Problem 24 - Master Resource Book in JEE Main Physics

Question

Taking the Bohr radius as \(a_{0}=53 \mathrm{pm}\), the radius of \(\mathrm{Li}^{\text {t+ }}\) ion in its ground state, on the basis of Bohr's model, will be about \(\quad\) [NCERT Exemplar] (a) \(53 \mathrm{pm}\) (b) \(27 \mathrm{pm}\) (c) \(18 \mathrm{pm}\) (d) \(13 \mathrm{pm}\)

Step-by-Step Solution

Answer

The radius of \( \mathrm{Li}^{2+} \) in its ground state is about \(18 \, \mathrm{pm}\).

Step 1: Understanding the Bohr Model Formula for Radius

According to the Bohr model, the radius of the orbit for a hydrogen-like ion is given by \( r_n = \frac{n^2 \cdot a_0}{Z} \), where \( n \) is the principal quantum number, \( a_0 \) is the Bohr radius for the hydrogen atom, and \( Z \) is the nuclear charge, which is the atomic number for the ion being considered. In our case, \(a_{0} = 53 \, \mathrm{pm}\), \( Li^{2+} \) has \( Z = 3 \), and in the ground state \( n = 1 \).

Step 2: Substitute Values into the Radius Formula

Substitute the known values into the Bohr model formula: \( r_1 = \frac{1^2 \cdot 53 \, \mathrm{pm}}{3} \).

Step 3: Calculate the Radius

Simplify the expression by performing the division: \[ r_1 = \frac{53 \, \mathrm{pm}}{3} \approx 17.67 \, \mathrm{pm} \].

Step 4: Round to the Nearest Option

Round the calculated radius value to the nearest whole number that matches the given options. The closest value is \(18 \, \mathrm{pm}\) as given in option (c).

Key Concepts

Bohr Radius

The Bohr radius is a fundamental concept in atomic physics. It represents the most probable distance between the nucleus and the electron in a hydrogen atom's ground state. In mathematical terms, it is defined as:\[a_0 = rac{ ext{h}^2}{4 ext{π}^2 m e^2}\]where:

• \( ext{h} \) is Planck's constant,
• \( m \) is the mass of the electron,
• \( e \) is the charge of the electron.

The standard value for the Bohr radius \( a_0 \) is approximately 53 pm (picometers). This measurement is crucial when calculating the atomic radii in Bohr's model for hydrogen-like atoms. The Bohr radius allows us to estimate how far an electron is likely to be found from the nucleus in these atomic models.
It's important because it serves as a unit of measurement for understanding atomic sizes, especially in cases involving hydrogen-like ions. When applying the Bohr model to predict the radii of these ions, knowing the Bohr radius helps in accurately determining how atomic size changes with different atomic numbers \( Z \).

Hydrogen-like Ion

Hydrogen-like ions are ions that have only one electron orbiting a nucleus, similar to a hydrogen atom. The key distinguishing factor is in the nucleus. While hydrogen has just one proton, a hydrogen-like ion has a nucleus that might include more protons and possibly more neutrons.
For example, \( Li^{2+} \) is a hydrogen-like ion of lithium that has lost two of its three electrons, leaving it with just one, similar to hydrogen.
These ions are important because they allow scientists to extend the Bohr model beyond hydrogen. By doing so, they can predict and understand the behavior of more complex atoms with a single electron. The Bohr model calculates properties such as radius and energy levels for these ions by adjusting parameters, specifically using the nuclear charge \( Z \).
Nuclear charge significantly affects the attractions between the nucleus and the lone electron, impacting both the energy levels and radii of the orbit. Thus, understanding hydrogen-like ions helps us study atomic spectra and chemical properties of not only simple ions but more complex atoms as well.

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a fundamental number in quantum mechanics that defines the energy level of an electron in an atom. It can be any positive integer (1, 2, 3, etc.) and directly relates to the size of the electron's orbit in atoms.
In Bohr's model, each \( n \) value corresponds to different energy levels, often termed as "shells". For a hydrogen-like atom or ion, the principal quantum number determines not only the size of the electron's orbit but also its energy level.

• A lower \( n \) means the electron is closer to the nucleus and the energy is lower.
• Conversely, a higher \( n \) signifies that the electron has more energy and is found further from the nucleus.

When calculating the orbit size for hydrogen or hydrogen-like ions (like \( Li^{2+} \)), the principal quantum number is crucial as it affects the formula for the radius: \( r_n = \frac{n^2 a_0}{Z} \). Here, \( a_0 \) is the Bohr radius, \( Z \) is the nuclear charge, and \( n \) plays a pivotal role in scaling the radius according to the electron's energy state. Understanding \( n \) helps predict how electrons occupy different levels within an atom.