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### Quantum numbers

## Principal Quantum Number (n):

### The number of the level (n):with their symbols

### The number of electrons in in each principal q. number

### Energy of the levels

## Angular Momentum Quantum Number (l):

## Magnetic Quantum Number (m_l):

## Spin Quantum Number (m_s):

### The relationships between these quantum numbers are as follows:

### What are the probable (l) values when n= 1

### The capacity of the sublevels with electrons

**Quiz**

**References of this lesson**

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Quantum numbers are a set of four parameters that describe the quantum state of an electron in an atom.

They provide important information about the electron’s energy, orbital shape, orientation in space, and spin. These quantum numbers are fundamental in understanding the electronic structure of atoms. The four quantum numbers are:

**Definition:** This quantum number represents the main energy level or shell in which an electron resides.

- It describes the distance of the electron from the nucleus.
- The larger the value of “n,” the higher the energy level and the farther the electron is from the nucleus.
- Their symbols: K,L,M,N,O,P,Q

**Examples: **

For n = 1, 2, 3, 4, and so on, we have the first, second, third, fourth, and higher energy levels, respectively.

The principal quantum number (n) represents the main energy level or shell in which an electron resides. Each energy level is associated with a specific symbol. Here are the symbols for the first few energy levels:

**n = 1:** The first energy level is often denoted as “K” shell.

**n = 2:** The second energy level is often denoted as “L” shell.

**n = 3:** The third energy level is often denoted as “M” shell.

**n = 4:** The fourth energy level is often denoted as “N” shell.

**n = 5:** The fifth energy level is often denoted as “O” shell.

**n = 6:** The sixth energy level is often denoted as “P” shell.

n** = 7:** The seventh energy level is often denoted as “Q” shell.

These symbols are used to label the different electron energy levels in an atom, and they are derived from the historical development of atomic theory. Each energy level can contain one or more subshells or orbitals (designated by the angular momentum quantum number, l) with varying shapes and orientations.

The number of electrons that can be accommodated in each principal quantum number (n) is given by the formula:

Number of Electrons in a Shell **(n) = 2n^2**

This formula provides the maximum number of electrons that can occupy a given energy level or shell (n) in an atom.

Here are the maximum numbers of electrons in the first few principal quantum numbers (n):

**For n = 1: 2 electrons**

**For n = 2: 8 electrons**

**For n = 3: 18 electrons**

**For n = 4: 32 electrons**

It’s important to note that this formula gives the maximum capacity of each energy level, but not all energy levels are fully occupied in real atoms.

The actual distribution of electrons in an atom depends on the atom’s atomic number and the filling of specific orbitals within each energy level, which is determined by the **Pauli Exclusion Principle **and **Hund’s Rule.**

**The rule 2n^2 not applied to the higher than fourth level ?**

Because the atom become unstable when it has more than 32 electrons

The formula “2n^2” for the maximum number of electrons in a given principal quantum level (n) is a simplified approximation and is most accurate for the first few energy levels.

However, it becomes less accurate for higher energy levels due to the more complex electron filling patterns in atoms as we move further from the nucleus. As we go beyond the fourth energy level, additional factors such as the presence of inner electron shells and electron-electron interactions come into play, making the actual electron distribution in these higher energy levels different from the simplified formula.

For example, while “2n^2” suggests that the fifth energy level (n = 5) should be able to hold a maximum of 50 electrons, the actual electron configuration of elements indicates that it can hold more than 50 electrons due to the presence of inner shells and electron interactions. In reality, the energy levels beyond the fourth can accommodate more electrons because of the complexity of electron distribution.

So, while **“2n^2” **serves as a helpful approximation for the first few energy levels, it’s important to recognize its limitations and refer to actual electron configurations and atomic structure principles for a more accurate understanding of electron distribution in atoms, especially for higher energy levels.

The energy of electron energy levels (shells) in an atom generally increases as you move farther from the nucleus. Electrons in the innermost energy levels have lower energy, while those in higher energy levels have higher energy. The exact energy levels of electrons in an atom are determined by quantum mechanics, and they depend on the atomic number of the element.

The energy levels can be labeled using the principal quantum number (n), where higher values of n correspond to higher energy levels.

The energy levels are usually expressed in terms of electron volts (eV) or joules (J) per electron.

The energy of the levels can vary for different elements, but the general trend is that they become further apart as you move away from the nucleus.

