Question

How many electrons in an atom can have the designation \(1 p\) \(6 d_{x^{2}-y^{2}}, 4 f, 7 p_{y}, 2 s, n=3 ?\)

Short answer

In summary, \(38\) electrons in an atom can have the given designations \(6d_{x^{2}-y^{2}}, 4f, 7p_{y}, 2s\), and \(n=3\).

Step-by-step solution

Determine if the given electron orbital notations are valid according to the quantum number rules

: To figure out whether each electron notation is valid, we have to check them against quantum number rules. There are four quantum numbers: principal (n), azimuthal (l), magnetic (ml), and spin (ms). Their rules are as follows: 1. n must be a positive integer (1,2,3,...) 2. l can be an integer ranging from 0 to n-1 3. ml can be an integer ranging from -l to l 4. ms can have values of +1/2 or -1/2 We will check each given electron orbital notation against these rules.

Check the validity of 1p notation

: For 1p, the principal quantum number n=1. This leads to the following possible values for l: l=0 to (n-1) => l=0 However, p orbitals have an l value of 1. Thus, the 1p notation is invalid as there are no p orbitals in the n=1 energy level.

Check the validity of 6d_(x^2-y^2) notation

: For 6d_(x^2-y^2), the principal quantum number is n=6, which leads to the following possible values for l: l=0 to (n-1) => l=0,1,2,3,4,5 Since d orbitals have an l value of 2, this notation is valid. There are five d orbitals with ml values ranging from -2 to 2, and the x^2-y^2 refers to one specific orbital. Each orbital can hold two electrons, one with ms=+1/2 and the other with ms=-1/2. So, there can be 2 electrons in this orbital.

Check the validity of 4f notation

: For 4f, the principal quantum number n=4, which leads to the possible values for l: l=0 to (n-1) => l=0,1,2,3 Since f orbitals have an l value of 3, this notation is valid. There are seven f orbitals with ml values ranging from -3 to 3, and each orbital can hold two electrons. So, there can be 2 x 7 = 14 electrons in this energy level.

Check the validity of 7py notation

: For 7py, the principal quantum number n=7, which leads to the possible values for l: l=0 to (n-1) => l=0,1,2,3,4,5,6 Since p orbitals have an l value of 1, this notation is valid. There are three p orbitals with ml values of -1, 0, and 1, and the py refers to one specific orbital with ml=1. Each orbital can hold two electrons, so there can be 2 electrons in this orbital.

Check the validity of 2s notation

: For 2s, the principal quantum number n=2, which leads to the possible values for l: l=0 to (n-1) => l=0,1 Since s orbitals have an l value of 0, this notation is valid. There is only one s orbital with ml=0, and it can hold two electrons. So, there can be 2 electrons in this energy level.

Check the validity of n=3 notation

: For n=3, there are no specific notations for orbitals (i.e., s, p, d, etc.), meaning we must consider all possible orbitals in this energy level. The principal quantum number n=3, which leads to the possible values for l: l=0 to (n-1) => l=0,1,2 So, in n=3, there are s, p, and d orbitals. The s orbital can hold 2 electrons, the three p orbitals can hold 2 x 3 = 6 electrons, and the five d orbitals can hold 2 x 5 = 10 electrons. In total, there can be 2 + 6 + 10 = 18 electrons in this energy level.

Calculate the total number of electrons that can have the given designations

: Now we just need to sum the number of electrons allowed in each of the valid electron orbital notations: 2 (from 6d_x^2-y^2) + 14 (from 4f) + 2 (from 7py) + 2 (from 2s) + 18 (from n=3) = 38 electrons So, 38 electrons in an atom can have the given designations.

Key concepts

Electron Configuration

When we talk about the electron configuration of an atom, we're describing the way in which electrons are arranged around the nucleus. Electrons occupy specific energy levels known as electron shells, and within these shells, they are found in subshells named s, p, d, and f.

Each subshell has a different capacity for electrons: an s subshell can hold up to 2 electrons, a p subshell can accommodate 6, a d subshell has room for 10, and an f subshell can contain 14 electrons. The order in which these subshells are filled is governed by the Aufbau principle, which states that electrons occupy the lowest energy orbital available first.

To understand an atom's electron configuration effectively, one must first grasp the principles of quantum mechanics that define these configurations, such as the principal quantum number, which indicates the relative size and energy of atomic orbitals.

Orbital Notations

Orbital notations extend the electron configuration concept by graphically representing the electrons in their respective orbitals. An orbital is depicted as a box or a line, and electrons are shown as arrows pointing upwards or downwards to represent their spin quantum number.

For instance, the notation for a 6d_x²-y² orbital includes five boxes (since there are five d orbitals) with one arrow in the box corresponding to the d_x²-y² orbital. It's important to remember that orbitals are filled according to Hund's Rule: each orbital in a given subshell gets one electron before any orbital gets a second one, and all the single electrons must have the same spin.

A clear understanding of orbital notations helps in visualizing the arrangement of electrons and is crucial for grasping the concept of electron pairing and the overall stability of atoms.

Principal Quantum Number

The principal quantum number, denoted as 'n', is a critical part of quantum mechanics that dictates the size and energy level of an electron's orbital. Simple put, it is the 'address' of the electron, pointing out which shell it's in. The value of 'n' starts at 1 and increases with the energy level of the electron.

For instance, in the 2s notation, 'n' is 2, indicating that the s orbital is in the second energy level. The range of 'l', the azimuthal quantum number, depends on the value of 'n' as l can be any integer from 0 to n-1. This implies that as 'n' increases, more types of subshells become available for electrons to occupy. For energy levels like 1 or 2 where 'n' is small, the types of subshells available are limited.

In addition to indicating the energy level, 'n' also indirectly informs us about the electron's distance from the nucleus—the higher the 'n', the farther away from the nucleus and higher the energy of the electron. This quantum number is pivotal when determining the possible electron configurations for an atom, especially when considering the more complex d and f orbitals.