Problem 1 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The difference between the radii of \(3^{\text {rd }}\) and \(4^{\text {th }}\) orbits of \(\mathrm{Li}^{2+}\) is \(\Delta \mathrm{R}_{1}\). The difference between the radii of \(3^{\text {rd }}\) and \(4^{\text {th }}\) orbits of \(\mathrm{He}^{+}\)is \(\Delta \mathrm{R}_{2}\). Ratio \(\Delta \mathrm{R}_{1}: \Delta \mathrm{R}_{2}\) is (a) \(8: 3\) (b) \(3: 8\) (c) \(2: 3\) (d) \(3: 2\)

Step-by-Step Solution

Answer

The ratio \( \Delta R_1 : \Delta R_2 \) is \( 2:3 \), option (c).

Step 1: Understanding the formula for orbit radius

In a hydrogen-like atom, the radius of the nth orbit is given by the formula \( r_n = \frac{a_0 n^2}{Z} \), where \( a_0 \) is the Bohr radius, \( n \) is the orbit number, and \( Z \) is the atomic number.

Step 2: Calculate the difference for \( \mathrm{Li}^{2+} \)

For \( \mathrm{Li}^{2+} \) where \( Z = 3 \), calculate the radii of the 3rd and 4th orbits: - \( r_3 = \frac{a_0 \cdot 3^2}{3} = 3a_0 \)- \( r_4 = \frac{a_0 \cdot 4^2}{3} = \frac{16}{3}a_0 \)- The difference \( \Delta R_1 = r_4 - r_3 = \frac{16a_0}{3} - 3a_0 = \frac{16a_0}{3} - \frac{9a_0}{3} = \frac{7a_0}{3} \).

Step 3: Calculate the difference for \( \mathrm{He}^{+} \)

For \( \mathrm{He}^{+} \) where \( Z = 2 \), calculate the radii of the 3rd and 4th orbits: - \( r_3 = \frac{a_0 \cdot 3^2}{2} = \frac{9a_0}{2} \)- \( r_4 = \frac{a_0 \cdot 4^2}{2} = 8a_0 \)- The difference \( \Delta R_2 = r_4 - r_3 = 8a_0 - \frac{9a_0}{2} = 8a_0 - 4.5a_0 = 3.5a_0 \).

Step 4: Calculate the ratio \( \Delta R_1 : \Delta R_2 \)

Determine the ratio \( \Delta R_1 : \Delta R_2 \):- \( \Delta R_1 = \frac{7a_0}{3} \) and \( \Delta R_2 = 3.5a_0 = \frac{7a_0}{2} \)- \( \Delta R_1 : \Delta R_2 = \frac{7a_0/3}{7a_0/2} = \frac{7a_0}{3} \times \frac{2}{7a_0} = \frac{2}{3} \).- Thus, the ratio is \( 2:3 \).

Key Concepts

Hydrogen-like Atom

A hydrogen-like atom is an atom or ion that has only one electron orbiting its nucleus. These types of atoms are special because they can be described using the Bohr model, similar to a hydrogen atom. In a hydrogen-like atom, the behavior of the electron is largely determined by the atomic number, represented by the symbol \( Z \). The atomic number indicates how many positive charges are in the nucleus, similar to the number of protons.

Examples of hydrogen-like atoms include \( ext{He}^+ \) and \( ext{Li}^{2+} \), where each has been stripped of all but one electron. This makes the electron experience a greater nuclear charge than hydrogen itself, as \( Z \) is greater than 1 for these ions.

To understand calculations involving hydrogen-like atoms, you need to be familiar with concepts such as orbitals, ionization, and energy levels. They follow specific equations derived from the Bohr model, allowing for precise calculations of various properties, like the orbit radius of the electron.

Atomic Radius

The atomic radius in the context of hydrogen-like atoms can be understood as the distance from the nucleus to the electron in a particular orbit. The Bohr model provides a useful formula for calculating this, especially for atoms or ions with a single electron, like \( ext{He}^+ \) or \( ext{Li}^{2+} \).

According to the Bohr model, the radius \( r_n \) of the \( n^{th} \) orbit is determined by:

• \( r_n = \frac{a_0 n^2}{Z} \), where \( a_0 \) is the Bohr radius (approximately equal to 0.529 Å),
• \( n \) is the principal quantum number indicating the orbit number,
• \( Z \) is the atomic number of the nucleus (number of protons)

This formula shows that the atomic radius is directly proportional to \( n^2 \) and inversely proportional to \( Z \).

For example, in a \( ext{Li}^{2+} \) ion (where \( Z = 3 \)), the radius of the third orbit (\( n = 3 \)) would be larger than the radius of the third orbit of \( ext{He}^+ \) (where \( Z = 2 \)), given the stronger nuclear pull in helium due to the higher effective nuclear charge.

Orbit Number

In the Bohr model, the orbit number, represented by \( n \), is a principal quantum number that dictates the electron's energy level and distance from the nucleus. Orbits are like the rings in which electrons move around in an atom, with each orbit corresponding to a different energy level.

Here's a simple breakdown of orbit number significance:

• As the orbit number \( n \) increases, the electron is at a higher energy level and further from the nucleus.
• The energy of an electron in a particular orbit is quantized, meaning it can only exist at specific levels, not in between these levels.
• The energy difference between orbits changes with the orbit number, being most significant in smaller orbits.

In a hydrogen-like atom, understanding the orbit number is crucial to predicting and calculating other properties such as atomic radius. For example, the difference in atomic radius between two consecutive orbits can be found by calculating the individual radii and subtracting them, as seen in the calculations for \( ext{Li}^{2+} \) and \( ext{He}^+ \). This helps understand how electron transitions between orbits impact atomic properties.