For example, the approximate energy levels for hydrogen, the simplest element, are as follows:

n = 1 (K shell): -13.6 eV

n = 2 (L shell): -3.4 eV

n = 3 (M shell): -1.5 eV

n = 4 (N shell): -0.85 eV

It’s important to note that the negative sign indicates that the electrons are bound to the nucleus, and energy must be added to remove them from the atom. These energy values are for hydrogen, and for other elements, the energy levels will be different due to the influence of the atomic nucleus and the presence of multiple electrons.

The energy levels for multi-electron atoms are more complex due to electron-electron interactions and shielding effects. Electrons in the same energy level do not necessarily have the same energy because of these interactions. The specific energies of energy levels for other elements can be calculated using quantum mechanics, but they are generally more challenging to predict compared to the simplified model for hydrogen.

- It is called The subsidiary quantum number (l)
- The subsidiary quantum number, often referred to as the angular momentum quantum number or azimuthal quantum number, is denoted by the letter “l.”
- This quantum number is one of the four quantum numbers used to describe the quantum state of an electron in an atom, and it provides information about the shape of the electron’s orbital.
- This quantum number defines the shape of the electron’s orbital.
- It is also called the azimuthal quantum number.

**Examples:**

For each value of “n,” there can be multiple values of “l” (0 to n-1), each corresponding to a specific type of orbital.

For example, when n = 2, l can be 0 or 1, corresponding to s and p orbitals, respectively.

**Here are the key characteristics of the subsidiary quantum number (l):**

- Definition: The subsidiary quantum number (l) defines the shape of the electron’s orbital within a particular energy level (principal quantum number, n).
- It specifies the type of subshell or suborbital in which the electron is found.
- Values: The possible values of the subsidiary quantum number (l) range from 0 to (n-1), where “n” is the principal quantum number of the energy level. In other words, if the principal quantum number (n) is 1, l can only be 0, and if n is 2, l can be 0 or 1, and so on.

Subshell Designations: Each value of “l” corresponds to a specific type of subshell or orbital.

**The designations are as follows:**

l = 0 corresponds to an s orbital.

l = 1 corresponds to a p orbital.

l = 2 corresponds to a d orbital.

l = 3 corresponds to an f orbital, and so on.

Number of Subshells:

- The number of possible subshells in an energy level
**is equal to the number of possible values of “l.”** - For example, in the second energy level (n = 2), there are two possible values for “l” (0 and 1), which means there are two types of subshells (s and p).
- The subsidiary quantum number (l) is essential in understanding the diversity of orbital shapes and orientations within an energy level.
- It plays a crucial role in determining the structure of the electron cloud around the nucleus and is a key component of the quantum mechanical model of the atom.

**Symbols of sublevels and the values of l : in table**

Here is a table that lists the symbols of sublevels (subshells) and their corresponding values of the subsidiary quantum number (l):

**Value of “l” Sublevel Symbol**

0 s

1 p

2 d

3 f

4 g

5 h

6 i

7 j

- These values of “l” represent different types of subshells or orbitals within an energy level (n).
- For example, when “l” is 0, it corresponds to an
**s subshell**, and when “l” is 1, it corresponds to a**p subshell**, and so on. - These sublevels define the shape and orientation of the electron orbitals within the energy level.

- This quantum number specifies the
**orientation**of the orbital in space. - It determines the number of orbitals of a particular type (given by “l”) and their spatial orientation.
- The magnetic quantum number, denoted by m_l, is one of the four quantum numbers that describe the quantum state of an electron in an atom.
- It specifies the orientation of an electron’s orbital within a particular subshell or sublevel defined by the subsidiary quantum number (l). In other words,
- it provides information about the spatial orientation of the electron’s probability cloud within a subshell

**Examples: **

For each value of “l,” there are 2l+1 possible values of m_l.

For example, if l = 1 (p orbital), m_l can be -1, 0, or 1, indicating the three different orientations of the p orbital.

**Values:** The values of m_l range from **-l to +l,** including zero, and they represent the different spatial orientations or magnetic sublevels within a given subshell.

**Examples: **

Here are some examples to illustrate the concept of the magnetic quantum number:

**l = 0 (s subshell):**

If l = 0 (s subshell), m_l can only be 0. This means there is only one spatial orientation or one spherical orbital within the s subshell.

