The energy of He+ ion in its first excited state is (The ground state energy of the hydrogen atom is

The energy of He+ ion in its first excited state is, (The ground state energy of the hydrogen atom is -13.6 eV) [JEE]
(a) -27.2 eV (b) -13.6 eV
(c) -54.4 eV (d) -3.4 eV
🔬 *Energy of He⁺ Ion in Its First Excited State | JEE Physics Explained* ⚛️
Welcome to a deep dive into *atomic physics**, where we explore the fascinating energy levels of **hydrogen-like ions* such as *He⁺**. This video simplifies the concepts of **energy quantization* and the relationship between energy levels and atomic structure. These ideas are critical for excelling in *JEE Physics* and understanding the behavior of single-electron systems.
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*Problem Statement*
Determine the energy of a *He⁺ ion* in its **first excited state**.
Given data:
Ground state energy of a hydrogen atom = \(-13.6 \, \text{eV}\).
Options:
(a) \(-27.2 \, \text{eV}\)
(b) \(-13.6 \, \text{eV}\)
(c) \(-54.4 \, \text{eV}\)
(d) \(-3.4 \, \text{eV}\)
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*Understanding Hydrogen-like Ions*
#### *What Are Hydrogen-like Ions?*
*Hydrogen-like ions* are atoms with only one electron, such as **He⁺**, **Li²⁺**, etc.
They behave similarly to hydrogen but with a stronger electrostatic attraction between the nucleus and the electron due to the increased **atomic number (\( Z \))**.
This makes their energy levels deeper (more negative) compared to hydrogen.
#### *Energy Levels in Hydrogen-like Atoms*
In these systems, electron energy is **quantized**, meaning it can only occupy specific levels corresponding to different principal quantum numbers (\( n \)).
The energy becomes less negative as \( n \) increases, showing that the electron is less tightly bound to the nucleus.
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*Why the Energy of He⁺ Is Different from Hydrogen*
In a hydrogen atom, the nucleus has a single proton, but in **He⁺**, the nucleus contains two protons. This increases the nuclear attraction felt by the electron, pulling it closer and making its energy levels more negative.
For a hydrogen atom, the ground state energy is \(-13.6 \, \text{eV}\). In **He⁺**, the ground state energy is scaled by the square of the atomic number \( Z \).
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*The First Excited State*
The *first excited state* of any atom corresponds to \( n = 2 \), where the electron is further from the nucleus compared to the ground state (\( n = 1 \)).
For **He⁺**, the energy in the first excited state is less negative than the ground state, reflecting reduced binding strength as the electron moves to a higher orbit.
By using the proportional relationship between energy and \( Z^2 \), the energy in the first excited state of *He⁺* is:
\[
\text{Energy of He⁺ at } n = 2 = -13.6 \, \text{eV}.
\]
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*Final Answer*
The energy of the He⁺ ion in its first excited state is:
*(b) \(-13.6 \, \text{eV}\)* 🎯
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*Key Insights for JEE Aspirants*
1️⃣ **Energy Quantization**: Bohr’s model reveals that energy is inversely proportional to \( n^2 \) and directly dependent on \( Z^2 \).
2️⃣ **Hydrogen-like Ions**: The behavior of single-electron ions like He⁺, Li²⁺, etc., provides a deeper understanding of atomic structure and electrostatic interactions.
3️⃣ **Excited States**: Understanding transitions between energy levels is essential for questions on atomic spectra and ionization energy.
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*Why This Concept Is Important for JEE?*
The problem demonstrates fundamental principles of quantum mechanics and atomic physics, helping aspirants:
Solve problems involving **energy transitions**.
Understand the effects of *nuclear charge* on energy levels.
Analyze hydrogen-like atoms, a recurring topic in **JEE Physics**.
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