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

Short answer

Option (c): 69% \(\text{Mg}^+\) and 31% \(\text{Mg}^{2+}\).

Step-by-step solution

Understanding the Problem

We need to find the percentage composition of \(\text{Mg}^+\) and \(\text{Mg}^{2+}\) in the final mixture. Magnesium vapour absorbs \(1200 \mathrm{~kJ} \mathrm{~mol}^{-1}\) of energy. The ionization energies are given as \(750 \mathrm{~kJ} \mathrm{~mol}^{-1}\) for the first ionization and \(1450 \mathrm{~kJ} \mathrm{~mol}^{-1}\) for the second ionization.

Determine Ionization States

Based on the absorbed energy \((1200 \mathrm{~kJ} \mathrm{~mol}^{-1})\), we can only fully ionize the first electron because the energy is insufficient to remove both electrons (requires \(750 \mathrm{~kJ} + 1450 \mathrm{~kJ} = 2200 \mathrm{~kJ}\)). Therefore, some magnesium will be in the \(\text{Mg}^+\) state and some in the \(\text{Mg}^{2+}\) state.

Energy Distribution

Subtract the first ionization energy from the total energy: \(1200 \mathrm{~kJ} - 750 \mathrm{~kJ} = 450 \mathrm{~kJ}\) left. This means the \(450 \mathrm{~kJ}\) remaining energy can partially convert \(\text{Mg}^+\) to \(\text{Mg}^{2+}\).

Calculate the Composition

To convert \(\text{Mg}^+\) to \(\text{Mg}^{2+}\), the energy required is the second ionization energy \((1450 \mathrm{~kJ}\)). With \(450 \mathrm{~kJ}\), the fraction of \(\text{Mg}^{2+}\) is given by \(\frac{450}{1450}\). Calculate this value to find \(31\%\) as \(\text{Mg}^{2+}\) and the remaining \(69\%\) as \(\text{Mg}^+\).

Choose the Correct Answer

The percentage calculation gives \(69\% \text{Mg}^+\) and \(31\% \text{Mg}^{2+}\). This matches option (c), which is \(69\% \text{Mg}^+\) and \(31\% \text{Mg}^{2+}\).

Key concepts

Mole Concept

The mole concept is a fundamental aspect of chemistry that acts like a bridge between the macroscopic world of visible substances and the atomic level. Simply put, one mole is a specific quantity used to count particles, much like a dozen represents 12 items. This is essential for performing chemical calculations, as the mole helps us understand and quantify the amount of substance involved in chemical reactions or changes.
A mole is equal to Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities, such as atoms, molecules, or ions. This concept allows chemists to equate the mass of a substance to its chemical identity by relating this number to the entity's atomic or molecular weight. When examining phenomena like ionization energies or energy absorption, using the mole concept becomes crucial as it ensures consistency in calculation, enabling chemists to accurately determine the involved energies per mole.

Magnesium Ionization

Magnesium ionization is the process of removing electrons from magnesium atoms. Like other elements, magnesium can lose its outermost electrons to form positive ions, also known as cations.
Magnesium has two electrons in its outer shell, and it can undergo ionization in two steps:

• The first ionization removes one electron, producing \(\text{Mg}^+\), which requires energy input, known as the first ionization energy.
• The second ionization removes a second electron, resulting in \(\text{Mg}^{2+}\). This also demands energy, termed the second ionization energy.

Ionization energies are measured in kilojoules per mole \(\text{kJ mol}^{-1}\). In chemical reactions or processes, understanding how and when magnesium ions form helps predict and calculate the energy changes involved. This is observed in this exercise, where magnesium under a certain energy absorption forms a specific percentage of \(\text{Mg}^+\) and \(\text{Mg}^{2+}\).

Energy Absorption

Energy absorption in the context of chemical reactions refers to the process by which energy is taken up by an atom or molecule, often leading to a state change. When an atom like magnesium absorbs energy, it may cause electrons to be removed, leading to ionization. This absorbed energy is crucial as it quantitatively determines how far the ionization process can progress.
In our example, magnesium vapor absorption of \(1200\, \text{kJ mol}^{-1}\) signals how much energy is available to facilitate the ionization of magnesium atoms. The available energy will dictate whether only the first ionization (\(\text{Mg}^+\)) occurs or if enough energy remains after the first ionization to enable the second electron removal (\(\text{Mg}^{2+}\)). Understanding how much energy it takes compared to how much is absorbed lets us calculate and predict the resulting composition of ions.

Chemical Calculations

Chemical calculations involve using various formulas and mathematical operations to determine quantities related to chemical reactions. These are essential for understanding how substances react and in what proportions. In this exercise, we are concerned with energy calculations needed for ionization.
The ionization requires specific energy amounts: the first requiring \(750\, \text{kJ mol}^{-1}\) and the second \(1450\, \text{kJ mol}^{-1}\). This guides us through the calculations where, after the first electron loss, we have \(450\, \text{kJ}\) left from \(1200\, \text{kJ mol}^{-1}\) of absorbed energy. This leftover energy is used to partially convert \(\text{Mg}^+\) to \(\text{Mg}^{2+}\). From there, calculating the percentage composition of each ion requires dividing \(450\) by \(1450\) and multiplying by \(100\) to obtain \(31\%\) \(\text{Mg}^{2+}\). The remaining \(69\%\) is \(\text{Mg}^+\).

This precise calculation is crucial for accurate prediction and comprehension of chemical processes.