Translational degeneracy

The energy corresponding to the wavefunction (2.7) is equal to (Na — l)eo + / and is Na degenerate (translational degeneracy) since the crystal energy does not depend on the choice of which of the molecules na is excited. 

We now apply the expression for U in Eq. (25.3-7) to obtain a formula to compare with Eq. (25.3-8). Atoms have only translational and electronic energy. The degeneracy of an energy level is the product of the translational degeneracy and the electronic degeneracy, and the energy is the sum of the translational and the electronic energy. The molecular partition function is... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

We have seen that for the electronic partition function there is no closed form expression (as there is for translation, rotation, and vibration) and one must know the energy and degeneracy of each state. That is. 

There may be no detectable effect because of the degeneracy of the code. This would be more likely if the changed base in the mRNA molecule were to be at the third nucleotide of a codon such mutations are often referred to as silent mutations. Because of wobble, the translation of a codon is least sensitive to a change at the third position. 

Molecular entropies For a perfect monoatomic gas, there is only translational motion. According to quantum mechanics, the translational energy of molecules in a box is quantized and the size of the quantum is proportional to the reciprocal of the atomic weight. Heavier gases have smaller gaps and the number of states available and degeneracies are greater. 

The most probable translational energy for an atom in a gas at temperature T is equal to 3/2 KT where K = R/Ny is Boltzmann s constant. Calculate the degeneracy of the most probable energy level for an argon atom at 300 K and 1 bar pressure, assuming that the atom can be treated as a particle in a three dimensional box. The volume of the gas at these conditions is 0.022 m ... 

The final set of thermodynamic quantities to illustrate is the entropy, also listed in Table 8.1. The largest contribution by far is from translation, calculated from Eq. 8.106. The portion of the entropy attributable to rotational and vibrational degrees of freedom are calculated by Eqs. 8.108 and 8.109, respectively. The electronic contribution to S from Eq. 8.101 is large (certainly relative to the role it played for the other thermodynamic functions just considered), 5.763 (= R In 2, from the ground-state degeneracy contribution) J/mole-K. Thus the net value of S at 298.15 K is calculated to be 198.542 J/mole-K, com-... 

As a check, we calculate the total vibrational degeneracy from eq. (4) as 6, which is equal, as it should be, to 3 N — 6. The arithmetic involved in the reduction of the direct sum for the total motion of the atoms can be reduced by subtracting the representations for translational and rotational motion from T before reduction into a direct sum of IRs, but the method used above is to be preferred because it provides a useful arithmetical check on the accuracy of T and its reduction. 

For the translational partition function, we first consider a particle in a onedimensional box of length /. The energy levels, with the zero of energy as the zero-point level, are En = (n2 — l)h2/(8ml2), with n = 1,2,..., and degeneracy wn = 1. The partition function takes the form... 

Here Ein is the energy of a molecule of species / in the state n, and gin is the degree of degeneracy of the given state. The energy of a molecule is usually represented as the sum of the relevant contributions of translational,... 

The vibronic exciton approximation restricts H to a subspace corresponding to a given vibronic molecular state. In this subspace the degeneracy of the localized vibronic states is lifted by the interactions JnmB Bm. Using the translational invariance, the eigenstates of the crystal are seen to be the vibronic excitons, or vibrons ... 

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