** ****l = 1 (p subshell):**

If l = 1 (p subshell), m_l can take on three values: -1, 0, and +1. This represents the three spatial orientations of the p orbitals along the x, y, and z axes.

**l = 2 (d subshell):**

If l = 2 (d subshell), m_l can take on five values: -2, -1, 0, +1, and +2. This represents the five spatial orientations of the d orbitals within the d subshell.

In summary, the magnetic quantum number (m_l) provides information about the spatial orientation of electron orbitals within subshells and is an important component of the quantum mechanical description of atomic and molecular behavior.

- Definition: The spin quantum number represents the intrinsic spin of the electron.
- Electrons can have one of two possible spin values: +1/2 or -1/2, denoted as “up” and “down” spins, respectively.
- This parameter is not related to the electron’s position in the atom but is essential for the Pauli Exclusion Principle, which states that
in an atom can have the__no two electrons__of quantum numbers.__same set__

- The principal quantum number (n) determines the energy level and is related to the
**average distance**of the electron from the nucleus. **The angular momentum quantum**number (l) is related to the**shape of the****electron’s orbital**within that energy level (n).**The magnetic quantum**number (m_l) specifies the**orientation**of the**orbital in space**and can take values from -l to +l.(m_s) is either +1/2 or -1/2 and describes the intrinsic spin of the electron.__The spin quantum number__

Electrons with opposite spin values can occupy the same orbital.

In summary, quantum numbers provide a comprehensive description of the quantum state of electrons in an atom, enabling us to understand their energy levels, orbital shapes, orientations, and spin behaviors within the atomic structure.

**The relation between the principal quantum number and the subsidiary quantum number (l):**

The principal quantum number (n) and the subsidiary quantum number (l) are related in that the principal quantum number defines the energy level (shell) in which an electron is located, and the subsidiary quantum number determines the shape of the electron’s orbital within that energy level. Here’s how they are related:

**Principal Quantum Number (n):**

Represents the main energy level or shell of an electron.

Determines the overall energy of the electron.

Defines the maximum number of subshells (sublevels) within an energy level.

The possible values for “n” are positive integers (1, 2, 3, 4, …).

Subsidiary Quantum Number (l):

Also known as the angular momentum quantum number.

Defines the shape of the electron’s orbital within a specific energy level (n).

Determines the type of subshell or suborbital.

The possible values for “l” range from 0 to (n-1).

l = 0 corresponds to an s subshell.

l = 1 corresponds to a p subshell.

l = 2 corresponds to a d subshell.

And so on.

Examples:

**Principal Quantum Number (n) = 1:**

In the first energy level (n = 1), there is only one subshell with “l = 0,” which corresponds to an s subshell. So, within this energy level, there is only an s orbital.

**Principal Quantum Number (n) = 2:**

In the second energy level (n = 2), there are two possible values for “l”: 0 and 1.

When “l = 0,” it corresponds to an s sub-shell, so there is an s orbital in this level.

When “l = 1,” it corresponds to a p sub-shell, so there are three p orbitals (px, py, pz) in this level.

**Principal Quantum Number (n) = 3:**

In the third energy level (n = 3), there are three possible values for “l”: 0, 1, and 2.

When “l = 0,” it corresponds to an s subshell, so there is an s orbital.

When “l = 1,” it corresponds to a p subshell, so there are three p orbitals.

When “l = 2,” it corresponds to a d subshell, so there are five d orbitals.

These examples illustrate how the values of the subsidiary quantum number (l) are limited by the principal quantum number (n) and determine the variety of orbital shapes and orientations within each energy level. The relationship between “n” and “l” is essential in understanding the electron distribution in atoms.

**The difference in the energy of sublevels in each principal level **

- The energy of sublevels within a principal energy level (shell) is not the same. Sublevels within a given energy level have different energies, and this energy difference is a result of the quantum mechanical behavior of electrons in atoms.
- The energy order of sublevels is based on the value of the subsidiary quantum number (l).
- In general, the energy order of sublevels within a given principal energy level follows this pattern:

**s sublevel (l = 0):** This is the lowest energy sublevel within the energy level.

**p sublevel (l = 1):** The p sublevel is higher in energy compared to the s sublevel.

**d sublevel (l = 2):** The d sublevel is higher in energy compared to both the s and p sublevels.

**f sublevel (l = 3):** The f sublevel is higher in energy compared to the s, p, and d sublevels.

The energy difference between these sublevels increases with the value of the subsidiary quantum number (l). For example, within the third principal energy level (n = 3), the energy order of sublevels is as follows:

**3s (l = 0) < 3p (l = 1) < 3d (l = 2) < 3f (l = 3)**

So, the energy difference between 3s and 3p is smaller than the energy difference between 3p and 3d, and so on.

