Question

Which of the following statements is false? (1) For azimuthal quantum number \(l=3\), the maximum number of electrons will be 14 . (2) The concept of electron spin was introduced by Uhlenbeck and Goudsmit. (3) The principal quantum number of an atom represents size of the orbit and distance of electron from nucleus. (4) The shape of an orbital is governed by magnetic quantum number.

Short answer

Statement (4) is false.

Step-by-step solution

- Understanding Electron Capacity for Azimuthal Quantum Number

For an azimuthal quantum number ( l = 3 ), the electrons occupy the f orbitals. Each f orbital can contain a maximum of 14 electrons because there are 7 orbitals and each orbital can hold 2 electrons. So, statement (1) is true.

- Electron Spin Concept

The concept of electron spin was indeed introduced by Uhlenbeck and Goudsmit. Hence, statement (2) is true.

- Principal Quantum Number

The principal quantum number ( n ) represents the size of the orbit and the average distance of the electron from the nucleus. As such, statement (3) is true.

- Shape of Orbitals and Magnetic Quantum Number

The shape of an orbital is actually governed by the azimuthal quantum number ( l ), not the magnetic quantum number ( m_l ), which dictates the orientation of the orbital in space. Therefore, statement (4) is false.

Key concepts

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number or secondary quantum number, is symbolized by the letter \( l \). This quantum number defines the shape of an electron's orbital and can take on any integer value from 0 to \( n-1 \), where \( n \) is the principal quantum number.

For each value of \( l \), there is a corresponding type of orbital:

• \( l = 0 \) corresponds to an s orbital
• \( l = 1 \) corresponds to a p orbital
• \( l = 2 \) corresponds to a d orbital
• \( l = 3 \) corresponds to an f orbital

In the given exercise, \( l = 3 \) for f orbitals. Each f orbital can hold a total of 14 electrons because there are 7 different orientations of an f orbital, each capable of holding 2 electrons (one with spin-up and one with spin-down). Thus, the statement regarding the maximum number of electrons being 14 is true.

Electron Spin

Electron spin, discovered by Samuel Goudsmit and George Uhlenbeck in 1925, is a fundamental property of electrons. This quantum number is often denoted as \( s \) or \( m_s \) and can have a value of either +1/2 or -1/2.

Electron spin describes the intrinsic angular momentum of electrons, which contributes to their magnetic moment. In simpler terms, it's as if the electron is spinning around its own axis, although this is a somewhat simplified picture.

The two possible spins (spin-up or spin-down) allow each orbital to hold two electrons, one with each spin type. This is key to understanding the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same four quantum numbers. Therefore, the statement that the concept of electron spin was introduced by Uhlenbeck and Goudsmit is true.

Principal Quantum Number

The principal quantum number, noted as \( n \), determines the overall size and energy level of an electron's orbital. It is always a positive integer (1, 2, 3, ...). The larger the value of \( n \), the larger and higher in energy the orbitals are.

Generally, the principal quantum number reflects how far the electron is from the nucleus. For example, electrons with a principal quantum number of 1 will be closer to the nucleus and lower in energy compared to electrons with \( n = 2 \) or \( n = 3 \). This quantum number is crucial for defining an electron's position within an atom.

Therefore, the statement in the exercise correctly identifies that the principal quantum number represents the size of the orbit and the average distance of the electron from the nucleus.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), specifies the orientation of an orbital in space relative to an external magnetic field. It can take on any integer value between \( -l \) and \( +l \), including zero.

For example, if \( l = 1 \) (a p orbital), \( m_l \) can be -1, 0, or +1. These values indicate that there are three different p orbitals, each oriented differently in space. However, the shape of the orbitals (spherical for s, dumbbell-shaped for p, etc.) is not defined by \( m_l \), but by the azimuthal quantum number \( l \).

In the given exercise, the statement that the shape of an orbital is governed by the magnetic quantum number is false. The correct determinant for the shape of an orbital is the azimuthal quantum number, as previously discussed.