Problem 26 - Master Resource Book in JEE Main Physics

Question

A hydrogen atom and a \(\mathrm{Li}^{2+}\) ion are both in second excited state. If \(l_{\mathrm{H}}\) and \(l_{\mathrm{Li}}\) are their respective electronic angular momenta and \(E_{\mathrm{H}}\) and \(E_{\mathrm{Li}}\) their respective energies, then (a) \(l_{H}>I_{\mathrm{U}}\) and \(\left|E_{\mathrm{H}}\right|>\left|E_{\mathrm{L}}\right|\) (b) \(L_{\mathrm{H}}=L_{i}\) and \(\left|E_{\mathrm{H}}\right|<\left|E_{\mathrm{U}}\right|\) (c) \(l_{H}>I_{\mathrm{L}}\) and \(\left|E_{\mathrm{H}}\right|>\left|E_{\mathrm{U}}\right|\) (d) \(l_{H}>l_{\mathrm{L}}\) and \(\left|E_{\mathrm{H}}\right| \ll\left|E_{\mathrm{L} \mid}\right|\)

Step-by-Step Solution

Answer

The correct option is (b): \( L_{\mathrm{H}} = L_{\mathrm{i}} \) and \( \left|E_{\mathrm{H}}\right| < \left|E_{\mathrm{Li}}\right| \).

Step 1: Determine Quantum Numbers

For an atom in the second excited state, the principal quantum number is given by \( n = 3 \). The angular momentum quantum number \( l \) can take on values from \( 0 \) to \( n-1 \). So, \( l = 0, 1, \) or \( 2 \).

Step 2: Calculate Angular Momentum

The angular momentum \( l \) for both the hydrogen atom and the \( \mathrm{Li}^{2+} \) ion is determined by the quantum number \( l \). For a given principal quantum number \( n = 3 \), \( l \) can have a maximum value of \( 2 \). Since \( l_H \) and \( l_{Li} \) are both in second excited state, then \( l_H = l_{Li} = 2 \).

Step 3: Calculate Energy Levels

Energy of a hydrogen-like atom is given by \( E_n = -\dfrac{13.6 \, \text{eV} \times Z^2}{n^2} \). For hydrogen \( (Z=1) \), \( E_H = -\dfrac{13.6}{3^2} = -1.51 \, \text{eV} \). For \( \mathrm{Li}^{2+} \) \( (Z=3) \), \( E_{Li} = -\dfrac{13.6 \times 9}{3^2} = -13.6 \, \text{eV} \).

Step 4: Compare Angular Momentum and Energy

Comparing \( l_H \) and \( l_{Li} \) we have \( l_H = l_{Li} \). In terms of energy, since \( |E_H| = 1.51 \, \text{eV} \) and \( |E_{Li}| = 13.6 \, \text{eV} \), we have \( |E_H| < |E_{Li}| \).

Step 5: Determine the Correct Statement

Based on the angular momentum and energy values calculated, the correct option is (b): \( L_{\mathrm{H}}=L_{\mathrm{i}} \) and \( \left|E_{\mathrm{H}}\right|<\left|E_{\mathrm{Li}}\right| \).

Key Concepts

Electronic Angular Momentum

In quantum mechanics, electronic angular momentum is a fundamental property of electrons orbiting a nucleus. It describes the motion of an electron around the nucleus. This property is quantized, meaning it can only take on specific discrete values.

The angular momentum quantum number, often denoted as \( l \), determines these values. For each electron, \( l \) ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number. This results in electrons having a certain angular momentum value determined by \( l \).

• If \( n = 3 \), as in our exercise, then \( l \) can be 0, 1, or 2.
• For the hydrogen atom and \( \mathrm{Li}^{2+} \) ion both in their second excited state, this means \( l \) can be 2.

Energy Levels

The energy levels of hydrogen-like atoms, such as a hydrogen atom or \( \mathrm{Li}^{2+} \) ion, are determined by their electron arrangements around the nucleus. Each energy level corresponds to a different orbit or shell of an electron.

These levels are influenced by both the principal quantum number, \( n \), and the atomic number, \( Z \). The formula to calculate these energy levels is given by:\[E_n = -\dfrac{13.6 \, \text{eV} \times Z^2}{n^2}\]

• For a hydrogen atom, with \( Z = 1 \), the energy of the second excited state \( (n = 3) \) is \( -1.51 \, \text{eV} \).
• For \( \mathrm{Li}^{2+} \), where \( Z = 3 \), the energy is \( -13.6 \, \text{eV} \).

The negative sign indicates that energy is released when moving to a lower energy level, making the electrons more tightly bound to the nucleus.

Quantum Numbers

Quantum numbers are integral to understanding the structure of atoms and the behavior of electrons. Each electron in an atom is described by a set of four quantum numbers that define its state. Let's explore these crucial identifiers:

• Principal Quantum Number \( n \): This number labels the electron's shell and determines its energy level. Higher \( n \) values mean larger orbitals and higher energy levels.
• Angular Momentum Quantum Number \( l \): This determines the shape of the electron's orbital and contributes to its angular momentum. For each \( n \), \( l \) can have integer values from 0 to \( n-1 \).
• Magnetic Quantum Number \( m_l \): This describes the orientation of the orbital in space and can range from \( -l \) to \( +l \).
• Spin Quantum Number \( s \): This refers to the intrinsic spin of the electron, which can be either +1/2 or -1/2.

A comprehensive understanding of these quantum numbers allows us to predict and explain the configuration of electrons in different atoms, influencing their chemical properties and interactions.