Ideal Bose & Fermi Gases - 1 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Gases
Here we consider systems with the kinetic energy exceeding the potential energy of inter-particle interactions: 

Ideal Gases

We start from neglecting the potential energy of interaction completely. Note though that quantum effect does not allow one to consider particles completely independent. The absence of any interaction allows one to treat any molecule as subsystem and apply to it the Gibbs canonical distribution: the average number of molecules in a given state is

which is called Boltzmann distribution. One can also use grand canonical ensemble considering all molecules in the same state as a subsystem with a non-fixed number of particles. Using the distribution (35) with N = na and  one expresses the probability of occupation numbers via the chemical potential: 

Consider now a dilute gas, when all . Then the probability of no particles in the given state is close to unity,   and the probability of having one particle and the average number of particles are given by the Boltzmann distribution in the form

  (39)

Boltzmann (classical) gas is such that one can also neglect quantum exchange interaction of particles (atoms or molecules) in the same state which requires the occupation numbers of any quantum state to be small, which in turn requires the number of states V p³/h³ to be much larger than the number of molecules N . Since the typical momentum is   we get the condition

 (40)

To get the feeling of the order of magnitudes, one can make an estimate with m = 1.6. 10^-24g (proton) and n = 10²¹cm^-3 which gives   Another way to interpret (40) is to say that the mean distance between molecules n^-1/3 must be much larger than the wavelength h/p. In this case, one can pass from the distribution over the quantum states to the distribution in the phase space:

(41)
In particular, the distribution over momenta is always quasi-classical for the Boltzmann gas. Indeed, the distance between energy levels is determined by the size of the box,

 which is much less than temperature according to (40). To put it simply, if the thermal quantum wavelength   is less than the distance between particles it is also less than the size of the box. We conclude that the Boltzmann gas has the Maxwell distribution over momenta. If such is the case even in the external field then 

 That gives, in particular, the particle density in space   where n₀ is the concentration without field. In the uniform gravity field we get the barometric formula   

Partition function of the Boltzmann gas can be obtained from the partition function of a single particle (like we did for two-level system and oscillator) with the only difference that particles are now real and indistinguishable so that we must divide the sum by the number of transmutations:

Using the Stirling formula ln   we write the free energy

   (42)
Since the motion of the particle as a whole is always quasi-classical for the Boltzmann gas, one can single out the kinetic energy: 

 If in addition there is no external field (so that  describes rotation and the internal degrees of freedom of the particle) then one can integrate over  and get for the ideal gas

To complete the computation we need to specify the internal structure of the particle. Note though that  depends only on temperature so that we can already get the equation of state 

Mono-atomic gas. At the temperatures much less than the distance to the first excited state all the atoms will be in the ground state (we put ε₀ = 0).
That means that the energies are much less than Rydberg   and the temperatures are less than   (otherwise atoms are ionized).
If there is neither orbital angular momentum nor spin (L = S = 0 — such are the atoms of noble gases) we get   as the ground state is non-degenerate and

Here  is called the chemical constant. Note that for F = AT + B T ln T the energy is linear  that is the specific heat, C_v = B , is independent of temperature. The formulas thus derived allow one to derive the conditions for the Boltzmann statistics to be applicable which requires . Evidently, it is enough to require exp where

Using such 

If there is a nonzero spin the level has a degeneracy 2S + 1 which adds

  to the chemical constant (45). If both L and S are nonzero then the total angular momentum J determines the fine structure of levels ε_J (generally comparable with the room temperature — typically,  Every such level has a degeneracy 2J + 1 so that the respective partition function

Without actually specifying ε_J we can determine this sum in two limits of large and small temperature. and z = (2S + 1)(2L + 1) which is the total number of components of the fine level structure. In this case 

In the opposite limit of temperature smaller than all the fine structure level differences, only the ground state with ε_J = 0 contributes and one gets

where J is the total angular momentum in the ground state.

Note that c_v = 3/2 in both limits that is the specific heat is constant at low and high temperatures (no contribution of electron degrees of freedom) having some maximum in between (due to contributions of the electrons).
We have already seen this in considering two-level system and the lesson is general: if one has a finite number of levels then they do not contribute to the specific heat both at low and high temperatures.

Specific heat of diatomic molecules. We need to calculate the sum over the internal degrees of freedom in (43). We assume the temperature to be smaller than the energy of dissociation (which is typically of the order of electronic excited states). Since most molecules have S = L = 0 in the ground state we disregard electronic states in what follows. The internal excitations of the molecule are thus vibrations and rotations with the energy characterized by two quantum numbers, j and K :

        (46)

We estimate the parameters here assuming the typical scale to be Bohr radius  and the typical energy to be Rydberg ε₀ =  Note that m = 9 . 10^-28g is the electron mass here. Now the frequency of the atomic oscillations is given by the ratio of the Coulomb restoring force and the mass of the ion:

Rotational energy is determined by the moment of inertia   We may thus estimate the typical energies of vibrations and rotations as follows:

 (47)

Since  then that both energies are much smaller than the energy of dissociation  and the rotational energy is smaller than the vibrational one so that rotations start to contribute at lower temperatures:

The harmonic oscillator was considered in in Sect. 1.4.2. In the quasiclassical limit,    the partition function of N independent oscillators is  

 the free energy  and the mean energy from (24): E = N T . The specific heat C_V = N .

For a quantum case, the energy levels are given by 

where one sees the contribution of zero quantum oscillations and the breakdown of classical equipartition. The specific heat (per molecule) of vibrations is thus as follows:   we have  At large T we have classical equipartition (every oscillator has two degrees of freedom so it has T in energy and 1 in C_V ).
To calculate the contribution of rotations one ought to calculate the partition function

Again, when temperature is much smaller than the distance to the first level,  the specific heat must be exponentially small. Indeed, retaining only two first terms in the sum (49), we get  which gives in the same approximation

We thus see that at low temperatures diatomic gas behaves an mono-atomic.

At large temperatures,  the terms with large K give the main contribution to the sum (49). They can be treated quasi-classically replacing the sum by the integral:

That gives the constant specific heat c_rot = 1. The resulting specific heat of the diatomic molecule, is shown on the figure:

Note that for the specific heat (weakly) decreases because the distance between rotational levels increases so that the level density (which is actually c_v ) decreases.
For (non-linear) molecules with   atoms we have 3 translations, 3 rotations and 6 — 6 vibrational degrees of freedom (3n momenta and out of total 3n coordinates one subtracts 3 for the motion as a whole and 3 for rotations). That makes for the high-temperature specific heat c_v = c_tr + c_rot + c_vib = 3=2 + 3=2 + 3 - 3 = 3 . Indeed, every variable (i.e. every degree of freedom) that enters ε(p, q), which is quadratic in p, q, contributes 1/2 to c_v . Translation and rotation each contributes only momentum and thus gives 1/2 while each vibration contributes both momentum and coordinate (i.e. kinetic and potential energy) and gives 1.