Problem 14 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

In an atom how many orbital(s) will have the quantum numbers; \(n=3, l=2\) and \(m_{l}=+2 ?\) [Main Online April 9, 2013] (a) 5 (b) 3 (c) 1 (d) 7

Step-by-Step Solution

Answer

The atom has 1 orbital with these quantum numbers.

Step 1: Understand the Quantum Numbers

The principal quantum number, \(n\), indicates the energy level of an electron in an atom. Here, \(n = 3\). The azimuthal quantum number, \(l\), defines the shape of the orbital, with \(l = 2\) corresponding to a 'd' orbital. The magnetic quantum number, \(m_l\), specifies the orientation of the orbital in space. Each value of \(l\) has specific \(m_l\) values ranging from \(-l\) to \(+l\).

Step 2: Identify Possible \(m_l\) Values

Since \(l = 2\) for a d-orbital, the values of \(m_l\) range from \(-2\) to \(+2\), including these integers: \(-2, -1, 0, +1, +2\). This gives us five possible orientations: one for each integer value.

Step 3: Determine Specific Orbital

We need to find the orbital that corresponds to the given \(m_l = +2\). For \(l = 2\), the value \(m_l = +2\) uniquely specifies one of the five possible orientations of a d-orbital.

Step 4: Count the Number of Orbitals

Since the value \(m_l = +2\) corresponds to exactly one of the possible orientations, there is only one orbital that fits all given conditions \(n = 3\), \(l = 2\), and \(m_l = +2\).

Key Concepts

Principal Quantum Number

The principal quantum number, represented by \( n \), is a fundamental concept in quantum mechanics that denotes the primary energy level of an electron within an atom. This number essentially describes how far an electron is from the nucleus. More simply, it tells us how big the electron cloud or the electron's orbit is.

• When \( n = 1 \), the electron is closest to the nucleus and is in the first energy level.
• When \( n = 2 \), the electron is in the second energy level, which is larger and further from the nucleus compared to \( n = 1 \).
• This pattern continues for higher values of \( n \), each indicating a higher energy level and a greater distance from the nucleus.

In our exercise, \( n = 3 \), meaning the electron is in the third energy level. This level can accommodate more orbitals compared to \( n = 1 \) or \( n = 2 \), allowing for a more complex arrangement of electrons.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \( l \), is also known as the angular momentum quantum number. This number determines the shape of an electron's orbital, which is where the probability of finding an electron is highest. The value of \( l \) ranges from 0 to \( n-1 \).

• When \( l = 0 \), the orbital is spherical and referred to as an 's' orbital.
• When \( l = 1 \), the orbital has a dumbbell shape, called a 'p' orbital.
• For \( l = 2 \), the shape is more complex, known as a 'd' orbital, which is the case in our exercise.

Since the given problem has \( l = 2 \), it indicates that the electron is in a 'd' orbital. 'D' orbitals are more intricate in shape and orientation, allowing for a diverse range of possible orientations.

Magnetic Quantum Number

The magnetic quantum number, \( m_l \), is crucial for identifying the orientation of an electron's orbital within a subshell. Understanding \( m_l \) completes the picture drawn by \( n \) and \( l \), giving us the electron's exact position.For a given azimuthal quantum number \( l \), the \( m_l \) can take integer values from \(-l\) to \(+l\). These include all the integers in this range:

• If \( l = 0 \), then \( m_l = 0 \).
• For \( l = 1 \), \( m_l \) can be \(-1, 0, +1\).
• In our exercise, \( l = 2 \), which means \( m_l \) can be \(-2, -1, 0, +1, +2\).

The magnetic quantum number gives us the specific orientation among these five possible orientations. In the case provided, \( m_l = +2 \), uniquely specifies one orientation of the 'd' orbital. This specificity is why, in the exercise, exactly one orbital corresponds to all the given conditions \( n = 3 \), \( l = 2 \), and \( m_l = +2 \).