L8 - Entropy(3) - PH2103 Thermal Physics Lecture 8...

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PH2103 Thermal Physics Lecture 8: Entropy & Microscopic view of Temperature Massimo Pia Ciamarra [email protected] SPMS-PAP-03-14 Textbook paragraph: 2.6; 3.1; 3.2 Discussion Forum
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1. Second law & ideal gas Multiplicity of a monoatomic Gas Interacting gas 2. Entropy Definition and properties Entropy of the ideal gas; of mixing Relation between entropy and temperature Entropy and heat Agenda
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1. Second law & ideal gas Multiplicity of a monoatomic ideal gas
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One particle - positions To specify the microstate of a single particle in a volume V, and with energy U, we need to specify its positions and its momentum. If V = L 3 , then 0 < x < L, 0 < y < L, and 0 < z < L. If we double L, the number of possible positions of the particle in each dimensions also doubles. This implies that the position phase space volume is proportional to the volume: Ω 1 V
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One particle - momentum What are the possible value of the momentum p x , p y , p z ? We can answer this question considering that the momentum is related to the energy. In fact: U = 1 2 m v 2 = 1 2 m ( p m ) 2 = 1 2 m ( p x 2 + p y 2 + p z 2 ) This is the equation of a sphere of radius The momentum volume V p is the surface of this sphere. ( p x 2 + p y 2 + p z 2 )= 2 m U = R 2 R = 2 mU Ω 1 V p
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Plank's constant, h Quantum mechanics give us a fundamental quantity that has the dimension we are looking for, Plank's constant h . Remember that we have related the energy of an oscillator to its frequency f using Plank's constant. [ energy ]= h [ frequency ]→ h =[ time x energy ] We have thus have: Ω 1 V V p h 3
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Heisenberg uncertainty principle A more intuitive explanations of why the multiplicity is the volume of the phase space expressed in units of h 3 , is provided by Heisenberg's uncertainty principle. This principle relates the uncertainties in the measure of conjugate variables, such as (x,p x ) or (y,p y ). For conjugate variables, one finds: And so x )(Δ p x )≈ h x )(Δ y )(Δ z )(Δ p x )(Δ p y )(Δ p z )≈ h 3 This indicates that the phase space volume of a particle should be measured in units of h 3 . It does not make sense to use a smaller unit, as we cannot known position and momentum with a greater precision.
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Two particles (indistinguishable) The result Ω 2 V 2 V p h 6 is correct as long as the two microstates Part 1: (x 1 ,y 1 ,x 1 ,p x1 ,p y1 ,p z1 ) Part. 1: (x 2 ,y 2 ,x 2 ,p x2 ,p y2 ,p z2 ) Part 2: (x 2 ,y 2 ,x 2 ,p x2 ,p y2 ,p z2 ) Part. 2: (x 1 ,y 1 ,x 1 ,p x1 ,p y1 ,p z1 ) are distinguishable. and This is not true in quantum mechanics (it is so in classical mechanics) To avoid overcounting, we thus define Ω 2 V 2 V p 2 h 6
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N particles If we have N particles in a volume V, the volume of phase space will be Where the hypershere has radius In d dimensions, the area of an hypersheres is So that we find Ω N V N N ! h 3 N x ( area of momentum hypersphere ) R = 2 mU 2 π d / 2 ( d / 2 1 ) !
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