Problem 177 - A Complete Resource Book in Chemistry for JEE Main

Question

The electrons identified by quantum numbers \(\mathrm{n}\) and \(l\) \([\mathbf{2 0 1 2}]\) (1) \(\mathrm{n}=4, l=0\) (2) \(\mathrm{n}=4, l=1\) (3) \(\mathrm{n}=3, l=1\) (4) \(\mathrm{n}=3, l=2\) can be placed in order of increasing energy as (a) \(4<2<3<1\) (b) \(2<4<1<3\) (c) \(1<3<2<4\) (d) \(3<4<2<1\)

Step-by-Step Solution

Answer

The correct order of increasing energy is option (d) \(3<4<2<1\).

Step 1: Understanding the Quantum Numbers

Quantum numbers are sets of values that describe the unique quantum state of an electron. Here we are using principal quantum number \(n\), which denotes the energy level, and azimuthal quantum number \(l\), which determines the subshell. The energy of electrons increases with increasing \(n+l\). For electrons with the same \(n+l\), the one with the smaller \(n\) has lower energy.

Step 2: Calculate \(n+l\) Values

Calculate \(n+l\) for each given set of quantum numbers:1. For \(\mathrm{n}=4,\ l=0\), \(n+l = 4+0 = 4\).2. For \(\mathrm{n}=4,\ l=1\), \(n+l = 4+1 = 5\).3. For \(\mathrm{n}=3,\ l=1\), \(n+l = 3+1 = 4\).4. For \(\mathrm{n}=3,\ l=2\), \(n+l = 3+2 = 5\).

Step 3: Determine Energy Using \(n+l\) Rule

Determine the relative energies of the electrons based on their \(n+l\) values and principal quantum numbers. Electrons with lower \(n+l\) values have lower energy. If \(n+l\) is the same, the one with a lower \(n\) has lower energy.- Both options (1) and (3) have \(n+l = 4\), but option (3) with \(n=3\) is lower in energy than option (1) with \(n=4\).- Both options (2) and (4) have \(n+l = 5\), but option (4) with \(n=3\) is lower in energy than option (2) with \(n=4\).

Step 4: Arrange in Increasing Energy Order

Based on the \(n+l\) rule:- Option (3) with \(n=3, l=1\) is the lowest energy.- Next is option (1) with \(n=4, l=0\).- Then comes option (4) with \(n=3, l=2\).- Lastly, option (2) with \(n=4, l=1\) has the highest energy.This sequence corresponds to choice (d) \(3<4<2<1\).

Key Concepts

The n+l Rule

The "n+l rule" is used to determine the energy levels of electrons in an atom. It's a handy method to find which electrons are more likely to be closer to the nucleus and therefore possess lower energy. Here's how it works:

• The sum of the principal quantum number ( defined as "n") and the azimuthal quantum number ( defined as "l") gives the term "n+l".
• Electrons with the lowest "n+l" value are at a lower energy state and are more stable.
• If two electrons have the same "n+l" value, the electron with the lower "n" value is considered lower in energy.

For example, in the step-by-step solution given for this exercise, electrons were compared according to their "n+l" values. If two electrons had the same "n+l", like options (1) and (3), option (3) was lower in energy because it had a smaller "n" value. Understanding and applying this rule helps in predicting electron configurations and the order in which electrons fill orbitals.

Principal Quantum Number (n)

The principal quantum number, often represented by the letter "n," plays a critical role in defining an electron's state within an atom. It primarily indicates the electron's energy level and distance from the nucleus. Let's break it down:

• The principal quantum number "n" can take positive integer values starting from 1 (e.g., 1, 2, 3, ...).
• Higher values of "n" mean the electron is in a higher energy level, which generally means a greater average distance from the nucleus.
• As "n" increases, the number of available subshells and orbitals for electrons also increases.

For the exercise, the variable "n" was used to characterize different electrons. For instance, electrons with the same "n" value but different "l" values (like options (1) and (2)) were higher in energy compared to those with a lower "n" value with the same "n+l" (like options (3) and (4)). This showcases how "n" effectively contributes to determining an electron's potential energy and position in an atom.

Azimuthal Quantum Number (l)

The azimuthal quantum number "l" is critical for understanding the shape and type of the electron's orbital within an atom. This number helps to specify the subshell an electron occupies, adding detail to what's initially described by the principal quantum number "n." Here's a closer look:

• The possible values of "l" range from 0 to (n-1), where "n" is the principal quantum number for that electron.
• Each value of "l" corresponds to a particular type of subshell or orbital: 0 = s, 1 = p, 2 = d, 3 = f, and so forth.
• The "l" value doesn't just designate the type of subshell but also indirectly influences the potential energy of an electron within its respective energy level.

In the exercise solution, the azimuthal quantum number was used to help differentiate between electrons with the same principal quantum number "n". It determined which electron configuration would have lower or higher energy when the principal quantum number and "n+l" values were the same. This makes "l" paramount in understanding not just electron placement but also subtle energy variations within an atom's electron cloud.