Let T1 and T2 be the energy of an electron in the first and second excited states of hydrogen atom

Let T1 and T2 be the energy of an electron in the first and second excited states of hydrogen atom, respectively. According to the Bohr’s model of
an atom, the ratio T1 : T2 is: [NEET]
(a) 4 : 1 (b) 4 : 9 (c) 9 : 4 (d) 1 : 4
🔬 *Understanding the Ratio of Electron Energies in Excited States | NEET Physics* ⚛️
Dive into this intriguing question from *atomic physics**, which highlights the relationship between electron energy levels in the hydrogen atom as described by **Bohr's model**. Mastering this concept is essential for excelling in **NEET Physics* and builds a strong foundation for understanding atomic structure.
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*Problem Statement*
The question is:
Let \( T_1 \) and \( T_2 \) represent the energy of an electron in the *first excited state* (\( n = 2 \)) and *second excited state* (\( n = 3 \)) of a hydrogen atom, respectively. According to **Bohr’s model**, find the ratio \( T_1 : T_2 \).
Options:
(a) \( 4 : 1 \)
(b) \( 4 : 9 \)
(c) \( 9 : 4 \)
(d) \( 1 : 4 \)
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*Key Concepts from Bohr’s Model*
1️⃣ **Energy Quantization**:
Electrons in a hydrogen atom occupy specific energy levels, defined by the principal quantum number (\( n \)). The energy of these levels decreases as the electron moves closer to the nucleus.
2️⃣ **Relationship Between Energy and \( n \)**:
The total energy of an electron in the \( n \)-th orbit is inversely proportional to the square of \( n \). This relationship helps us compare the energy at different energy levels without needing detailed calculations.
3️⃣ **Energy and Stability**:
Lower energy levels correspond to more stable orbits, while higher energy levels represent states where the electron is less tightly bound to the nucleus.
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*Theory-Driven Solution*
The energy levels in the Bohr model are represented by the formula:
\[
T_n \propto -\frac{1}{n^2}
\]
The *first excited state* corresponds to \( n = 2 \).
The *second excited state* corresponds to \( n = 3 \).
Using the proportionality above, we see that the energy decreases rapidly as \( n \) increases. The ratio of energies \( T_1 : T_2 \) is directly related to the squares of the quantum numbers \( n_2^2 \) and \( n_3^2 \).
Thus, the ratio becomes:
\[
T_1 : T_2 = 9 : 4
\]
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*Why the Ratio is 9:4?*
Energy is inversely proportional to the square of the principal quantum number.
For the first excited state (\( n = 2 \)), the denominator is smaller compared to the second excited state (\( n = 3 \)).
The larger denominator for \( n = 3 \) reduces the energy, making the energy of the second excited state less negative than that of the first excited state.
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*Final Answer*
The ratio of electron energies in the first and second excited states is:
*(c) 9 : 4* 🎯
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*Why This Concept Is Important for NEET?*
This question combines several essential aspects of **Bohr’s atomic model**, including:
The quantization of energy levels.
The inverse relationship between energy and \( n^2 \).
How energy levels determine the stability of electrons in an atom.
It strengthens your ability to analyze **atomic structure**, a topic frequently tested in **NEET Physics**.
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📌 *Key Learning Points*
1. **Bohr’s Model Limitations**: While it explains hydrogen-like atoms well, it cannot accurately describe multi-electron systems.
2. **Energy Levels in Hydrogen Atom**: Electrons transition between these levels by absorbing or emitting energy, producing characteristic spectral lines.
3. **NEET Relevance**: Understanding energy transitions is crucial for solving problems on spectra, ionization energy, and atomic models.
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