It’s important to note that the energy levels themselves are also different, with higher principal energy levels having higher energy compared to lower ones. However, the primary energy differences between sublevels are determined by the value of the subsidiary quantum number (l).

When the principal quantum number (n) is equal to 1, there is only one possible value for the subsidiary quantum number (l). The allowed values of l are limited by the rule:

**0 ≤ l < n**

So, in the case of n = 1, the only allowed value for l is 0. Therefore, when n = 1, there is only one sublevel (s sublevel) with l = 0.

**What are the probable (l) values when n= 2**

When the principal quantum number (n) is equal to 2, there are two probable values for the subsidiary quantum number (l). The allowed values of l are limited by the rule:

**0 ≤ l < n**

**So, for n = 2:**

- The first allowed value for l is 0. This corresponds to an s sublevel (l = 0), denoted as the 2s sublevel.
- The second allowed value for l is 1. This corresponds to a p sublevel (l = 1), denoted as the 2p sublevel.
- Therefore, when n = 2, there are two possible sublevels: 2s (l = 0) and 2p (l = 1).

**What are the probable (l) values when n= 3**

When the principal quantum number (n) is equal to 3, there are three probable values for the subsidiary quantum number (l). The allowed values of l are limited by the rule:

**0 ≤ l < n**

**So, for n = 3:**

- The first allowed value for l is 0. This corresponds to an s sublevel (l = 0), denoted as the 3s sublevel.
- The second allowed value for l is 1. This corresponds to a p sublevel (l = 1), denoted as the 3p sublevel.
- The third allowed value for l is 2. This corresponds to a d sublevel (l = 2), denoted as the 3d sublevel.
- Therefore, when n = 3, there are three possible sublevels: 3s (l = 0), 3p (l = 1), and 3d (l = 2).

**What are the probable (l) values when n= 4**

When the principal quantum number (n) is equal to 4, there are four probable values for the subsidiary quantum number (l). The allowed values of l are limited by the rule:

**0 ≤ l < n**

**So, for n = 4:**

- The first allowed value for l is 0. This corresponds to an s sublevel (l = 0), denoted as the 4s sublevel.
- The second allowed value for l is 1. This corresponds to a p sublevel (l = 1), denoted as the 4p sublevel.
- The third allowed value for l is 2. This corresponds to a d sublevel (l = 2), denoted as the 4d sublevel.
- The fourth allowed value for l is 3. This corresponds to an f sublevel (l = 3), denoted as the 4f sublevel.
- Therefore, when n = 4, there are four possible sublevels: 4s (l = 0), 4p (l = 1), 4d (l = 2), and 4f (l = 3).

**Mention the sublevels which exist in an atom if its last principal level is L:answer in a table**

If the last principal energy level in an atom is denoted as “L” (which corresponds to n = 2), the sublevels that exist in this atom are 2s and 2p. Here’s the information in table form:

**Principal Energy Level (n) Sublevels Present**

1(K) 1s

2 (L) 2s, 2p

In this case, for the principal energy level “L” (n = 2), there are two sublevels: 2s and 2p. These sublevels correspond to the subshells or orbitals available in the second energy level of the atom.

**How to detect the values of the subsidiary quantum number (l): with examples ?**

The values of the subsidiary quantum number (l) can be determined based on the principal quantum number (n) for a given energy level. The allowed values of l are limited by the rule:

**0 ≤ l < n**

Here are some examples of how to detect the values of l for different energy levels (n):

**Energy Level (n = 1):**

The only allowed value for l is 0.

Example: For n = 1, l = 0, and the only sublevel is 1s.

** ****Energy Level (n = 2):**

For n = 2, there are two possible values for l: 0 and 1.

Example: When n = 2, the values of l are 0 and 1, corresponding to 2s and 2p sublevels.

**Energy Level (n = 3):**

For n = 3, there are three possible values for l: 0, 1, and 2.

Example: When n = 3, the values of l are 0, 1, and 2, corresponding to 3s, 3p, and 3d sublevels.

** ****Energy Level (n = 4):**

For n = 4, there are four possible values for l: 0, 1, 2, and 3.

