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Chapter 8: Problem 63

Rationalize the following lattice energy values: $$ \begin{array}{|lc|} \hline \text { Compound } & \text { Lattice Energy (kJ/mol) } \\ \hline \text { CaSe } & -2862 \\ \mathrm{Na}_{2} \mathrm{Se} & -2130 \\ \mathrm{CaTe} & -2721 \\ \mathrm{Na}_{2} \mathrm{Te} & -2095 \\ \hline \end{array} $$

Short Answer

Expert verified
The lattice energy values can be rationalized by considering the charges of the ions and their relative sizes. CaSe has a more substantial lattice energy than Na鈧係e because Ca虏鈦� has a higher charge and smaller radius than Na鈦�. CaTe has a more significant lattice energy than Na鈧俆e due to the same reason. The compounds containing Se have higher lattice energy values than those containing Te because the Se虏鈦� ions have smaller ionic radii.

Step by step solution

01

Identify the ionic charges and radii

The first step in rationalizing the lattice energy values is to identify the charges of the ions involved and the sizes of these ions, called ionic radii. The ions and their ionic radii are as follows: - Ca虏鈦� ionic radius: 114 pm (picometers) - Na鈦� ionic radius: 116 pm - Se虏鈦� ionic radius: 198 pm - Te虏鈦� ionic radius: 221 pm Remember that higher charges and smaller ionic radii lead to more substantial lattice energies.
02

Understand the relationship between charges and lattice energy

The relationship between lattice energy and the ionic charges is given by the Born-Lande equation: \[E_l = -\frac{N_AMz^+z^-e^2}{4\pi\epsilon_0r}\] Where: - \(E_l\) is the lattice energy - \(N_A\) is the Avogadro's number - \(M\) is a coefficient called the Madelung constant - \(z^+\) is the charge of the cation (positive ion) - \(z^-\) is the charge of the anion (negative ion) - \(e\) is the elementary charge - \(r\) is the distance between the centers of the ions - \(\epsilon_0\) is the vacuum permittivity From the Born-Lande equation, we can see that larger charges will lead to a more significant negative lattice energy (i.e., more substantial lattice energy).
03

Understand the relationship between ionic radii and lattice energy

From the Born-Lande equation in Step 2, we also see that smaller ionic radii will lead to stronger lattice energy. If the ionic radii are smaller, the distance between the ions (\(r\)) will be shorter, resulting in a more significant negative lattice energy.
04

Compare the charges and sizes of the ions in the compounds

Using the information from Step 1, let's compare the charges and sizes of the ions in the given compounds: - In CaSe, the Ca虏鈦� ion has a higher charge and a smaller radius than the Na鈦� ion in Na鈧係e. Additionally, both compounds contain Se虏鈦� ions with the same charge and size. Hence, CaSe should have a more substantial lattice energy than Na鈧係e. - Similarly, CaTe has the Ca虏鈦� ion, and Na鈧俆e has the Na鈦� ion. The Te虏鈦� ions are larger than the Se虏鈦� ions, and therefore, the compounds containing Te should have lower lattice energy values than the ones containing Se.
05

Rationalize the given lattice energy values and compare

The provided lattice energy values are as follows: - CaSe: -2862 kJ/mol - Na鈧係e: -2130 kJ/mol - CaTe: -2721 kJ/mol - Na鈧俆e: -2095 kJ/mol As predicted in Step 4, CaSe has a higher lattice energy than Na鈧係e, and CaTe has a higher lattice energy than Na鈧俆e. Additionally, the compounds containing Se have higher lattice energy values than the ones containing Te. To summarize, the lattice energy values can be explained by considering the charges of the ions and their relative sizes. Compounds with higher charges and smaller ionic radii are associated with more substantial lattice energy values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ionic Radii
Understanding ionic radii is crucial when examining the stability and strength of ionic bonds. This term specifically refers to the measured size of an ion. An ionic radius is influenced by the ion's charge, with cations (positively charged ions) generally having smaller radii than anions (negatively charged ions) due to the loss of electrons and subsequent contraction of the electron cloud.

For example, when we consider the ionic lattice of calcium selenide (CaSe) versus sodium selenide (Na2Se), the size difference between the cations, calcium (Ca2+) and sodium (Na+), is instrumental in explaining the disparity in their lattice energies. The smaller calcium ion leads to a stronger attraction between ions in the crystal lattice structure. This stronger attraction results in a higher lattice energy compared to the larger sodium ion.

