To calculate the energy needed to ionize a hydrogen atom when an electron is present in the 2nd excited state, you need to consider the difference in energy between the initial state and the final ionized state.
In this case, the initial state is the second excited state of the hydrogen atom, and the final state is the ionized state with the electron completely removed. The ionization energy of a hydrogen atom refers to the energy required to completely remove the electron from the ground state (n=1).
The energy of an electron in a hydrogen atom is given by the formula:
E = -13.6 eV / n^2
where E is the energy, -13.6 eV is the ionization energy of the hydrogen atom from the ground state, and n is the principal quantum number.
For the second excited state (n=3), the energy is:
E_initial = -13.6 eV / 3^2
To calculate the energy needed to ionize the hydrogen atom when the electron is in the 2nd excited state, you subtract the energy of the initial state from the ionization energy:
E_ionization = Ionization energy - E_initial
Since the ionization energy is given in kilojoules per mole (kJ/mol), you need to convert the electron volt (eV) to joules (J) and then adjust for the number of moles of hydrogen atoms:
1 eV = 1.60218 x 10^-19 J 1 kJ/mol = 1000 J/mol
Therefore, the energy needed to ionize a hydrogen atom when the electron is in the 2nd excited state can be calculated as follows:
E_ionization = (1312 kJ/mol * 1000 J/kJ) / 6.022 x 10^23 (mol^-1) - (-13.6 eV / 3^2 * 1.60218 x 10^-19 J/eV)
The first term in the equation calculates the ionization energy in joules per mole, and the second term calculates the energy of the initial state in joules. Subtracting these values will give you the energy needed to ionize the hydrogen atom when the electron is in the 2nd excited state.