Quantum numbers: Bonus Concept

In this post we, will discuss about the quantum numbers and its types.

The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers. There are four quantum numbers, namely, principal, azimuthal, magnetic and spin quantum numbers.

Electronic quantum numbers (the quantum numbers describing electrons) can be defined as a group of numerical values which provide solutions that are acceptable by the Schrodinger wave equation for hydrogen atoms.

Types of quantum numbers

There are 4 types of quantum numbers, which are as follows:

• Principal quantum number
• Azimuthal quantum number
• Magnetic quantum number
• Spin quantum number

Representation and possible values

Principal quantum number

• Principal quantum numbers are denoted by the symbol ‘n’. They designate the principal electron shell of the atom. Since the most probable distance between the nucleus and the electrons is described by it, a larger value of the principal quantum number implies a greater distance between the electron and the nucleus (which, in turn, implies a greater atomic size).
• The value of the principal quantum number can be any integer with a positive value that is equal to or greater than one. The value n=1 denotes the innermost electron shell of an atom, which corresponds to the lowest energy state (or the ground state) of an electron.
• Thus, it can be understood that the principal quantum number, n, cannot have a negative value or be equal to zero because it is not possible for an atom to have a negative value or no value for a principal shell.
• When a given electron is infused with energy (excited state), it can be observed that the electron jumps from one principle shell to a higher shell, causing an increase in the value of n. Similarly, when electrons lose energy, they jump back into lower shells and the value of n also decreases.
• The increase in the value of n for an electron is called absorption, emphasizing the photons or energy being absorbed by the electron. Similarly, the decrease in the value of n for an electron is called emission, where the electrons emit their energy.

Azimuthal quantum number

• The azimuthal (or orbital angular momentum) quantum number describes the shape of a given orbital. It is denoted by the symbol ‘l’ and its value is equal to the total number of angular nodes in the orbital.
• A value of the azimuthal quantum number can indicate either an s, p, d, or f subshell which vary in shapes. This value depends on (and is capped by) the value of the principal quantum number, i.e. the value of the azimuthal quantum number ranges between 0 and (n-1).
• For example, if n =3, the azimuthal quantum number can take on the following values – 0,1, and 2. When l=0, the resulting subshell is an ‘s’ subshell. Similarly, when l=1 and l=2, the resulting subshells are ‘p’ and ‘d’ subshells (respectively). Therefore, when n=3, the three possible subshells are 3s, 3p, and 3d.
• In another example where the value of n is 5, the possible values of l are 0, 1, 2, 3, and 4. If l = 3, then there are a total of three angular nodes in the atom.

Magnetic Quantum number

• This quantum number describes the behaviour of electron in a magnetic field.
• This quantum number gives the number of orbitals in a given subshell. Its value is from -l to +l.
• If the value of l is given as 2, then the value of m is from -2 to +2, i.e., -2, -1, 0, +1, +2. It indicates that the number of orbitals in d-subshell are 5 called d-orbitals.

Spin Quantum Number

• The electron spin quantum number is independent of the values of n, l, and m_l. The value of this number gives insight into the direction in which the electron is spinning, and is denoted by the symbol m_s.
• The value of m_s offers insight into the direction in which the electron is spinning. The possible values of the electron spin quantum number are +½ and -½.
• The positive value of m_s implies an upward spin on the electron which is also called ‘spin up’ and is denoted by the symbol ↑. If m_s has a negative value, the electron in question is said to have a downward spin, or a ‘spin down’, which is given by the symbol ↓.
• The value of the electron spin quantum number determines whether the atom in question has the ability to produce a magnetic field. The value of m_s can be generalized to ±½.

Shapes of orbital

Some Important Questions

Question: In a multi electron atom which of the following orbital described by the three quantum number will have the same energy in the absence of magnetic and electric fields.

1. n=1, l=0, m=0

2. n=2, l=0, m=0

3. n=2, l=1, m=1

4. n=3, l=2, m=0

5. n=3, l=2, m=1

(a) 1 and 2    (b) 2 and 3    (c) 3 and 4    (d) 4 and 5

Solution: We know, l=0 represents s-subshell, l=1 represents p-subshell, l=2 represents d-subshell and l=3 represents f-subshell.

For energy, consider the Aufbau’s principle. Greater the n+l value, greater the energy. If n+l value is same, then greater the n value, greater than energy.

Considering only n and l values, so

1. 1s-subshell

2. 2s-subshell

3. 2p-subshell

4. 3d-subshell

5. 3d-subshell

Points 4 and 5 have same energy because both belongs to same subshell, i.e., 3d.

Hence, option (d) is correct.

Question: What are the Possible Subshells when n = 4? How Many Orbitals are Contained by Each of these Subshells?

Solution: When n = 4, the possible l values are 0, 1, 2, and 3. This implies that the 4 possible subshells are the 4s, 4p, 4d, and 4f subshells.

• The 4s subshell contains 1 orbital and can hold up to 2 electrons.
• The 4p subshell contains 3 orbitals and can hold up to 6 electrons.
• The 4d subshell contains 5 orbitals and can hold up to 10 electrons.
• The 4f subshell has 7 orbitals and can hold up to 14 electrons.

Thus, a total of 4 subshells are possible for n = 4.

Question: What are the Possible m_l values for l = 4?

Solution: Since the value of the magnetic quantum number ranges from -l to l, the possible values of m_l when l = 4 are: -4, -3, -2, -1, 0, 1, 2, 3, and 4.

Question: Which energy level has the least energy?

Solution: There is a single 1s orbital that can accommodate 2 electrons at the lowest energy level, the one nearest to the atomic core. There are four orbitals at the next energy level; a 2s, 2p1, 2p2 and a 2p3. Each of these orbitals can carry 2 electrons, so we can find a total of 8 electrons at this energy level.

Read More: Unique Guide to Organic Conversions (Cheat Sheet included)

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