Question

(a) Explain the following trend in lattice energy: \(\mathrm{BeH}_{2}\), \(3205 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{MgH}_{2}, 2791 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{CaH}_{2}, 2410 \mathrm{~kJ} / \mathrm{mol}\) \(\mathrm{Sr} \mathrm{H}_{2}, 2250 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{BaH}_{2}, 2121 \mathrm{~kJ} / \mathrm{mol}\). (b) The lattice energy of \(\mathrm{ZnH}_{2}\) is \(2870 \mathrm{~kJ} / \mathrm{mol}\). Based on the data given in part (a), the radius of the \(\mathrm{Zn}^{2+}\) ion is expected to be closest to that of which group \(2 \mathrm{~A}\) element?

Short answer

The trend in lattice energy for the given compounds (BeH2, MgH2, CaH2, SrH2, BaH2) decreases as we move down the Group 2A elements due to the increase in ionic radius. The lattice energy of ZnH2 (2870 kJ/mol) falls between the values for MgH2 and CaH2, suggesting that the ionic radius of Zn2+ is closest to either Mg2+ or Ca2+ ions of Group 2A elements.

Step-by-step solution

Define Lattice Energy

Lattice energy is the energy required to separate one mole of an ionic compound into its constituent gaseous ions. It is a measure of the strength of the electrostatic forces between the oppositely charged ions in the solid. The greater the lattice energy, the stronger the electrostatic forces and the more stable the ionic compound.

Understand Factors Influencing Lattice Energy

Lattice energy depends on the charge of the ions and the distance between the ions in the crystal lattice. According to Coulomb's Law, the lattice energy (U) is directly proportional to the product of charges (q1 and q2) and inversely proportional to the distance (r) between the ions: \(U \propto \frac{q_1 \cdot q_2}{r}\) In case of the compounds in this exercise, all have the same anion (H-), so the ionic charge should not affect the trend much. The other factor, ionic radius, will be the dominant factor affecting the trend in lattice energy.

Explain the Trend in Lattice Energy for the Given Compounds

As we move down the Group 2A elements, the atomic size increases due to the addition of electron shells. As a result, the cations (Be2+, Mg2+, Ca2+, Sr2+, and Ba2+) also have increasing ionic radii. Since lattice energy is inversely proportional to ionic radius, we would expect that lattice energy decreases with increasing ionic radius: BeH2 > MgH2 > CaH2 > SrH2 > BaH2 This matches the given data for lattice energies of the compounds.

Estimate the Ionic Radius of Zn2+ Based on Lattice Energy

The lattice energy of ZnH2 is given as 2870 kJ/mol. Looking at the data given in part (a), we see that this value is between the lattice energies of MgH2 and CaH2. Since lattice energy is inversely proportional to ionic radius, we can infer that the ionic radius of Zn2+ should be closest to the ionic radii of either Mg2+ or Ca2+. We can conclude that the Zn2+ ion has an ionic radius closest to either the Mg2+ or Ca2+ ions from Group 2A elements.

Key concepts

Ionic Radius

The ionic radius is a critical concept in understanding how ionic compounds behave. It refers to the size of an ion, which can vary depending on several factors. The ionic radius essentially determines how closely ions can pack in a lattice structure. This directly impacts the strength of the ionic bonds and, therefore, the compound's properties, such as its lattice energy.

As you move down a group in the periodic table, such as Group 2A, the ionic radius increases. This occurs because additional electron shells spread further from the nucleus are added with each successive element. Consequently, despite the increase in atomic number and nuclear charge, the outer electrons experience more shielding and are held less tightly.

When discussing lattice energy, the ionic radius is vital. According to Coulomb's Law, with everything else being equal, a larger ionic radius results in a smaller lattice energy. This inverse relationship means that as the ions in an ionic compound become larger, the attraction between the ions weakens.

Group 2A Elements

Group 2A elements, also known as the alkaline earth metals, play a significant role in the study of ionic compounds. These elements include Beryllium (Be), Magnesium (Mg), Calcium (Ca), Strontium (Sr), and Barium (Ba). Each of these elements forms divalent cations ( ext{Be}^{2+}, ext{Mg}^{2+}, etc.), which means they lose two electrons to achieve a stable electron configuration.

As we examine these cations, a trend emerges, especially in their ionic radii and resulting properties such as lattice energy. Moving down the group, there is a notable increase in atomic and ionic sizes. This is due to the addition of electron shells, making each successive cation larger than the one before.

These size variations lead to changes in their physical and chemical properties. For instance, larger cations tend to form weaker bonds in ionic compounds due to the increased distance between charged ions, hence a trend of decreasing lattice energy as seen in ext{BeH}_2 > ext{MgH}_2 > ext{CaH}_2 > ext{SrH}_2 > ext{BaH}_2.

Coulomb's Law

Coulomb's Law is a fundamental principle in understanding ionic bonds and lattice energy in ionic compounds. It states that the electrostatic force between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law can be mathematically represented as:

• Force (F) is proportional to \( \frac{q_1 \cdot q_2}{r^2} \)

Here, \( q_1 \) and \( q_2 \) are the charges on the ions, and \( r \) is the distance (ionic radius) between the centers of the ions.

In the context of lattice energy, this principle helps explain why larger ionic radii result in lower lattice energies. As the ions grow larger and \( r \) increases, the electrostatic attraction becomes weaker. Conversely, smaller ions are closer together, resulting in a stronger attraction and higher lattice energy.

This concept underpins why in Group 2A elements, as the ionic radii increase down the group, the lattice energy systematically decreases.

Ionic Compounds

Ionic compounds are composed of positive and negative ions held together by strong electrostatic forces known as ionic bonds. These bonds arise from the transfer of electrons from a metal to a non-metal, resulting in a positive ion (cation) and a negative ion (anion).

These compounds, such as ext{MgH}_2 and ext{CaH}_2, are characterized by their high melting and boiling points, and their ability to conduct electricity when in solution or molten. This conductivity is due to the movement of ions, which are the charge carriers.

The stability of ionic compounds is often a reflection of their lattice energy. A higher lattice energy indicates a more stable compound due to the stronger attraction between the ions within the crystal lattice. Factors such as ionic charge and ionic radius, which are explained by Coulomb’s Law, determine how tightly the ions can pack together.

For example, the similar lattice energies of ext{ZnH}_2 to those of ext{MgH}_2 and ext{CaH}_2 indicate these compounds share closely related ionic radii, hinting at comparable sizes of the respective ext{Zn}^{2+}, ext{Mg}^{2+}, and ext{Ca}^{2+} ions.