Explanation:

To solve the problem, we need to use the following equations:

- The binding energy of an electron in the nth energy level of a hydrogenic atom is given by:

E_n = -13.6/n^2 Z^2 eV

where Z is the atomic number of the nucleus.

- The energy required to excite an electron from the nth energy level to the mth energy level is given by:

E_mn = E_m - E_n

where E_m is the energy of the mth energy level and E_n is the energy of the nth energy level.

- The ionization energy of a hydrogenic atom is the energy required to completely remove an electron from the atom, and is given by:

E_i = -13.6 Z^2 eV

Solution:

We are given that the binding energy of the 2nd excited state of a hypothetical hydrogenic atom is 12 eV.

Step 1: Calculate the energy of the 2nd excited state

Using the first equation, we can calculate the energy of the 2nd excited state:

E_3 = -13.6/3^2 Z^2 eV

Since we are dealing with a hydrogenic atom, Z = 1, so we can simplify the equation to:

E_3 = -4.54 eV

Step 2: Calculate the energy required to excite an electron from the 2nd excited state to the 3rd excited state

Using the second equation, we can calculate the energy required to excite an electron from the 2nd excited state to the 3rd excited state:

E_32 = E_3 - E_2

where E_2 is the energy of the 2nd excited state.

E_32 = -4.54 - (-12) = 7.46 eV

Therefore, the energy required to excite an electron from the 2nd excited state to the 3rd excited state is 7.46 eV.

Step 3: Calculate the ionization energy

Using the third equation, we can calculate the ionization energy:

E_i = -13.6 Z^2 eV

E_i = -13.6 = -13.6 eV

Therefore, the ionization energy of the hydrogenic atom is 13.6 eV.

Step 4: Verify the options

Now we need to check which of the given options is correct.

Option A: 1st excitation potential is 81 eV

Using the first equation, we can calculate the energy of the 1st excited state:

E_2 = -13.6/2^2 Z^2 eV = -3.4 eV

Therefore, the energy required to excite an electron from the ground state to the 1st excited state is:

E_21 = E_2 - E_1 = -3.4 - (-13.6) = 10.2 eV

Option A is not correct.

Option B: 2nd excitation energy is 96 eV

We have already calculated the energy required to excite an electron from the 2nd excited state to the 3rd excited state, which is 7.46 eV.

Option B is not correct