Chapter 3 Classical Statistics of Maxwell-Boltzmann

1- Boltzmann Statistics - Goal: Find the occupation number of each energy level (i.e. find (N1,N2,…,Nn)) 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 N1 particles from a total of N to be placed in the first level is

1- Boltzmann Statistics We ask: In how many ways can these N1 particles be arranged in the first level such that in this level there are g1 quantum states? For each particle there are g1 choices. That is, there are possibilities in all. Thus the number of ways to put N1 particles into a level containing g1 quantum states is

1- Boltzmann Statistics For the second energy level, the situation is the same, except that there are only (N-N1) particles remaining to deal with: Continuing the process, we obtain the Boltzmann distribution ωB as:

1- Boltzmann Statistics

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

2- The Boltzmann Distribution Applying Stirling’s law:

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

2- The Boltzmann Distribution We will prove later that

2- The Boltzmann Distribution and hence where fi is the probability of occupation of a single state belonging to the ith energy level. 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 Zsp:

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, εs = 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 (ε1=0, ε2=ε, and ε3=2ε) with g1=2, and g2=g3=1. Calculate Zsp by using Eqs.(6) and (7)

2- The Boltzmann Distribution Macrostate Nb. microstates (N1,N2,N3) ωB εs (1,0,0) 2 (0,1,0) 1 ε (0,0,1) 2ε

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:

3- Dilute gases and the Maxwell-Boltzmann Distribution The word “dilute” means Ni << gi, 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:

3- Dilute gases and the Maxwell-Boltzmann Distribution The Maxwell-Boltzmann distribution corresponding to ωMB,max: ωMB and ωB 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:

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.

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.

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

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

4- Thermodynamic Properties from the Partition Function

4- Thermodynamic Properties from the Partition Function - Calculation of β

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

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:

5- Partition Function for a Gas γs = 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.

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

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

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