Which state of \(\mathrm{Be}^{3+}\) has the same orbit radius as that of the ground state of hydrogen atom? (a) 3 (b) 2 (c) 4 (d) 5

In the world of quantum physics, the principal quantum number, denoted as \( n \), plays a crucial role in determining the properties of electrons in an atom. It is a positive integer that represents the energy level in which an electron resides. The significance of \( n \) stems from its role in quantifying the energy of an electron and the size of the electron's orbit within an atom. As \( n \) increases, the energy levels become higher, and the electron’s orbit becomes larger and farther from the nucleus, implying that electrons are less tightly bound to the nucleus and have higher potential energy.

In the problem at hand, identifying the principal quantum number that corresponds to an electron orbit with the same radius as a hydrogen atom's ground state is key. If we recall, in the hydrogen ground state, \( n = 1 \). For a hydrogen-like atom, such as \( \text{Be}^{3+} \), we have to find the value of \( n \) that gives us an orbit of equivalent size. This value is determined through the relation between \( n \) and the atomic number, \( Z \), of the atom in question. Consequently, for multi-electron systems or ions, it's critical to understand how these quantum numbers correlate with the corresponding electron configurations and orbital sizes.

Bohr's model presents a way to determine the radius of an electron's orbit in a hydrogen-like atom. The formula \( r = \frac{{n^2h^2}}{{4\text{π}^2kme^2Z}} \) encapsulates several fundamental constants as well as variable parameters that directly affect the electron's orbit size. Here's a breakdown:

• \( h \) is Planck's constant, a fundamental quantity in quantum mechanics.
• \( k \) is Coulomb's constant, which pertains to the electrostatic interaction between charged particles.
• \( m \) is the mass of the electron.
• \( e \) is the electric charge of the electron.
• \( Z \) is the atomic number, which effectively determines the strength of the nuclear charge that the electron experiences.

The radius increases as \( n \), the principal quantum number, increases, allowing us to visualize electron shells as going further out from the nucleus with each step up in \( n \). In particular, this formula helps us understand how an ion with a higher atomic number like \( \text{Be}^{3+} \) can have an electron orbit matching the radius of a hydrogen atom's ground state merely by adjusting the principal quantum number.

A hydrogen-like atom refers to any single-electron ion that resembles hydrogen in its electron structure but contains a nucleus with a different atomic number, \( Z \). This term grants us the ability to apply Bohr's model beyond hydrogen, to ions such as \( \text{Be}^{3+} \), by simply adjusting the \( Z \) parameter to reflect the number of protons in the nucleus.

For such atoms, despite the varying nuclear charges, the electron's behavior can be understood similarly to that in a hydrogen atom by using Bohr's model. However, the increased nuclear charge (higher \( Z \)) typically means that electrons are drawn closer to the nucleus, leading to smaller radii for their orbits compared to a hydrogen atom at the same principal quantum number. Therefore, to find a matching radius with hydrogen's ground state, a hydrogen-like atom would require an electron to be in a higher principal quantum number, which effectively compensates for the stronger nuclear attraction due to the greater \( Z \). Understanding hydrogen-like atoms allows us to predict the properties of ions in states that would be difficult to analyze using more complex models.