Example: When n = 4, the values of l are 0, 1, 2, and 3, corresponding to 4s, 4p, 4d, and 4f sublevels.

These examples illustrate how the values of the subsidiary quantum number (l) are determined based on the principal quantum number (n). The relationship between n and l is essential in describing the variety of sublevels or subshells within different energy levels of an atom.

**How to detect the number of orbitals in each principal level: rule and examples**

The number of orbitals in each principal energy level (shell) is determined by the value of the principal quantum number (n). To find the number of orbitals in a given principal level, you can use the following rule:

Number of Orbitals in a Principal Energy Level (n) = n^2

**Here are some examples to illustrate this rule:**

**For n = 1:**

Number of Orbitals = 1^2 = 1

In the first energy level (n = 1), there is 1 orbital, which is the 1s orbital.

**For n = 2:**

Number of Orbitals = 2^2 = 4

In the second energy level (n = 2), there are 4 orbitals: 2s, 2p (consisting of three orbitals: 2px, 2py, 2pz).

**For n = 3:**

Number of Orbitals = 3^2 = 9

In the third energy level (n = 3), there are 9 orbitals: 3s, 3p (three orbitals), 3d (five orbitals).

**For n = 4:**

Number of Orbitals = 4^2 = 16

In the fourth energy level (n = 4), there are 16 orbitals: 4s, 4p (three orbitals), 4d (five orbitals), 4f (seven orbitals).

The rule “Number of Orbitals in a Principal Energy Level (n) = n^2” gives you the total number of orbitals in each energy level. This rule is useful for understanding the distribution of electrons in atoms and predicting the complexity of electron arrangements as you move to higher energy levels.

**How to detect the number of the sublevels orbitals ?**

The number of sublevel orbitals within a principal energy level (shell) is determined by the value of the subsidiary quantum number (l). To find the number of sublevel orbitals in a given principal level, you can use the following rule:

Number of Sublevel Orbitals in a Principal Energy Level (n) = 2l + 1

Here are some examples to illustrate this rule:

**For n = 1:**

No matter what value of l (which can only be 0 for n = 1), you will have only one sublevel orbital.

For l = 0, there is 2(0) + 1 = 1 sublevel orbital (1s orbital).

**For n = 2:**

Number of Sublevel Orbitals = 2l + 1

For the 2s sublevel (l = 0), there are 2(0) + 1 = 1 sublevel orbital (2s orbital).

For the 2p sublevel (l = 1), there are 2(1) + 1 = 3 sublevel orbitals (2px, 2py, 2pz orbitals).

** ****For n = 3:**

Number of Sublevel Orbitals = 2l + 1

For the 3s sublevel (l = 0), there are 2(0) + 1 = 1 sublevel orbital (3s orbital).

For the 3p sublevel (l = 1), there are 2(1) + 1 = 3 sublevel orbitals (3px, 3py, 3pz orbitals).

For the 3d sublevel (l = 2), there are 2(2) + 1 = 5 sublevel orbitals (3dx²-y², 3dz², 3dxy, 3dxz, 3dyz orbitals).

** ****For n = 4:**

Number of Sublevel Orbitals = 2l + 1

For the 4s sublevel (l = 0), there are 2(0) + 1 = 1 sublevel orbital (4s orbital).

For the 4p sublevel (l = 1), there are 2(1) + 1 = 3 sublevel orbitals (4px, 4py, 4pz orbitals).

For the 4d sublevel (l = 2), there are 2(2) + 1 = 5 sublevel orbitals (4dx²-y², 4dz², 4dxy, 4dxz, 4dyz orbitals).

For the 4f sublevel (l = 3), there are 2(3) + 1 = 7 sublevel orbitals (4f orbitals).

The rule “Number of Sublevel Orbitals in a Principal Energy Level (n) = 2l + 1” provides the count of sublevel orbitals within each energy level, and it is based on the value of the subsidiary quantum number (l).

**How to detect the values of the magnetic quantum number (ml) ? **

The values of the magnetic quantum number (m_l) are determined by the subsidiary quantum number (l).

The rule for determining the values of m_l is as follows:

**Values of m_l range from -l to +l, including zero.**

Here are examples to illustrate this rule for different values of the subsidiary quantum number (l):

**For l = 0 (s sublevel):**

Values of m_l can only be 0.

Example: In the 1s sublevel, m_l = 0.

