How does the angular momentum quantum number relate to orbital shape?

The angular momentum quantum number, l, describes the shape of the orbital that the electron occupies. It can have integer values from 0 to n – 1 for each value of n. Orbitals with l = 0 are s orbitals, l = 1 are p orbitals, l = 2 are d orbitals, and l = 3 are f orbitals, each with distinct shapes.

Why was a fourth quantum number needed to explain atomic spectra?

When hydrogen-line spectra were examined at extremely high resolution, some lines appeared as pairs of closely spaced lines rather than single peaks, revealing fine structure. This indicated additional small energy differences for electrons in the same orbital, leading scientists to propose the spin quantum number to account for these observations.

What are the two possible values for electron spin?

Electrons are spinning charges that behave like tiny bar magnets. The two possible spinning motions are clockwise, indicated by a spin quantum number ms of +1/2, and counterclockwise, indicated by ms of –1/2. These are the only two spin states available to an electron in an orbital.

How many orbitals exist in different subshells?

The number of orbitals in a subshell is determined by the formula 2l + 1. An s subshell (l = 0) contains one orbital, a p subshell (l = 1) contains three orbitals, a d subshell (l = 2) contains five orbitals, and an f subshell (l = 3) contains seven orbitals.

How are the first three quantum numbers related to each other?

The principal quantum number n determines the possible values of the angular momentum quantum number l, which ranges from 0 to n – 1. The magnetic quantum number ml then depends on l, ranging from –l to +l. Together, these three interrelated quantum numbers define where in space an electron is most likely to be located.

7.10

Quantum Numbers

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Each electron is described by four quantum numbers, three of which have interrelated values. These three quantum numbers define where in space the electron is most likely to be located.

The principal quantum number, n, denotes the energy level to which an electron belongs. n can have integer values of one, two, three, and so forth. The greater the value of n, the farther the electron is from the nucleus.

The angular momentum, or azimuthal, quantum number, l, describes the shape of the orbital that the electron occupies, which is related to the electron’s angular momentum.

l can have integer values from 0 to n – 1 for each value of n. When n is one, l is zero. When n is two, l has two possible values — zero and one. When n is three, l can be zero, one, or two. The number of possible l values indicates the number of groups of differently shaped orbitals, or subshells, at the nth energy level.

Subshells are designated by letters associated with the values of l. Zero, one, two, and three correspond to s, p, d, and f orbitals, respectively. These will be discussed in the next lesson.

The magnetic quantum number, m_l, describes the orientation in space of the orbital that the electron occupies, which is also related to the electron’s angular momentum. m_l can have any integer value between –l and +l, including zero. The number of orientations for a given subshell is equal to 2l + 1.

When n is two, there are two subshells, as l can be zero or one. These are the 2s and 2p subshells, respectively. The 2p subshell has three 2p orbitals, as there are three values for m_l, –1, 0, and +1.

The fourth quantum number, m_s, is the spin quantum number. Electrons are spinning charges and behave like tiny bar magnets. The two possible spinning motions for the electron are clockwise, indicated by an m_sof +1/2, and counterclockwise, indicated by an m_s of –1/2.

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It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one ener…