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Astronomical Distance Units .

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Celestial Coordinates .

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Celestial Navigation .

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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number .

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Balancing Chemical Equations

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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle

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Center of Mass Frame

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Comparison Between Gravitation and Electrostatics

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Compton Effect .

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Cyclotron Resonance

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Double Slit Experiment

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Elastic and Inelastic Collisions .

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Electric Fields

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Gauss's Law of Universal Gravity .

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Gravity - Force and Acceleration

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Ideal and Non-Ideal Gas Laws (van der Waal)

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Kinetic Theory of Gases .

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General Relativity

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Gravitational Waves

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Quantum Gravity

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The Area Metric

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Group Theory

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Basic Group Theory .

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Basic Representation Theory .

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Building Groups From Other Groups .

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Symmetric Groups .

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The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .

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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .

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Quaternions 1 .

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Quaternions 2 .

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Stirling's Approximation

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Volume Integrals

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Factor Analysis

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Quantum Computing

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Quantum Field Theory

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Path Integral Formulation

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Relativistic Quantum Field Theory

Quantum Mechanics

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Clebsch-Gordan Coefficients .

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Electron Orbital Angular Momentum and Spin

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Heisenberg Uncertainty Principle

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Ladder Operators .

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Multi Electron Wavefunctions .

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Pauli Spin Matrices

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Photoelectric Effect .

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Position and Momentum States .

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Probability Current

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Schrodinger Equation for Hydrogen Atom .

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Spin 1/2 Eigenvectors

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The Differential Operator

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The Essential Mathematics of Quantum Mechanics

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Time Evolution and Symmetry Operations

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Solid State Electronics

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Band Theory of Solids .

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Fermi-Dirac Statistics .

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Intrinsic and Extrinsic Semiconductors .

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The P-N Junction

Special Relativity

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Electromagnetic (Faraday) Tensor .

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Energy and Momentum in Special Relativity, E = mc² .

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Invariance of the Velocity of Light .

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Lorentz Invariance .

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Lorentz Transform .

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Lorentz Transformation of the EM Field .

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Newton versus Einstein

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Spinors - Part 1 .

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Spinors - Part 2 .

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The Continuity Equation .

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The Lorentz Group .

Statistical Mechanics

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Entropy and the Partition Function

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The Harmonic Oscillator

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String Theory

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Bosonic Strings

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Extra Dimensions

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Introduction to String Theory

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Kaluza-Klein Compactification of Closed Strings

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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory

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Introduction to Superconductivity .

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Superconductivity (Lectures 1 - 10)

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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields

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Generators of a Supergroup

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Grassmann Numbers

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Introduction to Supersymmetry

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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)

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Gauge Theories (Yang-Mills)

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1 .

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3 .

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: November 20, 2021 ✓


Kinetic Theory of Gases
-----------------------

An ideal gas is defined as one in which all 
collisions between atoms or molecules are 
perfectly elastic and in which there are no 
intermolecular attractive forces. One can 
visualize it as a collection of perfectly 
hard spheres which collide but which otherwise 
do not interact with each other. In such a gas, 
all the internal energy is in the form of 
kinetic energy and any change in internal 
energy is accompanied by a change in 
temperature.



Consider a molecule striking the wall of a box.  
Assume the collision with the wall is perfectly 
elastic so that the change in the x momentum is 
given by:

mv_x - (-mv_x) = 2mv_x

Now F_x = ma_x = 2mv_x/Δt

Now, if we assume the molecule makes a 
'round trip' we get: 

Δt = 2L/v_x

So,

F_x = 2mv_x/(2L/v_x) = mv_x²/L

The force due to N molecules is Nmv_x²/L

Now v² = v_x² + v_y² + v_z² = 3v_x²

∴ v_x² = v²/3

P = F/A =  Nmv²/3LA = Nmv²/3V

∴ PV = Nmv²/3 = nRT ... the Ideal Gas Law

Now,

n = N/N_A where N_A is Avogadro's Number

∴ Nmv²/3 = NRT/N_A

mv²/3 = RT/N_A

mv² = 3k_BT where k_B =  R/N_A = BOLTZMANN'S CONSTANT

∴ (1/2)mv² = 3k_BT/2 

The total INTERNAL ENERGY of the gas is thus:

U =  3Nk_BT/2 = 3nRT/2