Problem 48 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The \(1^{\text {st }}, 2^{\text {nd }}\), and the \(3^{\text {rd }}\) ionization enthalpies, \(I_{1}, I_{2}\), and \(I_{3}\), of four atoms with atomic numbers \(n, n+1, n+2\), and \(n+3\), where \(n<10\), are tabulated below. What is the value of \(n ?\) $$ \begin{array}{|c|c|c|c|} \hline {\begin{array}{c} \text { Atomic } \\ \text { number } \end{array}} & & {\text { Ionization Enthalpy }(\mathrm{kJ} / \mathrm{mol})} \\\ { 2 - 4 } & I_{1} & I_{,} & I_{3} \\ \hline n & 1681 & 3374 & 6050 \\ \hline n+1 & 2081 & 3952 & 6122 \\ \hline n+2 & 496 & 4562 & 6910 \\ \hline n+3 & 738 & 1451 & 7733 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The value of \(n\) is 3.

Step 1: Understand Ionization Enthalpy Definitions

Ionization enthalpy refers to the energy required to remove an electron from an atom in its gaseous state. The first ionization enthalpy \(I_1\) is the energy required to remove the first electron, the second ionization enthalpy \(I_2\) for the second electron, and so on. A significant jump in ionization enthalpy often indicates a change in electron shell or energy level.

Step 2: Identify Sudden Increase in Ionization Enthalpy

Analyze the given data to spot notable increases in enthalpy values. This usually suggests the removal of an electron from a stable electronic configuration, such as a noble gas configuration. Here, observe the increase from \(I_1\) to \(I_2\) or \(I_2\) to \(I_3\).

Step 3: Examine the Given Data

Check the ionization enthalpies provided in the table:- For \(n\), \(I_1 = 1681\), \(I_2 = 3374\), and \(I_3 = 6050\).- For \(n+1\), \(I_1 = 2081\), \(I_2 = 3952\), and \(I_3 = 6122\).- For \(n+2\), \(I_1 = 496\), \(I_2 = 4562\), and \(I_3 = 6910\).- For \(n+3\), \(I_1 = 738\), \(I_2 = 1451\), and \(I_3 = 7733\).

Step 4: Recognize a Nobility Jump

Focus on the entry where \(I_2\) jumps drastically after \(I_1\). For \(n+2\), the first ionization enthalpy \(I_1\) is 496 kJ/mol, which is significantly lower compared to \(I_2\) at 4562 kJ/mol. This suggests a stable configuration is reached by removing the first electron (similar to Noble gases).

Step 5: Conclude the Value of n

Based on the analysis, assume \(n+2\) has a noble gas configuration after the first ionization (indicated by the sudden jump). The known light noble gases in this range are Helium \(Z=2\) and Neon \(Z=10\). Comparing these values to the periodic table, \(n+2\) corresponds to \(Z=3\), which is Lithium.

Key Concepts

Periodic Table

The Periodic Table is a tabular arrangement of chemical elements, organized by their atomic numbers (number of protons in the nucleus), electron configurations, and recurring chemical properties. As you move across a period (row), elements gradually increase in atomic number. This progression results in elements with varied properties but also periodic patterns forming, which is reflected in the orders of ionization enthalpies. Understanding this table is crucial when studying elements' behaviors, such as their ionization energies. By moving down a group (column), elements typically show an increase in atomic size and often a decrease in ionization energy, due to the added electron shells making outer electrons easier to remove. When determining the atomic number in exercises like our original problem, it's important to compare ionization energies to those typical of known sets in the periodic table, especially around light elements or transitions, to pinpoint specific atomic numbers.

Noble Gas Configuration

A noble gas configuration refers to a stable electron arrangement found in noble gases, which are known for their lack of reactivity due to fully filled electron shells. These configurations confer stability and thus require more energy to disrupt. Therefore, atoms prefer reaching configurations similar to noble gases as their electronic end-goal. When electrons are removed or added to form ions, notable jumps in ionization energies often indicate the disruption of this stable configuration.

• First, noble gases like Helium (atomic number 2) and Neon (atomic number 10) stand out as having notably high ionization enthalpies with very little reactive tendencies, because their outmost electron shell is full.
• In our exercise, for an atom to show a sharp increase in ionization energies suggests that initially, its configuration was close to these stable adjustments.
• The noble gas configuration, therefore, acts somewhat like a barrier that, once crossed, prevents easy removal of electrons from the inner, more tightly bound shells.

Recognizing shifts toward or away from such configurations help conclude certain atomic identities, as seen when mapping values against noble gas traits.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an isolated atom in its gaseous state. This energy acts as an indicator of an atom's tendency to hold onto its electrons, impacted by both electron shell configuration and nuclear charge.

• The first ionization energy ( I_1 ) is usually less than subsequent ionization energies ( I_2, I_3 ) due to increased positive charge pulling on fewer remaining electrons.
• Particularly large ionization energy spikes can indicate that electrons are being removed from a stable electronic configuration, like a noble gas, hence requiring significantly more energy.
• These increases provide fingerprints for identifying departure points from stable configurations and can guide determinations like the atomic number in exercises.

This pattern explains why atoms such as Lithium ( Li, Z = 3 ) exhibit a large jump from the first to the second ionization energy, as removing one puts it into a stable Helium-like configuration. Understanding these principles in ionization energy aids in predicting atomic behavior and arranging elements logically in exercises.