Physical Chemistry III
- Structure of Matter -

Discussion Fr 17.6.2005 10:30 - 11:15 am lecture hall SN 20.2

Problem Set 8 (angular momentum, rotation)

Problem 1

The energy of a rotational state of a (linear) molecule is Erot = Bl(l+1), where l is the rotational quantum number and B = h²/2I . Formally B is measured in energy units, i.e. in Joule. Commonly, for B the unit of wavenumbers [cm-1] is used, with the conversion relation: B [cm-1] = h²/2Ihc = h/8π²cI , where c is the speed of light (in cm/s).

1. Calculate the moment of inertia I = µr² for the isotopes H35Cl, D35Cl and H37Cl. Use the reduced masses of the isotopes mHCl = mHmCl /(mH+mCl) where mH = 1 u, mD = 2 u, mCl = 35 u and mCl = 37 u. The equilibrium bond length is r = 127,5 pm.
2. Calculate B in wavenumbers for all isotopes.
3. What is the energy gap (in wavenumbers) between the states l=10 and l=11?
4. The mean rotational energy at room temperature is approximately 200 cm-1. Which l state is closest in energy?
5. The lowest l state of HCl is l=0. What is the corresponding rotational energy?

Problem 2

What is the frequency of rotation of molecular oxygen in the state J=11 (B = 1,446 cm-1; r = 120,7 pm)? Use the classical relationship Erot = 1/2Iw² for the determination of the rotational energy. What is the speed of the rotating oxygen atoms?

Problem 3

1. Normalize the spherical harmonic function Y2,0 = C (3 cos2θ - 1).
2. Create the (non-normalized) spherical harmonicsY2,+1 and Y2,+2 by applying the operator L+.

Problem 4

Are the following spherical harmonics orthogonal to each other: Y2,0, Y1,0, Y0,0? Consider all possible constellations.