Question

In the ground state of mercury, \(\mathrm{Hg}\), 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" \(\left(m_{s}=+\frac{1}{2}\right)\) ?

Short answer

a. 18 electrons occupy orbitals with n=3 b. 30 electrons occupy d orbitals c. 8 electrons occupy \(p_{z}\) orbitals d. 40 electrons have spin "up"

Step-by-step solution

Determine the electron configuration of Mercury

Mercury, \(\mathrm{Hg}\), has an atomic number of 80, meaning it has 80 electrons. The electron configuration can be determined using the periodic table by following the order of atomic orbitals (s, p, d, f) and the energy levels (n=1, 2, 3, ...). The electron configuration of Mercury is: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10} 5p^6 6s^2 4f^{14} 5d^{10} 6p^2\] Step 2: Determine the number of electrons in n=3 orbitals

Count electrons in orbitals with n=3

Using the electron configuration above, we can count the electrons in orbitals with n=3: \[3s^2 3p^6 3d^{10}\] Adding the superscripts or electron counts, we get 2+6+10 = 18 electrons. Step 3: Determine the number of electrons in d orbitals

Count electrons in d orbitals

Focus on the d orbitals in the electron configuration: \[3d^{10} 4d^{10} 5d^{10}\] Adding the superscripts, we get: 10+10+10 = 30 electrons. Step 4: Determine the number of electrons in p_{z} orbitals

Count electrons in p_{z} orbitals

In the p orbitals, each corresponds to a specific orientation (p_x, p_y, and p_z). Since the p orbitals can hold a maximum of 6 electrons (2 for each orientation), the p_z orbital will have one-third of the total p electrons. In the electron configuration for Mercury, the p orbitals have: \[2p^6 3p^6 4p^6 5p^6 6p^2\] Summing up the electrons: \[6 + 6 + 6 + 6 + 2 = 26\] Divide by 3 to find the number of electrons occupying p_{z} orbitals: \(\frac{26}{3}\approx 8.7\). Since electrons must be whole integers, we'll round down to 8 p_{z} electrons. Step 5: Determine the number of electrons with spin "up"

Count electrons with spin "up"

Each atomic orbital can hold two electrons with opposite spins (spin "up": +1/2, spin "down": -1/2). Since the total number of electrons is 80, half of them will have spin up. So, the number of electrons with spin "up" is: \[\frac{80}{2} = 40\] So, the answers are: a. 18 electrons occupy orbitals with n=3 b. 30 electrons occupy d orbitals c. 8 electrons occupy p_{z} orbitals d. 40 electrons have spin "up"

Key concepts

Atomic Orbitals

Atomic orbitals are the regions in an atom where there is a high probability of finding electrons. Each orbital is associated with a set of quantum numbers, and these numbers describe the energy level (), angular momentum (), magnetic moment (), and electron spin within the atom. Mercury, in its ground state, has electrons distributed across various orbitals from 1s to 6p.

According to the quantum mechanical model, orbitals are grouped into different 'shapes' known as s, p, d, and f orbitals, each with a specific number of orientations that dictate how the electrons are arranged around the nucleus. For example, in the solution provided, we find that the electron configuration for mercury involves multiple energy levels, including n=3, where 18 electrons are present in the 3s, 3p, and 3d orbitals.

Electron Spin

Electron spin refers to the intrinsic angular momentum of electrons. It is a quantum mechanical property that can take one of two possible values: spin 'up' () or spin 'down' (). When we talk about electron spin in an atomic orbital, we can visualize it as electrons spinning on their own axis, which generates a magnetic moment.

An atomic orbital can hold a maximum of two electrons with opposite spins. In the context of the provided exercise, we consider the electron spin when determining how many electrons have a spin 'up' state for mercury. Since there are 80 electrons in total, half of them, that is 40 electrons, would have a spin 'up' state, as electrons in orbitals are paired with opposite spins to maintain a balanced magnetic field in an atom.

Quantum Numbers

Quantum numbers are like an address for an electron, detailing their location and behavior within an atom. There are four types: the principal quantum number (n) indicates the electron's energy level and distance from the nucleus; the angular momentum quantum number () defines the shape of the orbital; the magnetic quantum number () specifies the orbital's orientation in space; and the spin quantum number () describes the electron's spin orientation.

In the case of mercury, one of the steps to finding how many electrons occupy atomic orbitals with n=3 involves using quantum numbers to sort out the electrons across the different sublevels and orbitals. The principal quantum number n indicates that orbitals 3s, 3p, and 3d are of interest, which helps in computing the 18 electrons associated with this energy level.

Periodic Table Electron Configuration

The periodic table is a powerful tool for predicting and understanding the electron configuration of an element. Electron configuration describes how electrons are distributed among atomic orbitals and is often determined by following the order of fill based on the Aufbau principle starting from the lowest energy levels to higher levels. The sequence in which electrons fill these orbitals is generally s, p, d, and f.

For mercury, the periodic table helps visualize its electron configuration which ultimately provides information on its chemical behavior and properties. As solved in the exercise, mercury's configuration is spread across different orbitals listed in ascending order from 1s up to 6p, showing how electrons are arranged and which orbitals (like d orbitals) are filled or partially filled. In practical terms, knowing an element's electron configuration is essential for understanding the element's reactivity and the types of bonds it can form.