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9. Quantum Mechanics
Wavelength and Frequency
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Which of the following wavelengths corresponds to a photon with just enough energy to ionize a hydrogen atom from its ground state (energy required: 13.6 eV)?
A
91.2 nm
B
121.6 nm
C
400 nm
D
656 nm
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Recall that the energy required to ionize a hydrogen atom from its ground state is given as 13.6 eV. This energy corresponds to the energy of a photon that can just remove the electron from the atom.
Use the relationship between the energy of a photon and its wavelength, which is given by the equation: \(E = \frac{hc}{\lambda}\), where \(E\) is the photon energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength.
Convert the energy from electron volts (eV) to joules (J) if necessary, using the conversion factor \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\), so that the units are consistent with \(h\) and \(c\).
Rearrange the equation to solve for the wavelength: \(\lambda = \frac{hc}{E}\). Substitute the known values for \(h\) (Planck's constant), \(c\) (speed of light), and \(E\) (energy in joules) to calculate the wavelength.
Compare the calculated wavelength to the given options to identify which wavelength corresponds to the photon with just enough energy to ionize the hydrogen atom from its ground state.
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