Question

In the ground state of mercury, Hg, a. how many electrons occupy atomic orbitals with $n=3?$ b. how many electrons occupy $d$ atomic orbitals? c. how many electrons occupy $p_z$ atomic orbitals? d. how many electrons have spin "up" $(m_s=+\frac{1}{2})?$

Short answer

There are $18$ electrons occupying orbitals with n=3, $30$ electrons occupying d orbitals, $8$ electrons occupying p_z atomic orbitals, and $40$ electrons with spin "up" (m_s = +1/2) in the ground state of mercury (Hg).

Step-by-step solution

a. Electrons occupying orbitals with n=3

To find the number of electrons in orbitals with n=3, we need to count all the electrons in the 3s, 3p, and 3d orbitals. From the electronic configuration, we can see that: 3s: 2 electrons, 3p: 6 electrons, 3d: 10 electrons. Hence, there are 2 + 6 + 10 = $18$ electrons occupying orbitals with n=3.

b. Electrons occupying d orbitals

To determine the number of electrons in d orbitals, we can examine the electronic configuration and sum up the electrons in the 3d, 4d, and 5d orbitals: 3d: 10 electrons, 4d: 10 electrons, 5d: 10 electrons. The total number of electrons in d orbitals is 10 + 10 + 10 = $30$ electrons.

c. Electrons occupying p_z orbitals

The p_z orbitals are a subset of the p orbitals, and there is one p_z orbital in each p subshell. Since each orbital can hold a maximum of 2 electrons, there will be 2 electrons in the 2p_z, 3p_z, 4p_z, and 5p_z orbitals. Therefore, there are $4$ sets of 2 electrons for a total of $8$ electrons occupying p_z atomic orbitals.

d. Electrons with spin "up" (m_s = +1/2)

Finally, to determine the number of electrons with spin "up" (m_s = +1/2), we can assume that, in each orbital, one electron has spin "up" and the other has spin "down". Since there are 80 electrons in mercury, half (40) will have spin "up" and half (40) will have spin "down". Thus, there are $40$ electrons with spin "up" (m_s = +1/2).

Key concepts

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. They are defined by the quantum numbers, which determine their size, shape, and orientation in space. The different types of atomic orbitals are designated as s, p, d, and f. These labels come from the words sharp, principal, diffuse, and fundamental, respectively.

Each type of orbital has a different shape and capacity for holding electrons:

• s orbitals: Spherical in shape and can hold up to 2 electrons.
• p orbitals: Dumbbell-shaped and come in sets of three (p_x, p_y, p_z), each capable of holding 2 electrons, for a total of 6 electrons per p subshell.
• d orbitals: More complex in shape and come in sets of five, each holding 2 electrons, totaling 10 electrons per d subshell.
• f orbitals: Even more complex, comprising of seven orbitals, and can hold a total of 14 electrons.

These orbitals play a crucial role in the arrangement of electrons in an atom, influencing its chemical properties and reactivity.

Electron Spin

Electron spin is an intrinsic form of angular momentum carried by electrons, described by the spin quantum number, $m_s$. Electrons can have a spin of either +$\frac{1}{2}$ or -$\frac{1}{2}$. These can be thought of as 'spin up' and 'spin down' states.

Having opposite spins allows two electrons to occupy the same orbital without repelling each other, due to the Pauli Exclusion Principle. This is an essential concept as it helps maximize electron occupancy in orbitals while maintaining stability within an atom.

When filling in electrons in an atom's orbitals, one electron will occupy an orbital with a 'spin up' configuration, and when paired, the second will be 'spin down'. This balanced arrangement minimizes energy and stabilizes the electron configuration.

Quantum Numbers

Quantum numbers are sets of numerical values that provide solutions to the Schrödinger equation, describing the state of an electron in an atom. These numbers are crucial for defining the electron configuration and behavior in an atom.

The four quantum numbers are:

• Principal quantum number (n): Indicates the energy level and size of the orbital. Larger values of n mean larger orbitals and higher energy.
• Azimuthal quantum number (l): Defines the shape of the orbital (s, p, d, f), with values ranging from 0 to n-1.
• Magnetic quantum number (m_l): Specifies the orientation of the orbital in space in relation to the other orbitals, with possible values from -l to +l.
• Spin quantum number (m_s): Describes the direction of electron spin, with only two possible values, +$\frac{1}{2}$ or -$\frac{1}{2}$.

These quantum numbers collectively explain the arrangement of electrons in atoms and the chemical properties that result from this arrangement.