Question

Out of four quantum numbers for an electron only spin quantum number is fractional because (1) Two consecutive values of any quantum number must differ by at least 1 . (2) The electrons complete half revolution during spin. (3) Fractional values assigned are arbitrary only. (4) None.

Short answer

(2) The electrons complete half revolution during spin.

Step-by-step solution

- Identify the Question

Clearly understand what is being asked: We need to determine why the spin quantum number for an electron is fractional.

- Recall Quantum Numbers

Recall the four quantum numbers: principal (\(n\)), azimuthal (\(l\)), magnetic (\(m_l\)) and spin (\(m_s\)). Understand that the spin quantum number (\(m_s\)) can have values of +1/2 or -1/2.

- Review Options

Look at each provided option. Options (1), (2), and (3) need careful consideration.

- Analyze Option 1

Option (1): Two consecutive values of any quantum number must differ by at least 1. Since this option talks about the difference, it doesn't explain why the spin quantum number itself is fractional.

- Analyze Option 2

Option (2): The electrons complete half revolution during spin. This is accurate because the spin quantum number indicates that an electron can spin half a revolution, giving it the values +1/2 or -1/2, hence they are fractional.

- Analyze Option 3

Option (3): Fractional values assigned are arbitrary only. This is not true as the values +1/2 and -1/2 have physical significance related to the electron's spin and aren't arbitrary.

- Conclusion

Option (2) is the correct answer as it accurately explains why the spin quantum number is fractional.

Key concepts

Quantum Numbers

Quantum numbers are essential for understanding the behavior and properties of electrons within atoms. There are four main quantum numbers:

* **Principal quantum number (n)**: Determines the energy level of an electron within the atom. Higher values of n correspond to higher energy levels.
* **Azimuthal quantum number (l)**: Defines the subshell or shape of the electron cloud. The value of l ranges from 0 to n-1.
* **Magnetic quantum number (m_l)**: Describes the orientation of the electron's orbital in space. The values range between -l and +l.
* **Spin quantum number (m_s)**: Indicates the direction of the electron's spin, with possible values of +1/2 or -1/2.

Together, these quantum numbers describe the unique position and energy of every electron in an atom, following the rules governed by quantum mechanics.

Electron Spin

Electron spin is a fundamental property of electrons. It is a type of angular momentum that is intrinsic to electrons. Unlike other quantum numbers, the spin quantum number ( m_s ) can only take on fractional values: +1/2 or -1/2.

An electron's spin contributes to its magnetic field and is a key part of the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of four quantum numbers. This means that in a single orbital, one electron will spin up (+1/2) and one electron will spin down (-1/2), ensuring each electron is unique.

The concept of spin helps explain many physical phenomena, such as the splitting of spectral lines in a magnetic field (Zeeman effect) and the operation of electron spin resonance (ESR) techniques in chemistry and physics.

Fractional Values

The spin quantum number is unique in that it is fractional, specifically +1/2 or -1/2. This fractionality arises because of the quantum mechanical nature of spin, which represents the intrinsic angular momentum of the electron.

A full revolution would imply a whole number quantum number. However, for the electron’s intrinsic spin, it makes only a half revolution, thus the values are +1/2 or -1/2. These fractions are not arbitrary; they correspond to physical properties observed experimentally.

The fractional spin quantum number plays a crucial role in determining the electron configuration in atoms. This directly influences chemical bonding and magnetic properties of substances.

Understanding the fractional values of the spin quantum number aids in the deeper comprehension of atomic structure, solid-state physics, and quantum computing, where spin states can represent qubits for information processing.