Question

The greatest wavelength of the radiation that will ionize unexcited hydrogen atom is

• A. \(1820\,\text{\AA} \)
• B. \(450\,\text{\AA} \)
• C. \(910\,\text{\AA} \)
• D. \(700\,\text{\AA} \)
• E. \(1400\,\text{\AA} \)

Hint:

Greater wavelength $\Rightarrow$ lower energy (use \( E \propto \frac{1}{\lambda} \)).

Answer 1

The Correct Option is C

Step 1: Understanding Ionization of Hydrogen:
An "unexcited" hydrogen atom is in its ground state, corresponding to the principal quantum number \(n_i = 1\). To ionize the atom means to remove the electron completely, which corresponds to transitioning it to the energy level \(n_f = \infty\). The minimum energy required to do this is the ionization energy.
Step 2: Calculating Ionization Energy:
The energy of an electron in the n-th state of a hydrogen atom is given by:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \] The energy required for the transition from \(n_i=1\) to \(n_f=\infty\) is:
\[ \Delta E = E_{final} - E_{initial} = E_\infty - E_1 \] \[ \Delta E = \left(-\frac{13.6}{\infty^2}\right) - \left(-\frac{13.6}{1^2}\right) = 0 - (-13.6 \, \text{eV}) = 13.6 \, \text{eV} \] This is the ionization energy of hydrogen.
Step 3: Relating Energy to Wavelength:
The energy of a photon (\(E\)) is inversely proportional to its wavelength (\(\lambda\)), according to the formula \(E = hc/\lambda\). Therefore, the minimum energy required for a process corresponds to the maximum (or greatest) wavelength.
We need to find the wavelength corresponding to the ionization energy, \(\Delta E = 13.6\) eV.
A useful shortcut formula for this conversion is:
\[ \lambda (\text{in Ångströms}) = \frac{12400}{E (\text{in eV})} \] Step 4: Detailed Calculation:
\[ \lambda_{max} = \frac{12400}{13.6} \approx 911.76 \, \text{Å} \] This value is closest to 910 Å. This specific wavelength is also known as the Lyman series limit, as it represents the shortest possible wavelength in the Lyman series.
Step 5: Final Answer:
The greatest wavelength of radiation that will ionize an unexcited hydrogen atom is approximately 910 Å.