Quantum Mechanical Model and Quantum Numbers in Hydrogen Atom

The quantum mechanical model describes the behavior of electrons in atoms. This model uses a set of four quantum numbers to define the unique state of an electron. These numbers include the principal quantum number (n), the angular momentum or azimuthal quantum number (l), the magnetic quantum number ($m_l$), and the spin quantum number ($m_s$). Understanding the relationship between these numbers is crucial for determining electron properties.

Angular Momentum Number (l)

The angular momentum number, denoted by 'l', describes the shape of an electron's orbital and the orbital angular momentum. Its values depend on the principal quantum number (n), ranging from 0 to n-1. Each 'l' value corresponds to a specific subshell: l=0 is an s orbital, l=1 is a p orbital, l=2 is a d orbital, and so on.

In this question, we are given that the angular momentum number is 4. This means we are considering an orbital with $l = 4$.

Orbital Magnetic Quantum Number ($m_l$)

The orbital magnetic quantum number, denoted by '$m_l$', describes the orientation of an orbital in space. For a given angular momentum number 'l', the possible values of '$m_l$' are integers ranging from $-l$ to $+l$, including zero.

The number of different permitted orbital magnetic quantum numbers for a given 'l' is determined by the formula:

Number of $m_l$ values $= 2l + 1$

Calculating Permitted Orbital Magnetic Quantum Numbers

Given the angular momentum number $l = 4$, we can calculate the number of different permitted orbital magnetic quantum numbers using the formula:

Number of $m_l$ values $= 2l + 1$

Substitute the given value of $l$ into the formula:

Number of $m_l$ values $= 2(4) + 1$

Number of $m_l$ values $= 8 + 1$

Number of $m_l$ values $= 9$

The permitted values for $m_l$ when $l=4$ are:

• $-4$
• $-3$
• $-2$
• $-1$
• $0$
• $1$
• $2$
• $3$
• $4$

Counting these values, we find there are exactly nine different permitted orbital magnetic quantum numbers.

Final Permitted Orbital Magnetic Quantum Numbers Count

Based on the calculations, if the angular momentum number for a hydrogen atom is 4, the number of different permitted orbital magnetic quantum numbers will be nine.