Question

Bohr's theory can also be applied to the ions like (a) \(\mathrm{He}^{+}\) (b) \(\mathrm{Li}^{2+}\) (c) \(\mathrm{Be}^{3+}\) (d) all of these.

Short answer

Bohr's model can be applied to hydrogen-like ions, which are ions with only one electron. Therefore, it can be applied to \(\mathrm{He}^{+}\), \(\mathrm{Li}^{2+}\), and \(\mathrm{Be}^{3+}\). The correct answer is (d) all of these.

Step-by-step solution

Understanding the Bohr Model

Bohr's atomic model proposes that electrons orbit the nucleus in specific energy levels and can only occupy certain allowed orbits. When applied to hydrogen-like ions (ions with only one electron), the model can be used to predict the electron's energy levels and spectral lines.

Applying Bohr's Model to Ions

Bohr's theory can be applied to any hydrogen-like ion (a single-electron system), which includes any ion where all but one of the electrons have been stripped away, such as \(\mathrm{He}^{+}\), \(\mathrm{Li}^{2+}\), and \(\mathrm{Be}^{3+}\). These ions have only one electron, just as a hydrogen atom does.

Conclusion

Since \(\mathrm{He}^{+}\), \(\mathrm{Li}^{2+}\), and \(\mathrm{Be}^{3+}\) are all hydrogen-like ions with one electron, Bohr's model can be applied to each of these to describe their spectral lines and energy levels. Therefore, the correct answer is (d) all of these.

Key concepts

Hydrogen-like ions

When we talk about hydrogen-like ions, we are referring to ions that have only one electron remaining, similar to a hydrogen atom. To visualize this better, imagine removing electrons from a helium atom (\texttt{He}) until only one electron is left, resulting in a \texttt{He\(^{+}\)} ion. The same process applies to lithium (\texttt{Li}), where two electrons are removed, creating a \texttt{Li\(^{2+}\)} ion and so on. These ions are special because they allow us to apply simplified models of atomic structure, such as Bohr's atomic model, which was originally developed for hydrogen atoms.

Bohr's model provides a valuable way to understand the behavior of these ions despite their simplistic approach. It assumes that electrons move in circular orbits around the atomic nucleus and only jump between these orbits, or energy levels, by emitting or absorbing energy in discrete packets called quanta. Therefore, hydrogen-like ions are important in studies of atomic physics because they offer a window into how electrons interact with the nucleus in a simple, one-electron system.

Spectral lines

Now let’s focus on spectral lines, which are unique patterns of light that are emitted or absorbed by electrons as they move between energy levels in an atom or ion. According to Bohr's atomic model, when an electron 'jumps' from a higher energy level to a lower one, it releases energy in the form of light — these are the spectral lines we can observe. Each transition corresponds to a specific wavelength or color of light. This process is like a fingerprint for elements and ions, allowing scientists to identify them based on their spectral lines.

Hydrogen-like ions emit their own distinctive spectral lines, which can be measured and used to understand the ion's structure. Bohr's model predicts these emissions by calculating the energy differences between allowed orbits for the electron. The precision of these calculations is integral for applications like astrophysics and plasma physics where spectral lines provide insights into the composition and conditions of distant stars and hot gases.

Electron energy levels

The concept of electron energy levels is a cornerstone in the Bohr's atomic model. Think of these energy levels like the rungs of a ladder where an electron can 'climb' up or 'fall' down, but it cannot rest in-between these specified levels. Bohr's theory postulates that each energy level in an atom corresponds to a particular orbital radius, with electrons residing in these discrete orbits. Transitions between these levels, as mentioned before, result in the emission or absorption of photons, producing the observed spectral lines.

The energy levels are quantized, meaning they are like steps on a staircase that you can count: the first energy level, the second, and so on. The model even allows us to compute the energy of an electron at a particular level using the formula \texttt{E\(_n\) = - \texttt{R\(_H\)} (Z^2/n^2)}, where \texttt{E\(_n\)} is the energy at level 'n', \texttt{R\(_H\)} is the Rydberg constant for hydrogen, 'Z' is the atomic number, and 'n' is the principle quantum number (level). These energy levels help us explain the chemical properties of elements and predict how atoms and ions will interact with radiation or with each other.