** ****For l = 1 (p sublevel):**

Values of m_l range from -1 to +1, including zero.

Examples: In the 2p sublevel, m_l can be -1, 0, or +1.

**For l = 2 (d sublevel):**

Values of m_l range from -2 to +2, including zero.

Examples: In the 3d sublevel, m_l can be -2, -1, 0, +1, or +2.

**For l = 3 (f sublevel):**

Values of m_l range from -3 to +3, including zero.

Examples: In the 4f sublevel, m_l can be -3, -2, -1, 0, +1, +2, or +3.

The values of m_l represent the different spatial orientations of the orbitals within a given sublevel. For each sublevel, there are 2l + 1 possible values of m_l. These values help distinguish individual orbitals within a sublevel and play a crucial role in understanding the spatial distribution of electrons in atoms.

**How to detect the values of subsidary quantum numer (l)?**

The values of the subsidiary quantum number (l) are determined by the principal quantum number (n) and range from 0 to (n-1).

The rule for determining the values of l is as follows:

**0 ≤ l < n**

Here are examples to illustrate how to detect the values of l for different principal quantum numbers (n):

**For n = 1:**

The only allowed value for l is 0.

Example: When n = 1, l = 0.

** ****For n = 2:**

Values of l can be 0 or 1.

Example: When n = 2, l can be 0 (2s sublevel) or 1 (2p sublevel).

** ****For n = 3:**

Values of l can be 0, 1, or 2.

Example: When n = 3, l can be 0 (3s sublevel), 1 (3p sublevel), or 2 (3d sublevel).

**For n = 4:**

Values of l can be 0, 1, 2, or 3.

Example: When n = 4, l can be 0 (4s sublevel), 1 (4p sublevel), 2 (4d sublevel), or 3 (4f sublevel).

The values of l represent the type of sublevel or subshell within a particular principal energy level (n) and determine the shape and orientation of the electron orbitals. The number of allowed values of l is equal to the number of different types of sublevels within that energy level.

Here’s a table that illustrates the values of the subsidiary quantum number (l) based on different principal quantum numbers (n):

Principal Quantum Number (n) Values of Subsidiary Quantum Number (l)

1 0

2 0, 1

3 0, 1, 2

4 0, 1, 2, 3

This table shows the allowed values of l for each principal quantum number (n). For example, when n = 2, the values of l can be 0 and 1, representing the 2s and 2p sublevels. Similarly, for n = 4, the values of l can be 0, 1, 2, and 3, corresponding to the 4s, 4p, 4d, and 4f sublevels.

The capacity of sublevels, which refers to the maximum number of electrons that can occupy a specific sublevel, is determined by the following rules:

**s sublevel:** The s sublevel has one orbital, and each orbital can hold a maximum of 2 electrons. Therefore, the s sublevel can hold a maximum of 2 electrons.

**p sublevel:** The p sublevel has three orbitals, and each orbital can hold a maximum of 2 electrons. Therefore, the p sublevel can hold a maximum of 6 electrons (3 orbitals x 2 electrons per orbital).

**d sublevel:** The d sublevel has five orbitals, and each orbital can hold a maximum of 2 electrons. Therefore, the d sublevel can hold a maximum of 10 electrons (5 orbitals x 2 electrons per orbital).

**f sublevel**: The f sublevel has seven orbitals, and each orbital can hold a maximum of 2 electrons. Therefore, the f sublevel can hold a maximum of 14 electrons (7 orbitals x 2 electrons per orbital).

These capacity rules are based on the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers (including the same values of n, l, m_l, and m_s). As a result, the capacity of each sublevel is determined by the number of orbitals it contains and the maximum number of electrons that each orbital can accommodate.

**what are the probable (ml) values when l=1 ?**

When the subsidiary quantum number (l) is equal to 1, the probable values of the magnetic quantum number (m_l) range from -1 to +1, including zero. The rule for determining these values is as follows:

-l ≤ m_l ≤ l

**For l = 1:**

The allowed values for m_l are -1, 0, and +1.

So, when l = 1, the probable values of m_l are -1, 0, and +1, representing three different spatial orientations for the p sublevel orbitals.

**what are the probable (ml) values when l=2 ?**

When the subsidiary quantum number (l) is equal to 2, the probable values of the magnetic quantum number (m_l) range from -2 to +2, including zero. The rule for determining these values is as follows:

-l ≤ m_l ≤ l

For l = 2:

The allowed values for m_l are -2, -1, 0, +1, and +2.

