Question

The hydrogen-like species \(\mathrm{Li}^{2+}\) is in a spherically symmetric state \(\mathrm{S}_{1}\) with one radial node. Upon absorbing light the ion undergoes transition to a state \(S_{2}\). The state \(S_{2}\) has one radial node and its energy is equal to the ground state energy of the hydrogen atom. Energy of the state \(S_{1}\) in units of the hydrogen atom ground state energy is A) \(0.75\) B) \(1.50\) C) \(2.25\) D) \(4.50\)

Short answer

The energy of state \(S_1\) is \(2.25\) times the ground state energy of hydrogen, so the answer is C) \(2.25\).

Step-by-step solution

Understand the Energy Levels of Hydrogen-like Ions

Hydrogen-like ions have energy levels described by the formula \( E_n = \frac{-Z^2 E_0}{n^2} \), where \( Z \) is the atomic number, \( E_0 \) is the ground state energy of hydrogen, and \( n \) is the principal quantum number. For \( \mathrm{Li}^{2+} \), \( Z = 3 \).

Determine the Principal Quantum Number for State \(S_1\)

State \(S_1\) for \(\mathrm{Li}^{2+}\) has one radial node. The number of radial nodes is given by \( n - l - 1 \), where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number. Since it's an S state, \( l = 0 \), and we have 1 radial node, so \( n - 1 = 1 \). Hence, \( n = 2 \) for state \(S_1\).

Calculate the Energy of State \(S_1\)

Using the formula from Step 1, the energy for state \(S_1\) with \( n = 2 \) and \( Z = 3 \) is calculated as \( E_1 = \frac{-3^2 E_0}{2^2} = \frac{-9 E_0}{4} = -2.25 E_0 \). Since the energy of states is negative and we're comparing magnitudes, the ground state energy is \(0.25 E_0\). Therefore state \(S_1\)'s energy is 9 times the ground state energy of hydrogen, not four times.

Compare With Given Answers

Given the magnitudes of energy, State \(S_1\) has an energy equal to \(9 \times\) the energy of the hydrogen ground state, meaning its energy is \(2.25 \times\) the hydrogen atom's ground state energy in magnitude. The correct answer is therefore C) \(2.25\).

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides the description of the physical properties of nature at the scale of atoms and subatomic particles. It establishes a different framework compared to classical mechanics by introducing quantized energy levels, probability waves, and uncertainty principles.

In the context of this exercise, quantum mechanics explains why hydrogen-like ions, such as \(\mathrm{Li}^{2+}\), can only absorb or emit light at certain wavelengths to transition between discrete energy states. Each state corresponds to a certain energy level that is determined by quantum numbers. Understanding these energy levels allows us to predict the behavior of electrons when they interact with light or transition between different states.

Atomic Structure

The atomic structure of any element is defined by its nucleus, consisting of protons and neutrons, and the electron cloud surrounding the nucleus. Electrons within this cloud exist in regions called orbitals, which are defined by their energy levels and the probabilities of the electrons occupying them.

For hydrogen-like ions, the idea is similar but simplified, considering there is only one electron. The energy levels of the ion are influenced by the effective nuclear charge (Z) which changes the energy required for an electron to transition between levels. In our exercise, \(\mathrm{Li}^{2+}\) has a \(Z\) value of 3, which significantly alters its energy levels compared to a neutral hydrogen atom, reflected in the calculation of state energies.

Spectroscopy

Spectroscopy is a technique used to measure and analyze the spectrum of light emitted or absorbed by substances. It's crucial in understanding the electronic transitions that occur in atoms and molecules. When an electron transitions between different energy levels, it either absorbs or emits photons with very specific energies, resulting in distinct spectral lines.

This process is what the exercise describes when \(\mathrm{Li}^{2+}\) absorbs light to move from state \(S_{1}\) to \(S_{2}\). The energy difference between the two states is directly related to the frequency of the light absorbed, as described by the Planck-Einstein relation, which connects the energy of a photon to its spectral frequency.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is one of the four quantum numbers used to describe the quantum state of an electron in an atom. It relates to the size of the orbital and therefore the energy level of an electron, with higher values of \(n\) corresponding to higher energy levels.

In the exercise, determining the principal quantum number for the state \(S_1\) was crucial for calculating the energy of the ion in that state. The principal quantum number provides insight into the number of radial nodes in an electron’s orbital, which in turn helped us find the correct energy level of the hydrogen-like ion. Using the value of the principal quantum number \(n\), we were able to apply the quantum mechanical formula for the energy levels of hydrogen-like ions, leading to the correct solution.