L7 - Second-Law_IdealGas(3) - PH2103 Thermal Physics...

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PH2103 Thermal Physics Lecture 7: Second law, Ideal Gas & Entropy Massimo Pia Ciamarra [email protected] SPMS-PAP-03-14 Textbook paragraph: 2.5, 2.6 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 Agenda
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1. Second law & ideal gas Multiplicity of a monoatomic ideal gas
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Multiplicity, The multiplicity is the number of microstates corresponding to a given macrostate. In a system of N coins, the macrostate is fixed by the number n of coins showing the head. The multiplicity is the number of way one can choose n coins out of N to be in state h . We have derived: In a system of N quantum oscillators, the total energy is E=qhf, where q is the total number of energy quanta. We have defined the multiplicty of a state of energy E as the number of way we can assign the q quanta to the oscillators. We have derived: Ω( n )= ( N n ) = N ! n! ( N n ) ! Ω( N ,q )= ( q + N 1 q ) = ( q + N 1 ) ! q! ( N 1 ) !
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Phase space of the ideal gas The microstate of an ideal gas is fixed specifying the position and the momentum of all particles. Position and momentum are continuous variables, not discrete. Classically, we say that the position and the momentum of each particle can acquire an infinite number of value. The only constraints we have is that the particle should be in the volume we consider, and that the total energy is constant. Previous example: our variables (coins, spin) can only acquire discrete values. Coin: head or tail Oscillator: 1xhf, 2xhf, 3xhf, … , 4xhf Ideal gas: Position and momentum can acquire continuous values. The novelty
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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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One particle – phase space Combining the dependence of the multiplicity of a single particle of an ideal gas on the volume and on the momemtum, we find: But the multiplicity is defined as the number of ways in which we can do something. It is a number, and so it must have the dimension of a number, i.e. it should be unitless.
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