So, when l = 2, the probable values of m_l are -2, -1, 0, +1, and +2, representing five different spatial orientations for the d sublevel orbitals.

**what are the probable (ml) values when l=3 ?**

When the subsidiary quantum number (l) is equal to 3, the probable values of the magnetic quantum number (m_l) range from -3 to +3, including zero. The rule for determining these values is as follows:

-l ≤ m_l ≤ l

For l = 3:

The allowed values for m_l are -3, -2, -1, 0, +1, +2, and +3.

So, when l = 3, the probable values of m_l are -3, -2, -1, 0, +1, +2, and +3, representing seven different spatial orientations for the f sublevel orbitals.

Here’s a short test to assess your understanding of the concepts related to electron configurations, quantum numbers, and sublevels. Choose the correct answers for each question:

1-Which of the following quantum numbers describes the shape of an electron’s orbital?

- Principal quantum number (n)
- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

2-For the 3d sub-level, how many possible values of the magnetic quantum number (m_l) are there?

- 1
- 3
- 5
- 7

3-What is the maximum number of electrons that can occupy the 4p sublevel?

- 2
- 4
- 6
- 8

4-In an atom, if an electron has n = 3 and l = 1, which sub-level is it in?

- 3s
- 3p
- 3d
- 3f

5-How many electrons can occupy the 4f- sub-level (n = 4) in total?

- 5
- 10
- 15
- 15

6-In the electron configuration 1s² 2s² 2p⁶ 3s¹, which sublevel is being filled after 2p⁶?

- 3s
- 3p
- 4s
- 4p

7-If the magnetic quantum number (m_l) for an electron is -2, what is the value of the subsidiary quantum number (l)?

- 0
- 1
- 2
- 3

8-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶, how many electrons have spin-up (m_s = +1/2)?

- 8
- 12
- 9
- 18

Answers:

- Subsidiary quantum number (l)
- 5
- 6
- 3p
- 25
- 3s
- 2
- 9

*****************

1-Which quantum number specifies the energy level or shell of an electron?

- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Principal quantum number (n)
- Spin quantum number (m_s)

2-If an electron has a magnetic quantum number (m_l) of 0, what is the shape of its orbital?

- Spherical (s)
- Dumbbell-shaped (p)
- Complex (d)
- Irregular (f)

3-How many electrons can occupy the 6p sub-level?

- 2
- 4
- 6
- 8

4-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹, which sub-level is being filled after 3p⁶?

- 3d
- 4s
- 4p
- 4d

5-What is the maximum number of electrons that can be accommodated in the 2nd principal energy level (n = 2)?

- 2
- 8
- 18
- 32

6-if an electron has a spin quantum number (m_s) of -1/2, what is the value of its counterpart’s spin quantum number in the same orbital?

- +1/2
- -1/2
- 0
- 1

7-What is the sublevel designation for a quantum state with n = 4 and l = 2?

- 4s
- 4p
- 4d
- 4f

Answers:

1-c. Principal quantum number (n)

2-Spherical (s)

3-6

4-4s

5-8

6-+1/2

7-4d

************

1-In an electron’s quantum description, which quantum number specifies the orientation of the electron’s intrinsic angular momentum (spin)?

- Principal quantum number (n)
- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

2-How many electrons can occupy the 3d sublevel?

- 2
- 6
- 10
- 14

3-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s², which sublevel is being filled after 3p⁶?

- 3d
- 4p
- 4d
- 4f

4-What is the maximum number of electrons that can occupy the 5f sublevel?

- 2
- 4
- 6
- 14

5-If the principal quantum number (n) is 5, how many different sublevels are available within that principal energy level?

- 3
- 4
- 5
- 6

6-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁴, how many electrons have a magnetic quantum number (m_l) of -1?

- 2
- 6
- 10
- 4

7-For a given electron in the 4p sub-level, what are the probable values of the magnetic quantum number (m_l)?

- -1, 0, 1
- -2, -1, 0, 1, 2
- -3, -2, -1, 0, 1, 2, 3
- -4, -3, -2, -1, 0, 1, 2, 3, 4

Answers:

1-d. Spin quantum number (m_s)

2-10

3-4p

4-14

5-4

6-10

7- -1, 0, 1

************

1-In an atom, which quantum number specifies the shape and orientation of an electron’s orbital within a sublevel?