It is essential to recognize that as the ionic radii decrease, the distance between the ions in the crystal lattice also decreases, leading to increased electrostatic forces of attraction and thus higher lattice energies.
Born-Lande Equation
The Born-Lande equation offers a theoretical means to calculate the lattice energy of an ionic solid. This equation is influential in predicting and explaining the factors that affect lattice energy:

\[\begin{equation}E_l = -\frac{N_AMz^+z^-e^2}{4\pi\epsilon_0r}\end{equation}\]
This formula encapsulates several variables.
  • NA is Avogadro's number, representing the number of pairs of ions in one mole of the solid.
  • M is the Madelung constant that accounts for the geometric arrangement of the ions.
  • z+ and z- are the charges on the cations and anions, respectively.
  • e is the elementary charge, equivalent to the charge of a proton.
  • r is the internuclear distance between the ions.
  • \(\epsilon_0\) is the permittivity of free space, a constant value in the equation.

Impact on Lattice Energy

With a fixed ionic arrangement, higher ionic charges and smaller ionic radii result in a more negative lattice energy according to the Born-Lande equation, which implies a stronger ionic bond. Hence, it's instrumental in explaining the differences in lattice energies for compounds like CaSe and Na2Se, where different ionic charges play a significant role.
Ionic Charges
The role of ionic charges is a fundamental aspect when evaluating lattice energies. Ions are atoms or molecules that have gained or lost electrons, resulting in a net positive or negative charge. A higher charge magnitude on an ion leads to stronger electrostatic forces of attraction within a crystal lattice, and therefore, a higher lattice energy.

For example, comparing calcium ions (Ca2+) and sodium ions (Na+) illustrates that calcium ions, with their +2 charge, create stronger ionic bonds in the lattice than the +1 charge of sodium ions. Consequently, calcium selenide (CaSe) has a more substantial lattice energy than sodium selenide (Na2Se).

The influence of ionic charges on lattice energy can be directly observed in the Born-Lande equation, where the product of the cationic and anionic charges significantly determines the magnitude of the lattice energy. Compounds with ions of higher charges should, in theory, possess larger lattice energies, which is typically observed in practice and reflected in exercises involving lattice energy comparisons, as seen in the example above.

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Most popular questions from this chapter

In general, the higher the charge on the ions in an ionic compound, the more favorable the lattice energy. Why do some stable ionic compounds have \(+1\) charged ions even though \(+4,+5\), and \(+6\) charged ions would have a more favorable lattice energy?

Consider the following reaction: $$ \mathrm{A}_{2}+\mathrm{B}_{2} \longrightarrow 2 \mathrm{AB} \quad \Delta H=-285 \mathrm{~kJ} $$ The bond energy for \(\mathrm{A}_{2}\) is one-half the amount of the \(\mathrm{AB}\) bond energy. The bond energy of \(\mathrm{B}_{2}=432 \mathrm{~kJ} / \mathrm{mol}\). What is the bond energy of \(\mathrm{A}_{2}\) ?

\(\mathrm{SF}_{6}, \mathrm{ClF}_{5}\), and \(\mathrm{XeF}_{4}\) are three compounds whose central atoms do not follow the octet rule. Draw Lewis structures for these compounds.

The compound \(\mathrm{NF}_{3}\) is quite stable, but \(\mathrm{NCl}_{3}\) is very unstable \(\left(\mathrm{NCl}_{3}\right.\) was first synthesized in 1811 by P. L. Dulong, who lost three fingers and an eye studying its properties). The compounds \(\mathrm{NBr}_{3}\) and \(\mathrm{NI}_{3}\) are unknown, although the explosive compound \(\mathrm{NI}_{3} \cdot \mathrm{NH}_{3}\) is known. Account for the instability of these halides of nitrogen.

Which of the following statements is(are) true? Correct the false statements. a. The molecules \(\mathrm{SeS}_{3}, \mathrm{SeS}_{2}, \mathrm{PCl}_{5}, \mathrm{TeCl}_{4}, \mathrm{ICl}_{3}\), and \(\mathrm{XeCl}_{2}\) all exhibit at least one bond angle, which is approximately \(120^{\circ}\). b. The bond angle in \(\mathrm{SO}_{2}\) should be similar to the bond angle in \(\mathrm{CS}_{2}\) or \(\mathrm{SCl}_{2}\). c. Of the compounds \(\mathrm{CF}_{4}, \mathrm{KrF}_{4}\), and \(\mathrm{SeF}_{4}\), only \(\mathrm{SeF}_{4}\) exhibits an overall dipole moment (is polar). d. Central atoms in a molecule adopt a geometry of the bonded atoms and lone pairs about the central atom in order to maximize electron repulsions.
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