Problem 166 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements in relation to the hydrogen atom is correct? \([2005]\) (a) \(3 \mathrm{~s}\) and \(3 \mathrm{p}\) orbitals are of lower energy than \(3 \mathrm{~d}\) orbital (b) 3p orbital is lower in energy than \(3 \mathrm{~d}\) orbital (c) 3 s orbital is lower in energy than \(3 \mathrm{p}\) orbital (d) \(3 \mathrm{~s}, 3 \mathrm{p}\) and \(3 \mathrm{~d}\) orbitals all have the same energy

Step-by-Step Solution

Answer

Option (d) is correct: all n=3 orbitals have the same energy in hydrogen.

Step 1: Understanding the Problem

Identify that we need to find the correct relation of energy levels among the orbitals in a hydrogen atom.

Step 2: Hydrogen Atom Energy Levels

Recall that in a hydrogen atom, the energy levels depend only on the principal quantum number n, and not on the azimuthal quantum number l. Therefore, all orbitals with the same n (principal quantum number) will have the same energy.

Step 3: Principal Quantum Number

Identify from the given options that all options refer to orbitals with n=3 (3s, 3p, 3d). Thus, according to the energy level rule for hydrogen, they should all have the same energy.

Step 4: Analyzing Options

Evaluate each option: - Option (a) suggests 3s and 3p have lower energy than 3d. This is incorrect because they all have the same energy. - Option (b) suggests 3p is lower than 3d. This is incorrect for the same reason. - Option (c) suggests 3s is lower than 3p, also incorrect. - Option (d) suggests they all have the same energy, which is correct for hydrogen atoms.

Step 5: Conclusion

Since the energy levels for all n=3 orbitals in hydrogen are the same, the correct statement is that all n=3 orbitals have the same energy level, as mentioned in option (d).

Key Concepts

Principal Quantum Number

In the world of quantum chemistry, the principal quantum number, denoted as \( n \), plays a crucial role. It essentially tells us the size and energy level of an electron's orbital within an atom. The principal quantum number is always a positive integer, starting from 1 and moving upwards. Each increase in \( n \) signifies a new energy level. For instance, electrons in the \( n = 1 \) level are closest to the nucleus and have the lowest energy.
Understanding \( n \) is important because it determines the major energy distinction between different orbitals. All orbitals belonging to the same \( n \) value are said to occupy the same energy shell. In the case of the hydrogen atom, this quantum number is particularly relevant, because the energy of an electron solely depends on \( n \). Other atoms are more complex, as they involve more factors such as the presence of multiple electrons and inter-electronic interactions, but hydrogen remains unique due to its simplicity.
In the context of our exercise, we are dealing with orbitals where \( n = 3 \). This tells us that all the orbitals mentioned — 3s, 3p, and 3d — are part of the third energy level, meaning they should have the same energy.

Orbital Energy Levels

The concept of energy levels in an atom describes the fixed energy that electrons possess within different orbitals. Orbital energy levels play a significant part in quantum chemistry, especially when analyzing atoms like hydrogen. In multi-electron atoms, energy levels are split based on the azimuthal quantum number \( l \), which gives rise to the commonly known s, p, d, and f orbitals.
However, the hydrogen atom simplifies this complexity. Rather than being affected by \( l \), energy levels in hydrogen atoms depend solely on the principal quantum number \( n \). This means that all orbitals, irrespective of their shape or type (s, p, d), will have identical energy levels if they share the same \( n \). For instance, in our exercise, since the 3s, 3p, and 3d orbitals all belong to \( n = 3 \), they share the same energy.
This simplification introduces a clear difference compared to other elements, where electron-electron interactions and varying \( l \) values alter the energy levels significantly. It's crucial for students to get comfortable with this concept to properly understand how hydrogen, despite its apparent simplicity, serves as a foundational principle for more complex atoms.

Quantum Chemistry

Quantum chemistry is a fascinating branch of chemistry that uses quantum mechanics to explain the behavior of electrons in atoms and molecules. It provides the theoretical framework to understand atomic and molecular structure, bonding, and the behavior of particles at the atomic level.
In its essence, quantum chemistry allows us to understand why atoms and molecules behave the way they do. This includes why certain elements form bonds in a particular way, why they adopt specific shapes in space, and how they interact with each other. The hydrogen atom, with its single proton and electron, is the simplest atom and serves as a basic model for understanding these concepts.
Through quantum chemistry, we learn how to utilize quantum numbers, like the principal quantum number \( n \), to predict the arrangement of electrons in atoms. By studying these quantum properties, chemists can determine how electrons will fill available energy levels and predict chemical reactivity and properties of elements.

• Understand electron configurations using quantum numbers.
• Examine periodic trends and chemical behaviors.
• Predict molecular geometry and bonding.

These insights are critical for advancing technology and material science and for developing new pharmaceuticals and energy solutions. Mastering quantum chemistry thus opens doors to a multitude of scientific discoveries and innovations.