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

The composition is 69% Mg⁺ and 31% Mg²⁺.

Step-by-step solution

Understand the Energy Involved

We are given that one mole of magnesium in the vapor state absorbs 1200 kJ/mol of energy. The task requires us to find out how this energy distributes into the ionization of magnesium atoms.

Energy Requirement for Ionization

The first ionization energy of magnesium is 750 kJ/mol, which is the energy needed to remove the first electron, making it Mg⁺. The second ionization energy is 1450 kJ/mol for making Mg²⁺, additional to the 750 kJ/mol already used. This means to convert Mg to Mg²⁺ (from neutral magnesium), it requires a total of 2200 kJ/mol (750 + 1450).

Calculate Contributions to Ionization

Given the total energy absorbed is 1200 kJ/mol, this amount of energy can fully ionize magnesium to Mg⁺ (750 kJ/mol), but not enough to go beyond that to completely ionize to Mg²⁺ (requires an additional 1450 kJ/mol). Since we only have 450 kJ/mol left after forming Mg⁺, insufficient energy is available to fully achieve Mg²⁺.

Consider Mg to Mg²⁺ Conversion

Calculate the fraction of magnesium that can reach Mg²⁺. 450 kJ/mol out of 1450 kJ/mol required for Mg⁺ to Mg²⁺ is approximately 31%, since 450 / 1450 = 0.31.

Deduce Composition of Mixture

Now, the remaining magnesium ions that weren't further ionized (100% - 31% = 69%) remain as Mg⁺. This results in 31% Mg²⁺ and 69% Mg⁺.

Key concepts

Magnesium Ionization

Magnesium ionization involves removing electrons from a magnesium atom to form positively charged ions. These ions are represented as Mg⁺ and Mg²⁺, depending on how many electrons are removed. The process of ionization requires energy because removing an electron opposes the attractive force of the protons in the nucleus. Hence, we measure this energy as ionization energy, expressed typically in kilojoules per mole.

• The first ionization energy: This is the energy required to remove the first electron from a neutral magnesium atom forming Mg⁺. For magnesium, this energy is 750 kJ/mol.
• The second ionization energy: After the first electron is removed, more energy is needed to remove a second electron and form Mg²⁺. For magnesium, this energy is 1450 kJ/mol.

Together, these energies give a clear picture of how much total energy is needed to achieve different states of ionization. Magnesium requires a total of 2200 kJ/mol to fully ionize into Mg²⁺, considering both ionization energy stages.

Energy Distribution

The distribution of energy plays a crucial role in understanding how much of a substance can be ionized. In this scenario, magnesium vapor absorbs 1200 kJ/mol of energy, which must be divided between the ionization processes. Initially, the 750 kJ/mol is required to remove the first electron, creating Mg⁺. This leaves us with 450 kJ/mol remaining from the original 1200 kJ/mol to attempt further ionization.

Limitations of Energy

Unfortunately, 450 kJ/mol isn't enough to fully ionize to Mg²⁺ since the second ionization requires 1450 kJ/mol. Therefore, only a fraction of the magnesium will be able to form Mg²⁺. This scenario demonstrates why knowing the precise energy distribution is crucial in predicting the composition of ionized substances. Thanks to these calculations, we determine the ion distribution:

• 100% of the magnesium can ionize to Mg⁺.
• 31% can further ionize to Mg²⁺ based on the available energy.
• The remaining 69% remains as Mg⁺, since there is insufficient energy for further ionization.

Vapour State Chemistry

In vapour state chemistry, atoms exist in a gaseous state, where they may gain energy quite differently than in solid or liquid states. This state is crucial when considering energy absorption for processes like ionization. With magnesium in its vapour state, it readily absorbs energy applied to it, an initial step for ionization to occur. This contrasts with solid or liquid states, where atoms are bound together, making energy absorption processes less direct.

Significance of Vapour State

Atomistic interactions are different in the vapor state as the atoms are more widely spaced and can more individually absorb energy. Therefore, when energy such as 1200 kJ/mol is introduced, the gas-phase atoms respond more predictively to energy inputs:

• Absorbing precise energy amounts means more individual atoms are affected, leading to clear predictions about ionization states.
• In this state, we efficiently witness the distribution of energy among atoms, allowing for clear calculations of resultant ion states.

Understanding these unique dynamics explains how vapor state chemistry allows us to predict and calculate outcomes, like the observed distribution of magnesium ions after energy absorption.