Consider an electron in the hydrogen atom. If you are able to excite its electron from the \(n=1\) shell to the \(n=2\) shell with a given laser, what kind of a laser (that is, compare wavelengths) will you need to excite that electron again from the \(n=2\) to the \(n=3\) shell? Explain.

To find the energy differences, start by calculating the energy level at the hydrogen atom's shells using the provided energy level formula: \(E_1 = -\frac{13.6 \text{ eV}}{1^2} = -13.6 \text{ eV}\) \(E_2 = -\frac{13.6 \text{ eV}}{2^2} = -3.4 \text{ eV}\) \(E_3 = -\frac{13.6 \text{ eV}}{3^2} = -1.51 \text{ eV}\)

Now that you have the energy levels, find the energy differences between the first and the second shell and between the second and the third shell: \(\Delta E_{12} = E_2 - E_1 = -3.4 \text{ eV} - (-13.6 \text{ eV}) = 10.2 \text{ eV}\) \(\Delta E_{23} = E_3 - E_2 = -1.51 \text{ eV} - (-3.4 \text{ eV}) = 1.89 \text{ eV}\)

Now you'll use the energy differences to find the required wavelengths for the lasers. To do this, you can use the formula relating energy and wavelength: \(E = \frac{hc}{\lambda}\), where \(h\) is the Planck's constant (\(6.626 x 10^{-34} \text{ Js}\)), \(c\) is the speed of light (\(3.0 x 10^8 \text{ m/s}\)), and \(\lambda\) is the wavelength. Solve the formula for wavelength: \(\lambda = \frac{hc}{E}\) Convert eV to Joules (1 eV = \(1.6 \times 10^{-19} \text{ J}\)): \(\Delta E_{12} = 10.2 \text{ eV} \times \frac{1.6 \times 10^{-19} \text{ J}}{1 \text{ eV}} = 1.63 \times 10^{-18} \text{ J}\) \(\Delta E_{23} = 1.89 \text{ eV} \times \frac{1.6 \times 10^{-19} \text{ J}}{1 \text{ eV}} = 3.02 \times 10^{-19} \text{ J}\) Finally, calculate the wavelengths: \(\lambda_{12} = \frac{(6.626 \times 10^{-34} \text{ Js})(3.0 \times 10^8 \text{ m/s})}{1.63 \times 10^{-18} \text{ J}} = 1.214 \times 10^{-7} \text{ m}\) \(\lambda_{23} = \frac{(6.626 \times 10^{-34} \text{ Js})(3.0 \times 10^8 \text{ m/s})}{3.02 \times 10^{-19} \text{ J}} = 6.562 \times 10^{-7} \text{ m}\) Since lasers are usually specified in terms of their wavelength in nanometers (nm), convert the values: \(\lambda_{12} \approx 121.4 \text{ nm}\) \(\lambda_{23} \approx 656.2 \text{ nm}\) So, you will need a laser with a wavelength of approximately 121.4 nm to excite the electron from the \(n=1\) shell to the \(n=2\) shell, and a laser with a wavelength of approximately 656.2 nm to excite the electron from the \(n=2\) shell to the \(n=3\) shell.