Question

If the ionisation energy for the hydrogen atom is \(13.6\) eV, then the energy required to excite it from the ground state to the next higher state is nearly

• A. \(-10.2\,eV\)
• B. \(-3.4\,eV\)
• C. \(10.2\,eV\)
• D. \(13.6\,eV\)

Hint:

Energy levels are -13.6/n^2 eV, so the first excitation goes from n = 1 to n = 2.

Answer 1

The Correct Option is C

Step 1: Understand the concept
The energy of the hydrogen atom in the \(n\)th level is \(E_n = -\dfrac{13.6}{n^2}\) eV. The ionisation energy of 13.6 eV is the energy needed to take the electron from \(n = 1\) to infinity.

Step 2: Energies of the two levels
\(E_1 = -13.6\) eV and \(E_2 = -\dfrac{13.6}{4} = -3.4\) eV.

Step 3: Excitation energy
\[ \Delta E = E_2 - E_1 = -3.4 - (-13.6) = 10.2\ \text{eV} \]

Step 4: Result
The energy required is \(+10.2\) eV, option (C). The value \(-3.4\) eV is the energy of the first excited state, not the energy needed, and \(-10.2\) eV has the wrong sign because the atom absorbs this energy.

Final Answer:
The required energy is 10.2 eV. This is option (C). \[ \boxed{\text{(C) }10.2\ \text{eV}} \]