Question

The "magic numbers" in the periodic table are the atomic numbers of elements with high stability (the noble gases): \(2,10,18,36,54\), and \(86 .\) In terms of allowed values of orbitals and spin quantum numbers, explain why these electron arrangements correspond to special stability.

Short answer

The special stability of noble gases (magic numbers) in the periodic table is due to their completely filled electron subshells as indicated by their electron configurations and allowed quantum number values. These filled subshells grant the noble gases a low reactivity and high stability because they are energetically favorable and represent a stable electron configuration.

Step-by-step solution

Recall quantum numbers and electron configurations

In atoms, electrons are organized into shells and subshells, each having specific energy levels. Electron configurations describe the distribution of electrons in these shells and subshells. There are 4 quantum numbers: the principal (n), azimuthal (l), magnetic (m_l), and the spin (m_s) quantum numbers. 'n' represents shell number, 'l' denotes the subshell, and (m_l) represents different orbital orientations within the subshells. The spin quantum number 'm_s' shows the electron's spin (+1/2 or -1/2).

Determine filled electron subshells

The enhanced stability of the noble gas elements is due to their filled electron subshells. For subshells, l takes values from 0 to n-1, where l = 0 represents the s subshells, l =1 corresponds to the p subshells, and so on. Each subshell can hold a maximum of 2(2l +1) electrons. The s subshell (l = 0) contains 2 electrons, p (l = 1) subshell can hold 6, d (l = 2) subshell contains 10, and the f (l = 3) subshell can take 14 electrons.

Identify electron configurations for the noble gases

We will analyze each magic number in the exercise and write the electron configuration for these nobles gases: 1. Helium (He, Z=2): 1s^2 2. Neon (Ne, Z=10): 1s^2 2s^2 2p^6 3. Argon (Ar, Z=18): 1s^2 2s^2 2p^6 3s^2 3p^6 4. Krypton (Kr, Z=36): 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5. Xenon (Xe, Z=54): 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5s^2 4d^10 5p^6 6. Radon (Rn, Z=86): 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5s^2 4d^10 5p^6 6s^2 4f^14 5d^10 6p^6

Explain the special stability of noble gases

As we see from the electron configurations of noble gases, they have completely filled subshells, which correspond to their enhanced stability. These filled subshells grant the noble gases a low reactivity and high stability because they are energetically favorable and represent a stable electron configuration. In conclusion, the special stability of noble gases (magic numbers) in the periodic table is due to their completely filled electron subshells as indicated by their electron configurations and allowed quantum number values. This plays a critical role in imparting their observed low reactivity and stability.