L 6,7 - "Boltzmann Factor and Partition Function"...

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“Boltzmann Factor and Partition Function” Prof. GianluigiVeglia
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Statistical Thermodynamics provides the link between the microscopic properties of matter and its bulk (macroscopic) properties. It answers one of the most important question in Chemistry: how are the molecules distributed over the energy states at a given temperature? Two major concepts: The Boltzmann Factor and the Partition Function
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The system is defined as an ensemble or macroscopic systems in thermal equilibrium with a heat reservoir. The number of the system in a state j with an energy associated E j (N,V) is a j and the total number of the systems is A . E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3
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The simplest case we consider is a monoatomic gas with only translational energy. For this system, we can define the total energy, E j (N,V), simply by adding the energy of each individual molecular energies: N j V N E ε + + + + = .... ) , ( 3 2 1 Where N is the total number of molecules and ε j the individual energies of the molecules
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For monoatomic gases, if we neglect the electronic contribution, the only energy term left is the translational energy. As we saw in the previous lectures, the translational energy is quantized (i.e. it occupies discrete levels defined by the translational quantum number n ): ( ) 2 2 2 2 2 , , 8 z y x n n n n n n ma h z y x + + = ε Note that in this expression a=V 1/3 (dimensions of the vessel)
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It is possible to demonstrate (see textbook) that the probability that a system in an ensemble in the state j and energy E j (N,V) is proportional to the Boltzmann factor. For two states (1,2), it is possible to describe the energy function, f (E), in the form ( ) 2 1 1 2 E E e a a = β Where a 1 and a 2 is the number of the systems in states 1 and 2
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In general, we can write: ( ) n m E E m n e a a = β Each of the number of the state of the system is given by j E j Ce a = To determine the constat, C, we sum over the total number of states j = j E j j j e C a Where the sum over all of the states is A
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Therefore, substituting the value of the constant we obtain = j E E j j j e e A a β Fraction of the system in our ensemble in the state j. For large A this ratio becomes the probability of the system ( p j ) that a randomly chosen system is in the state j:
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If we define Q as the denominator of this expression: And we set we obtain: ( ) ( ) = j N V E j e V N Q , , , β T k B 1 = k B is the Boltzmann constant and Q(N,V,T) or Q(N,V, β ) is the partition function and it can express the macroscopic properties of the systems.
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The expression allows one to calculate the average energy of a system in an ensemble of systems. One of the properties of the probability is to define the average properties of the system: ( ) ( ) ( ) T V N Q e T V N p T k V N E j B j , , , , / , = = = N i j i p x x 1 Where <x> is the average of a series of numbers x i associated with an outcome j.
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Using this criterion, it is possible to determine the average energy <E> of the system: Substituting the probability:
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L 6,7 - &quot;Boltzmann Factor and Partition Function&quot;...

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