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Chapter 3 Classical Statistics of Maxwell-Boltzmann

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1- Boltzmann Statistics - Goal: Find the occupation number of each energy level (i.e. find (N 1,N 2,…,N n )) when the thermodynamic probability is a maximum. - Constraints: N and U are fixed - Consider the first energy level, i=1. The number of ways of selecting N 1 particles from a total of N to be placed in the first level is

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1- Boltzmann Statistics We ask: In how many ways can these N 1 particles be arranged in the first level such that in this level there are g 1 quantum states? For each particle there are g 1 choices. That is, there are possibilities in all. Thus the number of ways to put N 1 particles into a level containing g 1 quantum states is

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1- Boltzmann Statistics For the second energy level, the situation is the same, except that there are only (N-N 1 ) particles remaining to deal with: Continuing the process, we obtain the Boltzmann distribution ω B as:

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1- Boltzmann Statistics

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2- The Boltzmann Distribution Now our task is to maximize ω B of Equation (3) At maximum,. Hence we can choose to maximize ln(ω B ) instead of ω B itself, this turns the products into sums in Equation (3). Since the logarithm is a monotonic function of its argument, the maxima of ω B and ln(ω B ) occur at the same point. From Equation (3), we have:

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2- The Boltzmann Distribution Applying Stirling’s law:

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2- The Boltzmann Distribution Now, we introduce the constraints Introducing Lagrange multipliers (see chapter 2 in Classical Mechanics (2)):

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2- The Boltzmann Distribution We will prove later that

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2- The Boltzmann Distribution and hence The sum in the denominator is called the partition function for a single particle (N=1) or sum-over-states, and is represented by the symbol Z sp : where f i is the probability of occupation of a single state belonging to the ith energy level.

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2- The Boltzmann Distribution or Ω = total number of microstates of the system, s = the index of the state (microstate) that the system can occupy, ε s = the total energy of the system when it is in microstate s. Example: A system possesses two identical and distinguishable particles (N=2), and three energy levels (ε 1 =0, ε 2 =ε, and ε 3 =2ε) with g 1 =2, and g 2 =g 3 =1. Calculate Z sp by using Eqs.(6) and (7)

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2- The Boltzmann Distribution MacrostateNb. microstates (N 1,N 2,N 3 )ωBωB εsεs (1,0,0)20 (0,1,0)1ε (0,0,1)12ε2ε

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2- The Boltzmann Distribution If the energy levels are crowded together very closely, as they are for a gaseous system: where g(ε)dε: number of states in the energy range from ε to ε+dε, N(ε)dε: number of particles in the range ε to ε+dε. We then obtain the continuous distribution function:

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3- Dilute gases and the Maxwell- Boltzmann Distribution The word “dilute” means N i << g i, for all i. This condition holds for real gases except at very low temperatures. The Maxwell-Boltzmann statistics can be written in this case as:

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3- Dilute gases and the Maxwell- Boltzmann Distribution The Maxwell-Boltzmann distribution corresponding to ω MB,max : - ω MB and ω B differ only by a constant – the factor N!, - Since maximizing ω involves taking derivatives and the derivative of a constant is zero, so we get precisely the Boltzmann distribution:

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3- Dilute gases and the Maxwell- Boltzmann Distribution - Boltzmann statistics assumes distinguishable (localizable) particles and therefore has limited application, largely solids and some liquids. - Maxwell-Boltzmann statistics is a very useful approximation for the special case of a dilute gas, which is a good model for a real gas under most conditions.

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4- Thermodynamic Properties from the Partition Function In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system.

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4- Thermodynamic Properties from the Partition Function - Calculation of average energy per particle: - Calculation of internal energy of the system:

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4- Thermodynamic Properties from the Partition Function - Calculation of Entropy for Maxwell-Boltzmann statistics: with

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4- Thermodynamic Properties from the Partition Function

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- Calculation of β

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4- Thermodynamic Properties from the Partition Function - Calculation of the Helmholtz free energy: - Calculation of the pressure:

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5- Partition Function for a Gas The definition of the partition function is For a sample of gas in a container of macroscopic size, the energy levels are very closely spaced. Consequences: - The energy levels can be regarded as a continuum. - We can use the result for the density of states derived in Chapter 2:

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5- Partition Function for a Gas γ s = 1 since the gas is composed of molecules rather than spin 1/2 particles. Thus Then The integral can be found in tables and is Partition function depends on both the volume V and the temperature T.

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6- Properties of a Monatomic Ideal Gas - Calculation of pressure: where N A is the Avogadro’s number and n the number of moles.

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6- Properties of a Monatomic Ideal Gas - Calculation of internal energy:

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6- Properties of a Monatomic Ideal Gas - Calculation of heat capacity at constant volume: C V is constant and independent of temperature in an ideal gas. - Calculation of entropy

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