Problem 129 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements is /are correct? (a) The energy of an electron is largely determined by its principal quantum number. (b) The energy of electron in an orbital in the hydrogen atom depends upon the value of principal quantum number only. (c) The value of principal quantum number for \(24^{\text {t }}\) electron is 3 . (d) The principal quantum number is a measure of the most probable distance of finding the electron around the nucleus

Step-by-Step Solution

Answer

All statements (a, b, c, d) are correct.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the properties of electrons in atoms. The principal quantum number, denoted as \(n\), is one of these quantum numbers and it primarily determines the energy level of an electron in an atom. It also helps in describing the size of the electron cloud.

Step 2: Analyzing Statement (a)

Statement (a) claims that the energy of an electron is largely determined by its principal quantum number. This statement is generally correct because the principal quantum number \(n\) largely influences the energy level and size of the electron's orbit.

Step 3: Analyzing Statement (b)

Statement (b) discusses electron energy in hydrogen and suggests it depends solely on the principal quantum number \(n\). For hydrogen, this is correct, as the energy in hydrogen-like atoms is determined only by \(n\) due to the presence of only one electron.

Step 4: Analyzing Statement (c)

Statement (c) claims the principal quantum number for the 24th electron is 3. In the case of iron (atomic number 26), the 24th electron is likely in the 3d orbital, which does indeed have a principal quantum number of 3. Thus, this statement is correct.

Step 5: Analyzing Statement (d)

Statement (d) states that the principal quantum number is a measure of the most probable distance of finding the electron around the nucleus. This is true, since \(n\) generally reflects the radial distance from the nucleus and helps describe the electron's probable location in space.

Key Concepts

Principal Quantum Number

The concept of the principal quantum number is fundamental in understanding how electrons are arranged in an atom. This quantum number is symbolized by \( n \), and it can take positive integer values like 1, 2, 3, etc.
It primarily determines the energy level of an electron in an atom. Higher values of \( n \) indicate electrons are located in higher energy levels. As \( n \) increases, the size of the electron cloud becomes larger, meaning that electrons are further away from the nucleus.

• For each atom, the principal quantum number dictates the shells or energy levels where electrons can reside.
• Electrons with the same \( n \) value have similar energy, but the energy can vary slightly due to other factors.

The principal quantum number also has implications for the electron's position. It gives a rough idea of the average distance of electrons from the nucleus. The higher the \( n \), the greater the average distance, contributing to the determination of the atom's size.

Electron Orbitals

Electron orbitals are regions in an atom where there is a high probability of finding electrons. These are defined by quantum numbers, including the principal quantum number \( n \).
Orbitals are depicted as clouds, with variations in size and energy based on their quantum numbers.

• Each orbital can hold a maximum of two electrons, according to the Pauli exclusion principle.
• Different types of orbitals, such as \( s \), \( p \), \( d \), and \( f \), have distinct shapes and orientations.

When it comes to electron configuration, considering both the principal quantum number and the type of orbital is crucial. Orbitals with lower \( n \) values fill up first, following the Aufbau principle. However, this filling order can vary slightly in multi-electron atoms due to energy differences amongst orbital types.

Hydrogen Atom Energy Levels

In a hydrogen atom, which contains only one electron, the energy levels are determined solely by the principal quantum number \( n \). This simplification occurs because there are no electron-electron interactions to consider.
As \( n \) increases, the energy levels become closer together, meaning the energy difference between consecutive orbits narrows.

• All orbitals within the same principal quantum level have the same energy, known as degeneracy, in a hydrogen atom.
• The energy of an electron in hydrogen is proportional to \(-\frac{1}{n^2}\), where \( n \) is the principal quantum number.

Unlike many-electron atoms, the hydrogen atom's energy levels are not affected by the angular momentum quantum number \( l \). This unique trait simplifies calculations and makes the hydrogen atom a model system in quantum mechanics.