- Mean energy of a quantum harmonic oscillator at a temperature T:
40)ha)
(E)— 2 ,^ (6.4c)
et.olkT^ — a a ' - Debye formula for molar vibrational energy of a crystal:
OIT
3 dx
U =9R6 F-L 8 1 +(-T )4 NO/ j ex—if S 1
where 0 is the Debye temperature,0 = hcontaxik.- Molar vibrational heat capacity of a crystal for T 0:
= _^12 T
5n4R (
0)3
- Distribution of free electrons in metal in the vicinity of the absolute
zero: - 1 / m- 3/2
dn nah3 E dE, (6.4g)
where do is the concentration of electrons whose energy falls within the inter-
val E, E dE. The energy E is counted off the bottom of the conduction band.- Fermi level at T = 0:
hz
EF =-2m (33t^2 n)2/3^7 (6.4h)
where n is the concentration of free electrons in metal.6.167. Determine the angular rotation velocity of an S2 molecule
promoted to the first excited rotational level if the distance between
its nuclei is d = 189 pm.
6.168. For an HCl molecule find the rotational quantum numbers
of two neighbouring levels whose energies differ by 7.86 meV. The
nuclei of the molecule are separated by the distance of 127.5 pm.
6.169. Find the angular momentum of an oxygen molecule whose
rotational energy is E = 2.16 meV and the distance between the
nuclei is d = 121 pm.
6.170. Show that the frequency intervals between the neighbour-
ing spectral lines of a true rotational spectrum of a diatomic molecule
are equal. Find the moment of inertia and the distance between the
nuclei of a CH molecule if the intervals between the neighbouring
lines of the true rotational spectrum of these molecules are equal to
Aw = 5.47.10 12 s- 1.
6.171. For an HF molecule find the number of rotational levels
located between the zeroth and first excited vibrational levels assum-
ing rotational states to be independent of vibrational ones. The
natural vibration frequency of this molecule is equal to
7.79.10 14 rad/s, and the distance between the nuclei is 91.7 pm.
(6.4d)(6.4e)(6.4f)