Problem 72 - Chemistry: The Central Science

Question

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) \(n=3, m_{l}=-2 ;\) (b) \(n=4\), \(l=3 ;(\mathrm{c}) n=5, l=3, m_{l}=2,(\mathrm{~d}) n=4, l=1, m_{l}=0\).

Step-by-Step Solution

Answer

The maximum number of electrons for each given set of quantum numbers is as follows: a) 2 electrons b) 14 electrons c) 2 electrons d) 2 electrons

Step 1: a) \(n=3\), \(m_l=-2\)

The given quantum numbers are \(n=3\) and \(m_l=-2\). Since there are no restrictions on the value of \(l\) and \(m_s\), Possible values of \(l\): \(0, 1, 2\) Possible values of \(m_s\): \(-1/2\), \(1/2\) Thus, the maximum number of electrons for the given values of \(n\) and \(m_l\) is 2.

Step 2: b) \(n=4\), \(l=3\)

The given quantum numbers are \(n=4\) and \(l=3\). For \(l=3\), the possible values of \(m_l\) are \(-3, -2, -1, 0, 1, 2, 3\). Each of these \(m_l\) values can have two \(m_s\) values, namely \(-1/2\) and \(1/2\). Thus, the maximum number of electrons for the given values of \(n\) and \(l\) is \(7 \times 2 = 14\).

Step 3: c) \(n=5\), \(l=3\), \(m_l=2\)

The given quantum numbers are \(n=5\), \(l=3\), and \(m_l=2\). Since \(m_l\) and \(l\) are specified, there is no freedom to choose other values. The only remaining quantum number is \(m_s\), which can be either \(-1/2\) or \(1/2\). Thus, the maximum number of electrons for the given values of \(n\), \(l\), and \(m_l\) is 2.

Step 4: d) \(n=4\), \(l=1\), \(m_l=0\)

The given quantum numbers are \(n=4\), \(l=1\), and \(m_l=0\). Since both \(l\) and \(m_l\) have specified values, there is no freedom to choose other values. The only remaining quantum number is \(m_s\), which can be either \(-1/2\) or \(1/2\). Thus, the maximum number of electrons for the given values of \(n\), \(l\), and \(m_l\) is 2.

Key Concepts

Atomic Orbitals

Understanding atomic orbitals is fundamental in exploring the arrangement of electrons within atoms. Orbitals are regions within an atom where there is a high probability of finding an electron. Each orbital corresponds to specific energy levels denoted by the quantum number 'n'.

For any value of the principal quantum number, 'n', there are 'n^2' orbitals. For instance, if 'n=3', there are 3^2, that is 9 orbitals in total. These orbitals are further categorized based on the angular quantum number 'l', which ranges from 0 to 'n-1' and designates the shape of the orbital, commonly known as 's', 'p', 'd', or 'f'.

The magnetic quantum number 'm_l', another crucial aspect, determines the orientation of the orbital in space and takes on values from '-l' to '+l', including zero. Therefore, for an orbital where 'n=3' and 'l=2' (a 'd' orbital), 'm_l' can be -2, -1, 0, 1, or 2, each orientation hosting up to 2 electrons due to electron spin—described by the spin quantum number 'm_s', which can be either -1/2 or +1/2.

Electron Configuration

Electron configuration outlines how electrons are distributed among the atomic orbitals in an atom. It follows the Pauli Exclusion Principle, which states that no two electrons can have the identical set of four quantum numbers. As a result, each electron in an atom has a unique 'address' consisting of 'n', 'l', 'm_l', and 'm_s'.

The arrangement of electrons starts with the lowest energy level, gradually filling up to higher levels—a process described by the Aufbau principle. Notably, the sublevels within a principal energy level can be of different energies, with 's' being the lowest, followed by 'p', then 'd', and 'f' respectively.

When looking at quantum numbers to determine the maximum number of electrons, for example in the case of 'n=4, l=3', this implies an 'f' orbital at the fourth energy level. Since 'l=3', the values of 'm_l' range from -3 to +3, making a total of 7 possible orientations, each capable of holding 2 electrons accounting for spins, resulting in a maximum of 14 electrons for this condition.

Quantum Mechanical Model

The quantum mechanical model of the atom presents a sophisticated framework for understanding atomic and subatomic behaviors. It portrays electrons not as particles in fixed orbits but as wave-like entities in orbitals with calculated probabilities of their presence at any given point.

This model is defined by the set of four quantum numbers: principal (n), angular momentum (l), magnetic (m_l), and spin (m_s), each detailing different characteristics and restrictions for electron placement within the atom. Together, they give an electron its unique state and help chemists predict chemical and physical properties of elements.

In exercises asking for the maximum number of electrons with specific quantum numbers, we apply all these principles. For example, given 'n=5, l=3, m_l=2', it specifies a single 'f' orbital at the fifth energy level, with a fixed orientation. With only two possible spin states, the maximum number of electrons here is restricted to just 2, reflecting the exclusivity and specificity imparted by quantum numbers in the realm of quantum mechanics.