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Introduction to Superconductors

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test

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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

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


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

An ideal gas is defined as one in which all collisions between atoms or molecules are 
perfectly eleastic 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