T_initial)

By substituting equation W = −p_ex . ΔV in the equation ΔU = q + W, we get

ΔU = q − p_ex . ΔV  ...(1)

If the reaction is carried out in a closed container so that the volume of the system is constant, then Δ = 0. In such a case, no work is involved.

The equation (1) becomes ΔU = q_v

Equation (1) suggests that the change in internal energy of the system is due to heat transfer. The subscript v indicates a constant volume process. As U is a state function, qv is also a state function. We see that an increase in the internal energy of a system is numerically equal to the heat absorbed by the system in a constant volume (isochoric) process.

Statement:
The net heat energy supplied to a system is equal to the sum of the change in internal energy of the system and the work done by the system. It is based on the law of conservation of energy.

Formula:

where Q = heat added, ΔU = change in internal energy, W = work done by the system.

The wavelength (λ_m​) for which the emissive power of a blackbody is maximum is inversely proportional to the absolute temperature of the blackbody:

With increase in temperature, λ_m​ decreases (shifts towards shorter wavelengths). Also, the energy E_max​ emitted at λ_m​ increases with the fifth power of temperature, i.e., E_max ∝ T⁵.

Statement:
For a gas in thermal equilibrium at temperature TT, the average energy associated with each molecule for each quadratic term (degree of freedom) is:

where k_B = 1.38 × 10⁻²³ J/K and T = absolute temperature.

Energy Expressions for Different Types of Motion:

1. Translational K.E.:
   ​\[\frac{1}{2}mv_x^2+\frac{1}{2}mv_y^2+\frac{1}{2}mv_z^2\] (3 degrees of freedom — along x, y, z axes)

2. Rotational K.E.:
   \[\frac{1}{2}I\omega_x^2+\frac{1}{2}I\omega_y^2+\frac{1}{2}I\omega_z^2\] (up to 3 degrees of freedom — rotation about x, y, z axes)

3. Vibrational K.E.:
   \[\frac{1}{2}m\dot{u}^2+\frac{1}{2}kr^2\] (2 terms — kinetic and potential energy of vibration)

Each quadratic term contributes \[\frac {1}{2}\]k_BT to the total average energy of the molecule.

First Law: Energy of system + surroundings remains constant → ΔU = q + W

ΔU: change in internal energy, q: heat, W: work done on system

Sign convention:

• Work by system (−)
• on system (+)
• Heat absorbed (+)
• released (−)

ΔU > 0: energy enters system; ΔU < 0: energy leaves system

• Isothermal: ΔU = 0 → q = −W
• Adiabatic: q = 0 → ΔU = W
• Isochoric: W = 0 → ΔU = q
• Isobaric: ΔU = q + W

• The greenhouse effect is a naturally occurring phenomenon that heats Earth's surface. Without it, Earth's temperature would be -18°C instead of 15°C.
• Greenhouse gases are transparent to solar radiation but retain and reflect back long-wave heat radiation. Main gases — CO₂ (60%), CH₄ (20%), CFCs (14%), N₂O (6%).
• Earth's surface re-emits heat as infrared radiation. Greenhouse gases like CO₂ and CH₄ absorb this and return heat to Earth's surface — causing the greenhouse effect.
• Rising CO₂ due to the burning of fossil fuels and deforestation intensifies the greenhouse effect, causing global warming.
• Global warming leads to melting of polar ice, rising sea levels, changes in rainfall patterns and loss of biodiversity.