Problem 44 - Master Resource Book in JEE Main Physics

Question

Which energy state of the triply ionized beryllium has the same electron orbital radius as that of ground state of hydrogen? Given \(\mathrm{Z}\) for \(\mathrm{Be}=4\) (a) \(n=4\) (b) \(n=3\) (c) \(n=2\) (d) \(n=1\)

Step-by-Step Solution

Answer

Energy state with n=2 of Be^{3+} has the same orbital radius as ground state hydrogen.

Step 1: Understanding the Problem

We need to find the energy state of the triply ionized beryllium atom that has the same electron orbital radius as the ground state of a hydrogen atom. Triply ionized beryllium (Be^{3+}) has a nuclear charge Z=4 which means it has the electron configuration of a hydrogen-like atom with one electron.

Step 2: Calculate Orbital Radius for Hydrogen

For a hydrogen atom, in the ground state (n=1), the electron orbital radius is given by the formula r_{n} = a_{0}n^{2}/Z, where a_{0} is the Bohr radius. Since Z=1 for hydrogen in its ground state, the radius is simply r_1 = a_{0}.

Step 3: Locate Formula for Radius in Be^{3+}

For Be^{3+}, a similar formula applies: r_{n} = a_{0}n^{2}/Z. Given Z=4 for Be^{3+}, we need to find n such that the radius r_{n} for Be^{3+} equals r_1 for hydrogen. That is a_{0}= a_{0}n^{2}/4.

Step 4: Calculate the Required n

Rearranging the equation a_{0} = rac{a_{0}n^{2}}{4} yields n^{2} = 4. Solving this gives n=2.

Step 5: Verify the Result

Upon solving n^{2} = 4, we determined n=2 is the only positive value possible. Thus, the energy state n=2 of Be^{3+} has the same orbital radius as the ground state of hydrogen.

Key Concepts

Electron Orbital Radius

The electron orbital radius is a fundamental concept in atomic physics that describes the average distance between an electron orbiting a nucleus and the nucleus itself. This radius is largely influenced by the energy level or state and the nuclear charge. The orbital radius of an electron determines the electron's potential energy and sense of "room" to move around the nucleus. For hydrogen-like atoms, the orbital radius in a specific energy level can be calculated using:

• \( r_n = a_0 \frac{n^2}{Z} \)

This formula highlights that the radius is dependent on:

• The principal quantum number \( n \)
• The Bohr radius \( a_0 \), which is a constant \( 0.529 \, \text{Å}\)
• The nuclear charge \( Z \)

In the given problem, when comparing the electron orbital radius in Be^{3+}, which is triply ionized with \( Z=4 \), we align the conditions with those in a hydrogen atom to find that the state \( n=2 \) for Be^{3+} matches the ground state orbital radius of hydrogen.

Bohr Model

The Bohr Model is a seminal theory in atomic physics, developed by Niels Bohr in 1913, to describe the structure of atoms. This model revolutionized our understanding by proposing that:

• Electrons orbit the nucleus in discrete paths or quantized orbits with fixed energy.
• These orbits are stable, and electrons do not radiate energy while they remain in a given orbit.

This model simplifies the complexity of an atom by focusing on specific allowed orbits rather than a cloud of probabilities.For a hydrogen atom and hydrogen-like systems, the Bohr Model provides a quantitative scope of electron radii and energy states. Each orbit is associated with a specific quantum number \( n \), which accounts for:

• The size of the orbit \( r_n \)
• The energy level of the electron

In cases like Be^{3+} in the Bohr Model, we calculate the electron's location and potential radius by taking into account the more significant nuclear charge which affects the orbital radius and energy states.

Energy States

In the context of atomic physics, energy states refer to the distinct energy levels that electrons can occupy around a nucleus. These states are quantized, which means they can only take on specific energy values, not a continuous range.The energy of an electron in a particular orbit or state can be calculated using:

• \( E_n = - \frac{Z^2 R_H}{n^2} \)

where \( Z \) is the atomic number, \( n \) is the principal quantum number, and \( R_H \) is the Rydberg constant. Each energy state is associated with a different \( n \), with the ground state having the lowest energy level at \( n=1 \).As shown in the problem, the triply ionized beryllium \( Be^{3+} \) adapts these calculations to account for its higher charge \( Z = 4 \). When we find that \( n=2 \), it means this specific energy level where Be^{3+} shares the same orbital characteristics as the hydrogen ground state. Adjusting the energy states by such calculations is crucial for understanding atomic behavior.

Ionized Beryllium

Ionized beryllium, particularly triply ionized \( Be^{3+} \), provides an interesting analogy with hydrogen due to its single remaining electron orbiting around a nucleus with a nuclear charge \( Z = 4 \). Such atoms are analyzed in the same manner as hydrogen atoms due to the simplicity of the single electron configuration, however, the higher nuclear charge makes calculations more complex.In the case of \( Be^{3+} \), it behaves similarly to a hydrogen atom but with heightened effects due to the increased charge.

• Influence on the electron orbit: the singular electron feels a significantly stronger attraction to the nucleus, affecting both orbital radius and energy states.

The challenge often lies in comparing such ions to hydrogen's ground state conditions. In the exercise, we equate this stronger attraction by looking for an energy state \( n \) where the reduced radius induced by increased nuclear charge matches hydrogen’s orbital radius.