C. \( \lambda=c / v \) D. \( \lambda=v / c \) E. \( \lambda=c-v \) In an investigation of the electronic absorption spectrum of a particular element, it is found at a photon having \( \lambda=500 \mathrm{~nm} \) provides just enough energy to promote an electron from the cond quantum level to the third. From this information, we can deduce A. The difference in energies between \( n=2 \) and \( n=3 \) B. The energy of the \( n=3 \) level C. The sum of the energies of \( n=2 \) and \( n=3 \) D. The energy of the \( n=2 \) level E. None of these

To analyze the information given, we start with the relationship between the energy of a photon and its wavelength. The energy \( E \) of a photon can be calculated using the formula: \[ E = \frac{hc}{\lambda} \] where: - \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)), - \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)), - \( \lambda \) is the wavelength of the photon. Given that \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \), we can calculate the energy of the photon: \[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^8 \, \text{m/s})}{500 \times 10^{-9} \, \text{m}} \] Calculating this gives: \[ E = \frac{(6.626 \times 3.00)}{500} \times 10^{-34 + 8 + 9} = \frac{19.878}{500} \times 10^{-17} \approx 3.9756 \times 10^{-19} \, \text{J} \] This energy corresponds to the energy difference between the two quantum levels involved in the transition, specifically from \( n=2 \) to \( n=3 \). Now, let's analyze the options: A. The difference in energies between \( n=2 \) and \( n=3 \) - **This is correct** because the energy of the photon corresponds to the energy difference between these two levels. B. The energy of the \( n=3 \) level - **This cannot be determined** from the information given, as we only know the difference in energy. C. The sum of the energies of \( n=2 \) and \( n=3 \) - **This cannot be determined** from the information given, as we only know the difference in energy. D. The energy of the \( n=2 \) level - **This cannot be determined** from the information given, as we only know the difference in energy. E. None of these - **This is incorrect** because option A is correct. Thus, the correct answer is: **A. The difference in energies between \( n=2 \) and \( n=3 \)**.