Question

Calculate the energies needed to remove an electron from the \(n=1\) state and the \(n=5\) state in the \(\mathrm{Li}^{2+}\) ion. What is the wavelength (in \(\mathrm{nm}\) ) of the emitted photon in a transition from \(n=5\) to \(n=1 ?\) Solving Equation 6.4 for energy gives \(\Delta E=R_{\infty} h c\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right),\) where the Rydberg constant \(R_{\infty}\) for hydrogen-like atoms is \(1.097 \times 10^{7} \mathrm{~m}^{-1} \cdot Z^{2},\) and \(Z\) is the atomic number.

Short answer

The energies are \(-1.96 \times 10^{-17} \mathrm{J}\) for \(n=1\) and \(-0.0785 \times 10^{-17} \mathrm{J}\) for \(n=5\). The photon's wavelength is \(106 \, \mathrm{nm}\).

Step-by-step solution

Find Rydberg constant for \( \text{Li}^{2+} \)

The Rydberg constant \( R_\infty \) for hydrogen-like atoms is given by \( 1.097 \times 10^7 \, \text{m}^{-1} \times Z^2 \). For the \( \text{Li}^{2+} \) ion, the atomic number \( Z = 3 \). Thus, \( R_\infty = 1.097 \times 10^7 \, \text{m}^{-1} \times 3^2 = 9.873 \times 10^7 \, \text{m}^{-1} \).

Calculate energy for \( n = 1 \)

Using the equation \( E_n = -R_\infty h c \left( \frac{1}{n^2} \right) \), for \( n = 1 \):\[ E_1 = -9.873 \times 10^7 \, \mathrm{m}^{-1} \cdot 6.626 \times 10^{-34} \, \mathrm{Js} \cdot 3 \times 10^8 \, \mathrm{ms}^{-1} \cdot \frac{1}{1^2} \= -1.96 \times 10^{-17} \, \mathrm{J}\]

Calculate energy for \( n = 5 \)

For \( n = 5 \), use the same energy formula:\[ E_5 = -9.873 \times 10^7 \, \mathrm{m}^{-1} \cdot 6.626 \times 10^{-34} \, \mathrm{Js} \cdot 3 \times 10^8 \, \mathrm{ms}^{-1} \cdot \frac{1}{5^2} \= -0.0785 \times 10^{-17} \, \mathrm{J}\]

Calculate energy difference \( \Delta E \) for transition

The energy difference \( \Delta E \) for a transition from \( n = 5 \) to \( n = 1 \) is given by \[ \Delta E = E_1 - E_5 \ = (-1.96 + 0.0785) \times 10^{-17} \, \mathrm{J} \ = -1.8815 \times 10^{-17} \, \mathrm{J} \]

Calculate wavelength of emitted photon

The energy of a photon is related to its wavelength by \( \Delta E = \frac{hc}{\lambda} \). Solve for \( \lambda \):\[ \lambda = \frac{hc}{\Delta E} \ \lambda = \frac{6.626 \times 10^{-34} \, \mathrm{Js} \cdot 3 \times 10^8 \, \mathrm{ms}^{-1}}{1.8815 \times 10^{-17} \, \mathrm{J}} \ = 1.06 \times 10^{-7} \, \mathrm{m} \ = 106 \, \mathrm{nm}\]

Key concepts

Hydrogen-like Atoms

Hydrogen-like atoms are special types of ions that possess only one electron, similar to the hydrogen atom. However, unlike hydrogen, these ions have more protons in their nucleus. Popular examples include the \( \text{He}^{+} \) and \( \text{Li}^{2+} \) ions.
The single electron experiences the full nuclear charge since there are no other electrons to shield this attraction. This makes the electron behave in a manner reminiscent of a hydrogen atom.

• The electron's movement is influenced by a stronger nuclear charge due to the increased number of protons.
• This strength alters the energy levels of the electron compared to a standard hydrogen atom.

In this structure, these hydrogen-like atoms follow the Rydberg formula, making them significant in spectroscopic studies.

Atomic Number Z

The atomic number, denoted as \( Z \), is crucial in determining an element's chemical properties. It tells us the number of protons present in an atom's nucleus. In the case of hydrogen-like atoms, \( Z \) plays a pivotal role in calculating the ion's Rydberg constant.
For example, the \( \text{Li}^{2+} \) ion has an atomic number \( Z = 3 \) indicating it contains three protons.

• The Rydberg constant for these atoms is modified by \( Z^2 \), enhancing the energy levels by increasing the attraction between the electron and the nucleus.
• This boost in electrostatic pull results in a higher energy needed to detach an electron from its orbit.

Thus, the atomic number serves as a fundamental parameter in the study of hydrogen-like atoms and their spectral lines.

Energy Levels n

Energy levels (denoted by \( n \)) are fundamental concepts in quantum mechanics, providing insight into an electron's possible states within an atom. Each energy level is characterized by specific values of energy, with the principal quantum number \( n \) describing these levels.
In hydrogen-like atoms, the energy required to shift an electron from one level to another can be calculated using the modified Rydberg formula.

• The ground state (\( n = 1 \)) represents the most stable and lowest energy position of an electron.
• Excited states such as \( n = 5 \) exhibit higher energy and are less stable.

When computing energy differences between levels, transitions release photons whose energies match the difference between these levels.

Photon Wavelength

Wavelength is an essential property of waves and can specifically describe the nature of emitted or absorbed photons during atomic transitions. In simpler terms, the wavelength tells us the distance between successive peaks of a light wave. When an electron in a hydrogen-like atom undergoes a transition from one energy level to another, a photon is either absorbed or emitted.

• The energy of this photon can be determined using \( \Delta E = \frac{hc}{\lambda} \), where \( h \) is the Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
• Mathematically manipulating this equation helps us derive the wavelength of the emitted photon.

For instance, during an electron transition from \( n = 5 \) to \( n = 1 \) in a \( \text{Li}^{2+} \) ion, the emitted photon has a wavelength of 106 \( \text{nm} \). This specific value indicates the characteristics of the light observed.