Chapter 1: Problem 14

Define quantum numbers and explain the significance of each of these numbers.

Short Answer

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Quantum numbers collectively describe the unique quantum state of an electron in an atom. They include the principal quantum number (n) which explains the energy level or shell, the azimuthal quantum number (l) which describes the shape of the orbital, the magnetic quantum number (m) determining the orientation of the orbital, and the spin quantum number (s) showing the spin direction of the electron.

Step by step solution

Defining Quantum Numbers

Quantum numbers are a set of four numbers that physicists use to describe the unique quantum state of an electron. These include the principal quantum number (n), the azimuthal or angular quantum number (l), the magnetic quantum number (m), and the spin quantum number (s).

Principal Quantum Number (n)

The principal quantum number, often symbolized by \(n\), is a positive integer that determines the energy level or shell in which the electron resides in an atom. It also indicates the size of the orbital. As \(n\) increases, electrons are further from the nucleus, and their energy levels increase.

Azimuthal Quantum Number (l)

The azimuthal quantum number, also known as the angular momentum quantum number, describes the shape of the orbital. Symbolized as \(l\), this quantum number varies from 0 to \(n-1\). For instance, when \(l=0\), it denotes the 's' orbital (spherical), \(l=1\) denotes 'p' orbital (dumbbell-shaped), \(l=2\) represents 'd' orbital, and so on.

Magnetic Quantum Number (m)

Represented as \(m_l\), the magnetic quantum number determines the orientation of the orbital in 3D space. Its values can range from \(-l\) to \(l\), including zero. Each value corresponds to a different orientation.

Spin Quantum Number (s)

The spin quantum number denotes the direction of the electron's intrinsic or 'spin' angular momentum. It can only have two possible values, +1/2 (spin-up) or -1/2 (spin-down), indicating that the electron is spinning clockwise or counterclockwise, respectively.

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Define quantum numbers and explain the significance of each of these numbers.

Short Answer

Step by step solution

Defining Quantum Numbers

Principal Quantum Number (n)

Azimuthal Quantum Number (l)

Magnetic Quantum Number (m)

Spin Quantum Number (s)

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