Question

question 17) just to make sure you understand this model, set the light source to 97 nm. this provides photons of just the right energy to raise (excite) the electron from n=1 to n=4 shell (1→ 4 transition). can you make sense of the atom diagram, the energy level diagram, and the spectrometer? what spectral lines do you observe? (4 pts)

Explanation:

Step1: Recall Bohr - model energy levels

The energy of a photon absorbed or emitted during an electron transition in a hydrogen - like atom is given by $\Delta E=E_f - E_i$, and for a hydrogen atom, the energy of the $n^{th}$ level is $E_n=-\frac{13.6}{n^{2}}\text{ eV}$. For a transition from $n = 1$ to $n = 4$, $\Delta E=E_4 - E_1$.
$E_1=-\frac{13.6}{1^{2}}=- 13.6\text{ eV}$ and $E_4=-\frac{13.6}{4^{2}}=-0.85\text{ eV}$. So, $\Delta E=-0.85-(-13.6)=12.75\text{ eV}$.

Step2: Relate energy to wavelength

The energy of a photon is $E = h
u=\frac{hc}{\lambda}$, where $h = 6.63\times10^{-34}\text{ J}\cdot\text{s}$, $c = 3\times10^{8}\text{ m/s}$. First, convert the energy from eV to J: $12.75\text{ eV}\times1.6\times10^{-19}\text{ J/eV}=2.04\times10^{-18}\text{ J}$. Then, $\lambda=\frac{hc}{E}=\frac{6.63\times10^{-34}\times3\times10^{8}}{2.04\times10^{-18}}\text{ m}\approx97.2\text{ nm}$. This is in the ultraviolet region. In the hydrogen spectrum, the transition from $n = 1$ to $n = 4$ corresponds to a Lyman - series line.

Step3: Analyze the spectrometer and diagrams

The atom diagram shows the position of the nucleus and the electron orbits. The energy - level diagram clearly shows the discrete energy levels of the atom and the transition from $n = 1$ to $n = 4$. The spectrometer would detect a single spectral line at approximately $97\text{ nm}$ corresponding to the $1
ightarrow4$ transition.

Answer:

A single spectral line in the ultraviolet region at approximately $97\text{ nm}$ corresponding to the Lyman - series transition from $n = 1$ to $n = 4$ would be observed. The atom diagram shows the electron's initial and final positions relative to the nucleus, the energy - level diagram shows the energy levels involved in the transition, and the spectrometer records the spectral line at the appropriate wavelength.