- Principal quantum number (n)
- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

2-How many electrons can occupy the 4d sublevel?

- 2
- 6
- 10
- 14

3-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁵, which sublevel is being filled after 4s²?

- 3d
- 4p
- 4d
- 4f

4-What is the maximum number of electrons that can occupy the 6p sublevel?

- 2
- 4
- 6
- 8

5-If the principal quantum number (n) is 6, how many different sublevels are available within that principal energy level?

- 3
- 4
- 5
- 6

6-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶, how many electrons have a magnetic quantum number (m_l) of -2?

- 2
- 6
- 5
- 8

7-For a given electron in the 5f sublevel, what are the probable values of the magnetic quantum number (m_l)?

- -4, -3, -2, -1, 0, 1, 2, 3, 4
- -3, -2, -1, 0, 1, 2, 3
- -2, -1, 0, 1, 2
- -1, 0, 1

Answers:

1-b. Subsidiary quantum number (l)

2-10

3-3d

4-6

5-6

6-5

7–4, -3, -2, -1, 0, 1, 2, 3, 4

*********

1-Which of the following quantum numbers uniquely identifies the electron within an atom?

- Principal quantum number (n)
- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

2-How many electrons can occupy the 5d sublevel?

- 2
- 6
- 10
- 14

3-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s², which sublevel is being filled after 4p⁶?

- 4d
- 5p
- 5d
- 5f

4-What is the maximum number of electrons that can occupy the 3p sublevel?

- 2
- 4
- 6
- 8

5-If the principal quantum number (n) is 7, how many different sublevels are available within that principal energy level?

- 3
- 4
- 5
- 6

6-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁹ 4s², how many electrons have a magnetic quantum number (m_l) of +1?

- 2
- 6
- 10
- 4

7-For a given electron in the 6p sublevel, what are the probable values of the magnetic quantum number (m_l)?

- -1, 0, 1
- -2, -1, 0, 1, 2
- -3, -2, -1, 0, 1, 2, 3
- -4, -3, -2, -1, 0, 1, 2, 3, 4

Answers:

- d. Spin quantum number (m_s)
- 10
- 5d
- 8
- 4
- 10
- -1, 0, 1

**********

1-Which quantum number specifies the main energy level or shell of an electron in an atom?

- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Principal quantum number (n)
- Spin quantum number (m_s)

2-How many sublevels (subshells) are there in the 4th principal energy level (n = 4)?

- 3
- 4
- 5
- 6

3-If an electron has a principal quantum number (n) of 3 and a subsidiary quantum number (l) of 2, what is the sublevel designation?

- 3s
- 3p
- 3d
- 3f

4-What is the maximum number of electrons that can occupy the 2p sublevel?

- 2
- 4
- 6
- 8

5-Which quantum number specifies the shape of an electron’s orbital within a sublevel?

- Principal quantum number (n)
- Subsidiary quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

6-In the electron configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s², which principal energy level is being filled after 3p⁶?

- 3
- 4
- 5
- 6

7-How many electrons can occupy the 5f sublevel?

- 2
- 4
- 6
- 14

8-If an electron has a magnetic quantum number (m_l) of -2, what is the value of the subsidiary quantum number (l)?

- 0
- 1
- 2
- 3

Answers:

- Principal quantum number (n)
- 4
- 3d
- 4
- Subsidiary quantum number (l)
- 4
- 14
- 2

**“Chemistry: The Central Science”** by Theodore L. Brown, H. Eugene LeMay, and Bruce E. Bursten – This widely used chemistry textbook covers atomic structure and electron configurations.

**“General Chemistry”** by Raymond Chang and Kenneth A. Goldsby – This textbook provides a comprehensive introduction to chemistry, including atomic theory and electron configurations.

**“Principles of Modern Chemistry”** by David W. Oxtoby, H. Pat Gillis, and Laurie J. Butler – This textbook covers modern principles of chemistry and includes discussions of quantum numbers and electron structure.

**Chemguide:** Quantum Numbers and Electron Configurations – An online educational resource that explains quantum numbers and electron configurations in detail.

**Khan Academy:** Quantum Numbers and Electron Configurations – Khan Academy offers free online lessons on this topic, which include video lectures and practice exercises.

**ChemCollective:** Electron Configurations – An interactive web resource that allows you to practice writing electron configurations for various elements.

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