Essential Principles of Thermodynamics and Quantum Physics

Mean Free Path in Kinetic Theory

Mean free path is the average distance traveled by a gas molecule between two successive collisions with other molecules.

• It is denoted by the Greek letter λ (lambda).
• It depends on the size of the molecules, pressure, and temperature.
• A higher pressure decreases the mean free path, while a higher temperature generally increases it.
• Formula:
  [ λ = \frac{1}{\sqrt{2}\pi d^2 n} ]
  where d represents the diameter of a molecule and n represents the number of molecules per unit volume.

Gibbs Free Energy and Thermodynamic Stability

Gibbs free energy (G) is the thermodynamic function that indicates the maximum useful work obtainable from a system at constant temperature and pressure.

• It is defined by the equation:
  [ G = H - TS ]
  where H is enthalpy, T is absolute temperature, and S is entropy.
• If ΔG < 0, the process is spontaneous.
• If ΔG = 0, the system is in equilibrium.
• If ΔG > 0, the process is non-spontaneous.

Electron Gas Model in Metallic Solids

An electron gas is a collection of free electrons moving randomly inside a metal.

• Electrons behave like gas molecules confined within the boundaries of the metal.
• The positive metal ions remain fixed in a lattice, while free electrons move throughout the material.
• The electron gas model explains electrical conductivity, thermal conductivity, and other fundamental properties of metals.
• It forms the basis of the free-electron theory of metals.

Planck's Law of Black Body Radiation

Planck's law describes the distribution of electromagnetic radiation emitted by a perfect black body at a given temperature.

• It explains how the intensity of radiation varies with wavelength and temperature.
• It introduced the revolutionary concept that energy is emitted in discrete packets called quanta.
• The energy of each quantum is calculated as:
  [ E = hν ]
  where h is Planck's constant and ν (nu) is the frequency of radiation.
• Planck's law successfully explains black-body radiation and laid the foundation for quantum mechanics.

The following sections provide short notes on additional thermodynamic topics for further study.

Viscosity and Fluid Transport Phenomena

Viscosity is the property of a fluid that opposes the relative motion between its layers. It is caused by internal friction between molecules. When one layer of a fluid moves faster than another, a resisting force acts between them. According to Newton's law of viscosity:

[ F = η A \frac{dv}{dx} ]

In this formula:

• F = viscous force
• η (eta) = coefficient of viscosity
• A = area of the layer
• dv/dx = velocity gradient

The SI unit of viscosity is the Pascal-second (Pa·s). Viscosity typically decreases with increasing temperature in liquids and increases in gases.

The Unattainability of Absolute Zero

The unattainability principle states that absolute zero (0 K or -273.15°C) cannot be reached by any finite number of physical processes. As the temperature approaches 0 K, the entropy change becomes extremely small, making further cooling increasingly difficult. This is a core statement of the Third Law of Thermodynamics. While scientists can cool substances very close to absolute zero, reaching it exactly remains physically impossible.

Spectral Energy Distribution of Black Bodies

A black body is an ideal body that absorbs all incident radiation and emits the maximum possible radiation at every temperature. The spectral distribution of energy describes how the emitted energy is distributed among different wavelengths. As the temperature increases:

• The total emitted energy increases significantly.
• The wavelength corresponding to maximum intensity shifts toward shorter wavelengths, a phenomenon known as Wien's displacement law.

This distribution is accurately explained by Planck's radiation law, which served as the foundation of quantum theory.

The Second Law of Thermodynamics and Entropy

Statement: Heat cannot flow spontaneously from a colder body to a hotter body, and no heat engine can convert all absorbed heat into useful work.

Explanation: The second law introduces the concept of entropy, which always increases in an isolated system. It demonstrates that natural processes are irreversible and that some energy is always lost as waste heat during energy conversion. Consequently, no heat engine can achieve 100% efficiency.