The ion \(\mathrm{Be}^{3+}\) makes an atomic transition from an \(n=3\) state to an \(n=2\) state. (a) What is the energy of the photon emitted during the transition? (b) What is the wavelength of the photon?

To calculate the energy of the photon emitted during the transition, we use the energy level formula for hydrogen-like ions, which is given by: \( E_n = -\cfrac{Z^2Ke^2}{2a_0n^2} \), where \(E_n\) is the energy at energy level n, \(Z\) is the atomic number of the ion, \(K\) is Coulomb's constant, \(e\) is the elementary charge, and \(a_0\) is the Bohr radius. First, let's calculate the energy levels for both the initial state (\(n=3\)) and the final state (\(n=2\)) of the \(\mathrm{Be}^{3+}\) ion. For Be³⁺, Z = 4 (since Be has 4 protons). We also have the following values for constants: \(K = 8.9875 \times 10^9\, N\,m^2 / C^2\) \(e = 1.602 \times 10^{-19}\, C\) \(a_0 = 5.2918 \times 10^{-11}\, m\) Now, let's calculate the energy levels: \( E_3 = -\cfrac{(4)^2(8.9875 \times 10^9)(1.602 \times 10^{-19})^2}{2(5.2918 \times 10^{-11})(3)^2} \) \(E_3 = -1.0177 \times 10^{-18}\, J \) Similarly, \( E_2 = -\cfrac{(4)^2(8.9875 \times 10^9)(1.602 \times 10^{-19})^2}{2(5.2918 \times 10^{-11})(2)^2} \) \(E_2 = -2.5433 \times 10^{-18}\, J \) Now we can find the energy of the emitted photon by taking the difference between the initial and final energy levels: \( ΔE = E_2 - E_3 = -2.5433 \times 10^{-18}\, J - (-1.0177 \times 10^{-18}\, J) \) \(ΔE = 1.5256 \times 10^{-18}\, J\)

With the energy of the emitted photon, we can now calculate its wavelength using the following equation derived from Planck's relation: \( λ = \cfrac{hc}{E} \), where \(λ\) is the wavelength, \(h\) is Planck's constant, \(c\) is the speed of light, and \(E\) is the energy of the photon. Using the following constants: \(h = 6.626 \times 10^{-34}\, Js\) \(c = 2.998 \times 10^8\, m/s\) Now, we can calculate the wavelength: \( λ = \cfrac{(6.626 \times 10^{-34})(2.998 \times 10^8)}{1.5256 \times 10^{-18}} \) \(λ = 1.3059 \times 10^{-8} m\) The wavelength of the emitted photon is approximately \(1.3059 \times 10^{-8}\, m\), or \(13.06\, nm\). (a) The energy of the photon emitted during the transition is \(1.5256 \times 10^{-18}\, J\). (b) The wavelength of the photon is approximately \(13.06\, nm\).