Problem 170 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following sets of quantum numbers represents the highest energy of an atom? [2007] (a) \(\mathrm{n}=3, l=2, m=1, \mathrm{~s}=+1 / 2\) (b) \(\mathrm{n}=4, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (c) \(\mathrm{n}=3, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (d) \(\mathrm{n}=3, l=1, \mathrm{~m}=1, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (a) \( (n=3, l=2) \) has the highest energy.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the position and energy of an electron in an atom. The principal quantum number, \( n \), indicates the main energy level, the azimuthal quantum number, \( l \), indicates the subshell or orbital shape, the magnetic quantum number, \( m \), specifies the orientation of the orbital, and the spin quantum number, \( s \), describes the direction of the electron's spin.

Step 2: Applying the n+l Rule

To compare the energy levels, we use the \( n+l \) rule. The sum of \( n+l \) determines the energy level, where a higher \( n+l \) corresponds to higher energy. If two options have the same \( n+l \), the orbital with the higher \( n \) has higher energy.

Step 3: Calculating for Option (a)

For the set \(( n=3, l=2)\), \( n+l = 3+2 = 5 \).

Step 4: Calculating for Option (b)

For the set \(( n=4, l=0)\), \( n+l = 4+0 = 4 \).

Step 5: Calculating for Option (c)

For the set \(( n=3, l=0)\), \( n+l = 3+0 = 3 \).

Step 6: Calculating for Option (d)

For the set \(( n=3, l=1)\), \( n+l = 3+1 = 4 \).

Step 7: Comparing n+l Values

Comparing the calculated \( n+l \) values, option (a) has the highest value \( n+l = 5 \). Therefore, option (a) represents the highest energy.

Key Concepts

n+l rule

In quantum chemistry, the "n+l rule" is a simple yet powerful guideline that helps us determine the energy levels of electrons within an atom. The "n" stands for the principal quantum number, while "l" is the azimuthal quantum number. By summing these two values, we get the "n+l" total. This sum allows us to predict which electron configurations have higher energy. The general principle is: the larger the "n+l" value, the higher the energy level of that electron.

• If two electron configurations have the same "n+l" total, the configuration with the higher principal quantum number "n" will be at a higher energy level.
• This rule is fundamental in determining the order in which orbital energies increase as one considers the position of electrons within an atom.

Understanding the "n+l" rule is crucial for grasping the behavior of electrons, especially for those just starting to dive into quantum mechanics. It provides a clear-cut method to compare and analyze different sets of quantum numbers in terms of their energy.

principal quantum number

The principal quantum number, often symbolized as "n," is one of the four quantum numbers that describe the unique quantum state of an electron. It defines the primary energy level of the electron, essentially the shell in which the electron resides. "n" can have positive integer values: 1, 2, 3, and so forth.

• Each increase in "n" represents a higher energy level and typically a greater distance of the electron from the nucleus.
• The principal quantum number influences the size of the electron cloud, with higher values indicating a larger cloud size.

This number gives insight into the general energy and radial distance an electron might inhabit in an atom. A higher "n" often means that the electron is less tightly bound to the nucleus, which can affect its chemical behavior and reactivity.

azimuthal quantum number

The azimuthal quantum number, designated as "l," determines the shape of the electron's orbital. It is also known as the angular momentum quantum number. For any given principal quantum number "n," the azimuthal quantum number "l" can range from 0 to (n-1).

• When "l" is 0, the shape is a spherical "s" orbital.
• When "l" is 1, it represents a dumbbell-shaped "p" orbital.
• "l" being 2 indicates a more complex "d" orbital shape.
• Higher values such as "l = 3" indicate "f" orbitals which have even more complex shapes.

The azimuthal quantum number therefore plays a crucial role in defining the characteristics of electrons within their orbits. It helps in determining the subshells and thus further divides the energy levels within a primary energy level.

magnetic quantum number

The magnetic quantum number, symbolized as "m," specifies the orientation of an electron's orbital relative to the other orbitals in space constrained within an atom. For a given azimuthal quantum number "l," "m" can take on integer values ranging from -l through 0 to +l.

• For instance, if "l" is 1 (a "p" orbital), then "m" might be -1, 0, or +1, indicating different orientations of these "p" orbitals in three-dimensional space.
• This quantum number is vital for understanding how magnetic fields influence electron positions, as its values imply the different potential orientations that orbitals may assume in these fields.

The concept of the magnetic quantum number allows chemists and physicists to deeply understand the three-dimensional arrangement of electron orbitals, a key to understanding many atomic interactions and behaviors.