To determine the greatest wavelength (i.e., the lowest frequency) of radiation that can ionize unexcited hydrogen atoms, we need to consider the ionization energy of hydrogen. The ionization energy is the amount of energy required to remove an electron from an unexcited hydrogen atom.
1. Identify the Ionization Energy:
The ionization energy for hydrogen is 13.6 electron volts (eV).
2. Planck's Relation:
The energy ([tex]\(E\)[/tex]) of a photon is related to its wavelength ([tex]\(\lambda\)[/tex]) by Planck's equation:
[tex]\[
E = \frac{hc}{\lambda}
\][/tex]
Here, [tex]\(h\)[/tex] is Planck's constant and [tex]\(c\)[/tex] is the speed of light.
3. Rearrange to Find Wavelength:
To find the wavelength, we rearrange Planck's equation:
[tex]\[
\lambda = \frac{hc}{E}
\][/tex]
4. Constants:
- Planck's constant ([tex]\(h\)[/tex]) is [tex]\(4.135667696 \times 10^{-15}\)[/tex] eV·s.
- Speed of light ([tex]\(c\)[/tex]) is [tex]\(3.0 \times 10^8\)[/tex] m/s.
5. Calculate the Wavelength:
Plugging in the values:
[tex]\[
\lambda = \frac{4.135667696 \times 10^{-15} \,\text{eV·s} \times 3.0 \times 10^8 \,\text{m/s}}{13.6 \,\text{eV}}
\][/tex]
This calculation results in the wavelength value in meters. The obtained wavelength is approximately [tex]\(9.122796388235295 \times 10^{-8}\)[/tex] meters.
6. Convert to Nanometers:
To convert the wavelength from meters to nanometers (nm):
[tex]\[
1 \,\text{meter} = 10^9 \,\text{nanometers}
\][/tex]
Hence, the wavelength in nanometers is:
[tex]\[
9.122796388235295 \times 10^{-8} \, \text{meters} \times 10^9 = 91.22796388235295 \, \text{nanometers}
\][/tex]
From the calculation, the greatest wavelength of radiation that can ionize unexcited hydrogen atoms is approximately [tex]\(91.2 \, \text{nm}\)[/tex].
Answer: A. 91.2nm
