What is the wavelength separation of the two fine structure

Question

1. The first excited state of potassium has the spectroscopic term symbol ²P_{1 /2},_3/2 Spin^-orbit coupling splits the energy levels of this state so they have energies with respect to the ground state of 12985.186 cm^-I and 13042.896 cm^-I, respectively.

a. What is the wavelength separation of the two fine structure lines that arise from transitions between the first excited state and the ground state?

b. In the presence of an external magnetic field, the Zeeman effect causes further splitting of the energy levels. Sketch an energy level diagram for the transition between the first excited state and ground state of potassium in the presence of a magnetic field that is weaker than the internal magnetic fields within the atom.

c. Label the relevant quantum numbers for each energy level found in part (b) and mark the transitions that are allowed by the various selection rules. Give the value of the energy level splittings due to an external field of 0.05T.

2. Find the energy required to dissociate a KCI molecule into a K atom and a Cl atom. Use the following data:

• The first ionisation potential of K is 4.34 eV
• The electron affinity of Cl is 3.82 eV
• The equilibrium separation of KCI is 2.79 x 10^-10 m

Hint : Show that the mutual potential energy of K⁺ and Cl is -1.44/R eV (where R is the separation in nm).

3. i. The two stable isotopes of chlorine ¹⁷C1³⁵ (75.4%) and ¹⁷C1³⁷ (24.6%) have masses of 34.96885u and 36.9659u respectively. Calculate the difference in the lowest energy rotational transitions for the two HC1 isotopomers involving ¹H¹ (1.007825u) and the stable chlorine isotopes. The mean separation of the hydrogen and chlorine atoms in HC1 is 1.27 x 10^-10 m. (1 u = 1.66054 x 10^-27 kg)

ii. The lowest energy vibrational transition of HCl is observed to have a wavelength of 3.46 um. What is the force constant for the HC1 bond?

iii. Band spectrum measurements can be used to determine the nuclear spin of atoms through measurements of the ratio of the number of pars and ortho transitions. For Diatomic nuclei containing ¹⁷C1³⁵ this ratio is observed to be 3/5. What is the spin of the ¹⁷C1³⁵ and does this agree with the predicted value of 3/2?