Problem 11 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The ground state energy of hydrogen atom is \(-13.6 \mathrm{eV}\). The energy of second excited state of \(\mathrm{He}^{+}\)ion in \(\mathrm{eV}\) is: (a) \(-54.4\) (b) \(-3.4\) (c) \(-6.04\) (d) \(-27.2\)

Step-by-Step Solution

Answer

The energy of the second excited state is \(-6.04\) eV, which is option (c).

Step 1: Understand the Formula

The energy of an electron in a hydrogen-like atom is given by the formula: \[ E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV} \] where \(Z\) is the atomic number, and \(n\) is the principal quantum number of the state.

Step 2: Identify Parameters

For the helium ion (\(\text{He}^+\)), the atomic number \(Z\) is 2. The second excited state corresponds to \(n = 3\).

Step 3: Plug Values into the Formula

Substituting \(Z = 2\) and \(n = 3\) into the formula, we calculate the energy:\[ E_3 = -13.6 \times \frac{2^2}{3^2} \text{ eV} \] .

Step 4: Calculate the Energy

Calculate the value:\[ E_3 = -13.6 \times \frac{4}{9} \] = \(-13.6 \times 0.4444\) eV = \(-6.04\) eV.

Step 5: Assess the Result with Given Options

The calculated energy for the second excited state of \(\text{He}^+\) ion is \(-6.04\) eV, which corresponds to option (c).

Key Concepts

Atomic Structure

All matter is composed of atoms, making this concept the basic building block of chemistry and physics. An atom consists of three main components: neutrons and protons located in the nucleus, and electrons orbiting around the nucleus.

The atomic structure is critical because it determines the chemical properties and reactivity of an element.

• Neutrons have no electrical charge and influence the atomic mass.
• Protons have a positive charge and determine the element's identity, aligning with the atomic number denoted by \(Z\).
• Electrons are negatively charged particles that move in orbitals around the nucleus.

An understanding of atomic structure allows us to explain phenomena such as bonding and atomic interactions.

In the context of quantum chemistry, it's essential to recognize that electrons exist in defined energy levels, which we refer to as orbitals, and their behavior is described by quantum mechanics.

Hydrogen-like Atom

Hydrogen-like atoms offer a simpler model for studying quantum characteristics. These are ions that contain only one electron, similar to the hydrogen atom, but may have more than one proton.

In a hydrogen atom, there's one proton in the nucleus and one electron. For hydrogen-like ions such as helium ions \((\text{He}^+)\), electron behavior models those of the hydrogen atom due to similar structural simplicity. Here, helium has a nucleus composed of two protons, but still only one electron after ionization.

This simplicity allows the use of hydrogen calculation models, making it easier to understand quantum states. The atomic number \(Z\) plays a critical role here:

• It defines the electrostatic force acting on the electron, since protons determine this attractive force.
• Higher \(Z\) values mean greater attraction, influencing the electron's energy in different states.

Understanding hydrogen-like atoms enables us to analyze how these forces influence the electron's position and energy levels, contributing to the broader study of atomic behavior.

Energy Levels

In atomic physics, particularly when discussing quantum systems, energy levels are the discrete energies that the electrons of an atom or ion can possess. Electrons move between these levels by absorbing or emitting photons of specific energies.

The formula used to determine the energy of hydrogen-like atoms is \[ E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV} \]where \(Z\) is the atomic number and \(n\) is the principal quantum number.

This principal quantum number \((n)\) is critical as it indicates the electron's energy level or shell.

• A lower \(n\) means the electron is closer to the nucleus and more tightly bound, having more negative energy.
• A higher \(n\) suggests that the electron is further out, with higher (less negative) energy.

In the solved problem, \(n = 3\) was used to determine the energy in a second excited state for \(\text{He}^+\), showcasing how different states correspond to specific configurations.

Understanding these energy levels helps in explaining spectral lines, the light absorbed and emitted by electrons, and plays a significant role in technological applications like lasers and quantum computing.