Question

The principal quantum number of an atom is related to the: (a) Orientation of the orbital in space (b) Spin angular momentum (c) Orbital angular momentum (d) Size of the orbital

Short answer

The principal quantum number is related to the size of the orbital, so the answer is (d).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers are used to describe the properties of electrons in atoms. The principal quantum number, denoted by \( n \), indicates the main energy level or shell of an electron around the nucleus. It tells us how far the electron's orbital is from the nucleus.

Principal Quantum Number and Its Role

The principal quantum number \( n \) is directly related to the size and energy of the orbital. As \( n \) increases, the orbital becomes larger, and the electron resides further from the nucleus. This means that \( n \) is indicative of the size of the orbital.

Evaluating Options

Given the options: (a) Orientation of the orbital in space, (b) Spin angular momentum, (c) Orbital angular momentum, and (d) Size of the orbital, we need to determine the correct association with the principal quantum number \( n \).

Testing Each Option

1. Option (a), orientation of the orbital, is determined by the magnetic quantum number \( m_l \), not \( n \).2. Option (b), spin angular momentum, is independent of quantum numbers \( n, l, \) and \( m_l \), and is described by the spin quantum number \(s\).3. Option (c), orbital angular momentum, depends on the azimuthal quantum number \( l \) rather than \( n \).4. Option (d), size of the orbital, is directly related to the principal quantum number \( n \).

Conclusion

The principal quantum number \( n \) is associated with the size of the orbital, making option (d) the correct answer.

Key concepts

Exploring Quantum Numbers

Quantum numbers are essential in understanding the behavior and arrangement of electrons within an atom. Each electron in an atom is described by four quantum numbers, which together provide a unique address or identifier, helping us pinpoint its precise location and state.

• Principal Quantum Number ( extbf{n} ). This number identifies the main energy level or shell where the electron resides. It plays a crucial role in determining the electron's energy and its distance from the nucleus.
• Azimuthal Quantum Number ( extbf{l} ). This quantum number defines the subshell or sublevel, which offers insight into the shape of the electron's orbital.
• Magnetic Quantum Number ( extbf{m} _l). It determines the orientation of the orbital in space, relative to an external magnetic field.
• Spin Quantum Number ( extbf{s} ). This describes the spin orientation of the electron, either up or down.

Understanding these quantum numbers is key to accurate predictions of electronic configurations and the chemical behavior of elements.

Energy Levels and Their Significance

In an atom, electrons are organized into various energy levels or shells, signified by the principal quantum number ( extbf{n} ). These energy levels are akin to the floors of a building, with each increase in ( extbf{n} ) indicating a progression to a higher level or floor. The energy associated with these levels increases as ( extbf{n} ) rises, meaning electrons in higher energy levels hold more energy.

• Low ( extbf{n} ). Electrons in lower energy levels are closer to the nucleus and have lower energy. They also experience a stronger attractive force from the nucleus.
• High ( extbf{n} ). Higher energy levels are filled as more electrons are added, placing these electrons farther from the nucleus, resulting in increased energy but decreased nuclear attraction.

This hierarchical arrangement is critical in understanding atomic structures and chemical reactions.

The World of Electron Orbitals

An electron orbital is a region in space where there is a high probability of finding an electron. The shape and size of these orbitals are defined by the quantum numbers, particularly the principal quantum number ( extbf{n} ) and azimuthal quantum number ( extbf{l} ).

• s-Orbitals. These are spherical and appear as simple spheres around the nucleus, increasing in size with larger ( extbf{n} ).
• p-Orbitals. Shaped like dumbbells, these have different orientations in space, defined by the magnetic quantum number ( extbf{m} _l).
• d and f-Orbitals. These orbitals have more complex shapes, found in transition metals and deeper elements of the periodic table.

Understanding the nature of electron orbitals is essential for predicting how atoms will bond and react with each other.

Nuclear Distance and Atomic Structure

The concept of nuclear distance is fundamental in understanding how electron configurations shape the size and properties of an atom. As electrons fill higher energy levels, the average distance between the nucleus and the electron increases, which is closely tied to the principal quantum number ( extbf{n} ).

• Increasing Distance. As ( extbf{n} ) increases, electrons occupy orbitals that are further from the nucleus, expanding the atomic size.
• Electron Shielding. Inner electrons can shield outer electrons from the full effect of the nucleus's positive charge, affecting the actual nuclear attraction felt by those outer electrons.
• Periodic Trends. Understanding nuclear distance and electron shielding helps explain trends in atomic and ionic radii across the periodic table.

This distance concept is pivotal for chemists and physicists to grasp how atoms interact in the vast realm of chemistry and physical science.