Problem 123 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of magnesium in the vapour state absorbed \(1200 \mathrm{~kJ} \mathrm{~mol}^{-1}\) of energy. If the first and second ionization energies of \(\mathrm{Mg}\) are 750 and \(1450 \mathrm{~kJ} \mathrm{~mol}^{-1}\) respectively, the final composition of the mixture is (a) \(86 \% \mathrm{Mg}^{+}+14 \% \mathrm{Mg}^{2+}\) (b) \(36 \% \mathrm{Mg}^{+}+64 \% \mathrm{Mg}^{2+}\) (c) \(69 \% \mathrm{Mg}^{+}+31 \% \mathrm{Mg}^{2+}\) (d) \(31 \% \mathrm{Mg}^{+}+69 \% \mathrm{Mg}^{2+}\)

Step-by-Step Solution

Answer

(c) 69% \(\mathrm{Mg}^{+}\) and 31% \(\mathrm{Mg}^{2+}\).

Step 1: Identify the Energy Requirements for Ionization

The first ionization energy is the energy required to remove the first electron from magnesium (\( \mathrm{Mg} \)). It is given as \( 750 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \). The second ionization energy is the energy required to remove the second electron from magnesium to form \( \mathrm{Mg}^{2+} \); it is given as \( 1450 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \).

Step 2: Calculate Total Energy Required for Full Ionization

To completely ionize magnesium to \( \mathrm{Mg}^{2+} \), the total energy required is the sum of the first and second ionization energies: \(750 + 1450 = 2200 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \).

Step 3: Compare Absorbed Energy with Required Energy

The magnesium absorbs \( 1200 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \) of energy. This energy is enough for the first ionization (750 kJ/mol) and partially contributes to the second ionization, but it does not fully meet the required 1450 kJ/mol for the second ionization. Total energy absorbed is \(750 + (1200 - 750) = 1200 \, \mathrm{kJ} \, \mathrm{mol}^{-1}\).

Step 4: Determine the Fraction of Ions Formed

Since \( 750 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \) is used for forming \( \mathrm{Mg}^{+} \) and the remaining \( 450 \, \mathrm{kJ} \, \mathrm{mol}^{-1} \) (1200 - 750) contributes to forming \( \mathrm{Mg}^{2+} \), calculate the percentage contribution to \( \mathrm{Mg}^{2+} \): \( \frac{450}{1450} \times 100\% \approx 31\% \), which translates approximately to \( 31\% \; \mathrm{Mg}^{2+} \). Consequently, \( 69\% \; \mathrm{Mg}^{+} \).

Step 5: Choose the Correct Answer Based on Calculations

Based on the calculation from Step 4, the composition of the mixture is approximately \( 69\% \; \mathrm{Mg}^{+} \) and \( 31\% \; \mathrm{Mg}^{2+} \), aligning to option (c).

Key Concepts

Magnesium Ionization

When magnesium atoms ionize, they lose electrons to form positively charged ions. This process is crucial for several reactions and bonding in chemistry. Ionization of magnesium involves two main steps. - In the first step, magnesium (\( \text{Mg} \)) loses one electron to form magnesium ions with a single positive charge (\( \text{Mg}^+ \)).- In the second step, it can lose another electron to achieve a doubly charged positive ion, known as magnesium ion (\( \text{Mg}^{2+} \)).This transition is important because the electrons in the outer shell are the ones most easily lost or shared, enabling magnesium to participate in various chemical and physical processes. The ionization process requires energy to overcome the attraction between the electron and the nucleus.

First Ionization Energy

First ionization energy refers to the energy needed to remove the outermost electron from a neutral atom in its gaseous state. For magnesium, its first ionization energy is \( 750 \, \text{kJ mol}^{-1} \). This energy value indicates how strongly an atom holds onto its outer electron. - This value is important because it gives us an idea about the reactivity of magnesium. - The lower this energy, the easier it is for the atom to lose its electron, increasing reactivity.The ionization energy can be thought of as a measure of the atom's happiness in holding on to its electrons. Magnesium, with relatively low first ionization energy, tends to lose an electron readily to form compounds, contributing significantly to its role in chemistry and materials science.

Second Ionization Energy

The second ionization energy is a measure of the energy required to remove an electron from a singly charged ion. For magnesium, it is \( 1450 \, \text{kJ mol}^{-1} \).- The higher value compared to first ionization energy indicates that it's more difficult to remove the second electron from a positively charged ion. - This energy is necessary to form magnesium ions (\( \text{Mg}^{2+} \)) from \( \text{Mg}^{+} \), reflecting stronger electrostatic attractions in a positively charged ion.Understanding the second ionization energy is crucial for predicting the formation of ions in different conditions. Magnesium's chemistry largely depends on its ability to form doubly charged ions, which plays a significant role in materials like alloys or biological systems where magnesium is an essential element.