2.6: Electronic Structure of Atoms

Learning Objectives

• Learn the quantum numbers that are assigned to electrons.
• Describe how electrons are grouped within atoms.

In 1913, the Danish scientist Niels Bohr suggested that the electron in a hydrogen atom could not have any random energy, having only certain fixed values of energy that were indexed by the number n (the same n in the equation above and now called a quantum number). Quantities that have certain specific values are quantized values. Bohr suggested that the energy of the electron in hydrogen was quantized because it was in a specific orbit. Figure $$\PageIndex{1}$$ shows a model of the hydrogen atom based on Bohr's ideas. Figure $$\PageIndex{1}$$: Bohr's Model of the Hydrogen Atom. Bohr's description of the hydrogen atom had specific orbits for the electron, which had quantized energies.

Bohr's ideas were useful but were applied only to the hydrogen atom. However, later researchers generalized Bohr's ideas into a new theory called quantum mechanics, which explains the behavior of electrons as if they were acting as a wave, not as particles. Key to this theory is the introducion of quantum numbers to dscribe each electron in the atom.

Principal Quantum Number

In the quantum-mechanical model of an atom, the state of an electron is described by four quantum numbers, not just the one predicted by Bohr. The first quantum number is called the principal quantum number ($$n$$). The principal quantum number largely determines the energy of an electron. Electrons in the same atom that have the same principal quantum number are said to occupy an electron shell of the atom. The principal quantum number can be any nonzero positive integer: 1, 2, 3, 4,….

Angular Momentum Quantum Number

Within a shell, there may be multiple possible values of the next quantum number, the angular momentum quantum number (). The ℓ quantum number has a minor effect on the energy of the electron but also affects the spatial distribution of the electron in three-dimensional space-that is, the shape of an electron's distribution in space. The value of the ℓ quantum number can be any integer between 0 and n − 1:

$ℓ = 0, 1, 2,…, n − 1$

Thus, for a given value of n, there are different possible values of ℓ:

If n equals ℓ can be
1 0
2 0 or 1
3 0, 1, or 2
4 0, 1, 2, or 3

and so forth. Electrons within a shell that have the same value of ℓ are said to occupy a subshell in the atom. Commonly, instead of referring to the numerical value of ℓ, a letter represents the value of ℓ (to help distinguish it from the principal quantum number):

If ℓ equals The letter is
0 s
1 p
2 d
3 f

Magnetic Quantum Number

The next quantum number is called the magnetic quantum number ($$m_ℓ$$). For any value of $$ℓ$$, there are $$2ℓ + 1$$ possible values of $$m_ℓ$$, ranging from $$−ℓ$$ to $$ℓ$$:

$−ℓ \lt m_ℓ \lt ℓ$

or

$|m_ℓ|≤ ℓ$

The following explicitly lists the possible values of $$m_ℓ$$ for the possible values of $$ℓ$$:

If $$ℓ$$ equals The m values can be
0 0
1 −1, 0, or 1
2 −2, −1, 0, 1, or 2
3 −3, −2, −1, 0, 1, 2, or 3

The particular value of $$m_ℓ$$ dictates the orientation of an electron's distribution in space. When $$ℓ$$ is zero, $$m_ℓ$$ can be only zero, so there is only one possible orientation. When $$ℓ$$ is 1, there are three possible orientations for an electron's distribution. When $$ℓ$$ is 2, there are five possible orientations of electron distribution. This goes on and on for other values of $$ℓ$$, but we need not consider any higher values of $$ℓ$$ here. Each value of $$m_ℓ$$ designates a certain orbital. Thus, there is only one orbital when $$ℓ$$ is zero, three orbitals when $$ℓ$$ is 1, five orbitals when $$ℓ$$ is 2, and so forth. The $$m_ℓ$$quantum number has no effect on the energy of an electron unless the electrons are subjected to a magnetic field-hence its name.

The $$ℓ$$ quantum number dictates the general shape of electron distribution in space (Figure $$\PageIndex{2}$$ ). Any s orbital is spherically symmetric (Figure $$\PageIndex{4a}$$, and there is only one orbital in any s subshell. Any p orbital has a two-lobed, dumbbell-like shape (Figure $$\PageIndex{4b}$$ ); because there are three of them, we normally represent them as pointing along the x-, y-, and z-axes of Cartesian space. The d orbitals are four-lobed rosettes (Figure $$\PageIndex{4c}$$ they are oriented differently in space (the one labeled $$d_{z^2}$$ has two lobes and a torus instead of four lobes, but it is equivalent to the other orbitals). When there is more than one possible value of m, each orbital is labeled with one of the possible values. It should be noted that the diagrams in Figure $$\PageIndex{2}$$ are estimates of the electron distribution in space, not surfaces electrons are fixed on. Figure $$\PageIndex{2}$$: Electron Orbitals. (a) The lone s orbital is spherical in distribution. (b) The three p orbitals are shaped like dumbbells, and each one points in a different direction. (c) The five d orbitals are rosette in shape, except for the $$d_{z^2}$$ orbital, which is a "dumbbell + torus" combination. They are all oriented in different directions.

Spin Quantum Number

The final quantum number is the spin quantum number ($$m_s$$). Electrons and other subatomic particles behave as if they are spinning (we cannot tell if they really are, but they behave as if they are). Electrons themselves have two possible spin states, and because of mathematics, they are assigned the quantum numbers +1/2 and −1/2. These are the only two possible choices for the spin quantum number of an electron.

Example $$\PageIndex{1}$$

Of the set of quantum numbers {n, ℓ, m, ms}, which are possible and which are not allowed?

1. {3, 2, 1, +1/2}
2. {2, 2, 0, −1/2}
3. {3, −1, 0, +1/2}

Solution

1. The principal quantum number n must be an integer, which it is here. The quantum number ℓ must be less than n, which it is. The m quantum number must be between −ℓ and ℓ, which it is. The spin quantum number is +1/2, which is allowed. Because this set of quantum numbers follows all restrictions, it is possible.
2. The quantum number n is an integer, but the quantum number ℓ must be less than n, which it is not. Thus, this is not an allowed set of quantum numbers.
3. The principal quantum number n is an integer, but ℓ is not allowed to be negative. Therefore this is not an allowed set of quantum numbers.

Exercise $$\PageIndex{1}$$

Of the set of quantum numbers {n, ℓ, m, ms}, which are possible and which are not allowed?

1. {4, 2, −2, 1}
2. {3, 1, 0, −1/2}

Spin must be either +1/2 or −1/2, so this set of quantum number is not allowed.

allowed

The modern theory quantum mechanics makes the following statements about electrons in atoms:

• Electrons in atoms can have only certain specific energies. We say that the energies of the electrons are quantized.
• Electrons are organized according to their energies into sets called shells (labeled by the principle quantum number, n). Generally the higher the energy of a shell, the farther it is (on average) from the nucleus. Shells do not have specific, fixed distances from the nucleus, but an electron in a higher-energy shell will spend more time farther from the nucleus than does an electron in a lower-energy shell.
• Shells are further divided into subsets of electrons called subshells. The first shell has only one subshell, the second shell has two subshells, the third shell has three subshells, and so on. The subshells of each shell are labeled, in order, with the letters s, p, d, and f. Thus, the first shell has only a single s subshell (called 1s), the second shell has 2s and 2p subshells, the third shell has 3s, 3p, and 3d and so forth.