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Should the lattice energy for \(\mathrm{Al}_{2} \mathrm{O}_{3}\) be greater or less than the lattice energy for \(\mathrm{Na}_{2} \mathrm{O} ?\) By how much? Justify your answers.

Short Answer

Expert verified
The lattice energy of Al鈧侽鈧 is approximately 1.96 times greater than the lattice energy of Na鈧侽. This is because the higher charges on Al鲁鈦 and O虏鈦 ions lead to stronger electrostatic attraction between them, causing more energy to be released when they form the crystal lattice. Additionally, the total distance between the ions in Al鈧侽鈧 is smaller than the distance between the ions in Na鈧侽, which also contributes to the higher lattice energy in Al鈧侽鈧.

Step by step solution

01

The lattice energy (U) of an ionic compound can be defined as the energy released when oppositely charged ions come together to form a solid. It can be calculated using the following equation known as the Born-Lande equation: \(U = \dfrac{N_A * M * Z_+ * Z_- * e^2}{4 * \pi * \varepsilon_0 * r}\) Where: U 鈫 Lattice energy N_A 鈫 Avogadro's number (6.022 x 10虏鲁 mol鈦宦) M 鈫 Madelung constant, which depends on the crystal structure Z_+ 鈫 Charge of the cation Z_- 鈫 Charge of the anion e 鈫 Elementary charge (1.60 x 10鈦宦光伖 C) 蟺 鈫 Pi (3.1415) 蔚鈧 鈫 Permittivity of free space (8.85 x 10鹿虏 C虏J鈦宦筸鈦宦) r 鈫 Distance between the ions The lattice energy is directly proportional to the product of the charges (Z_+ * Z_-) and inversely proportional to the distance between ions (r). #Step 2: Determine the charges and radii of ions in Al鈧侽鈧 and Na鈧侽#

For Al鈧侽鈧: Aluminum (Al) has a charge of +3 (loses 3 electrons to form Al鲁鈦 ions) Oxygen (O) has a charge of -2 (gains 2 electrons to form O虏鈦 ions) For Na鈧侽: Sodium (Na) has a charge of +1 (loses 1 electron to form Na鈦 ions) Oxygen (O) has a charge of -2 (gains 2 electrons to form O虏鈦 ions) We also need to find the ionic radii for these ion pairs. We can look up these values in a reference table: Al鲁鈦: 0.054 nm O虏鈦: 0.140 nm Na鈦: 0.102 nm The distance between ions (r) is equal to the sum of their ionic radii. #Step 3: Compare the lattice energies of Al鈧侽鈧 and Na鈧侽#
02

Since the Born-Lande equation contains constants such as Avogadro's number, Madelung constant, elementary charge, and permittivity of free space, we can ignore these constants while comparing the lattice energies of Al鈧侽鈧 and Na鈧侽. Now let's compare the lattice energies using the equation: \(U \propto \dfrac{Z_+ * Z_-}{r}\) For Al鈧侽鈧: \(U_{Al2O3} \propto \dfrac{(-3) * (-2)}{(0.054 + 0.140)}\) For Na鈧侽: \(U_{Na2O} \propto \dfrac{(-1) * (-2)}{(0.102 + 0.140)}\) #Step 4: Analyze the results and justify the answer#

Comparing the proportional expressions for lattice energies of Al鈧侽鈧 and Na鈧侽: \(U_{Al2O3} \propto \dfrac{6}{(0.194)}\) \(U_{Na2O} \propto \dfrac{2}{(0.242)}\) Dividing the two proportional expressions: \(\dfrac{U_{Al2O3}}{U_{Na2O}} = \dfrac{ \tfrac{6}{0.194} }{ \tfrac{2}{0.242} }\) After solving the equation, our result is: \(\dfrac{U_{Al2O3}}{U_{Na2O}} \approx 1.96\) This means that the lattice energy of Al鈧侽鈧 is approximately 1.96 times greater than the lattice energy of Na鈧侽. The reason for this difference is that the higher charges on Al鲁鈦 and O虏鈦 ions lead to a stronger electrostatic attraction between them, causing more energy to be released when they come together to form the crystal lattice. Additionally, the total distance between the ions in Al鈧侽鈧 is smaller than the distance between the ions in Na鈧侽, which also contributes to the higher lattice energy in Al鈧侽鈧.

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Key Concepts

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

Born-Lande equation
Understanding the Born-Lande equation is pivotal for students studying ionic compounds and solid-state chemistry. It offers a quantitative view of the energies involved when ions arrange into a crystalline structure, commonly found in salts. The equation combines fundamental constants and crystal-specific variables to produce an estimate for lattice energy.

The formula incorporates Avogadro's number, reflecting that lattice energy accounts for the formation of one mole of ionic solid. The Madelung constant represents the geometry of the crystal. Different crystal structures have unique arrangements, hence distinct Madelung constants, which affect electrostatic interactions between ions.

The charges of the cation and anion, denoted as Z鈧 and Z鈧 respectively, are pivotal in the equation; the greater the charges, the stronger the electrostatic attraction and thus the higher the energy released upon formation. The distance between the ions, r, also contributes significantly鈥攃loser ions result in a stronger attraction and more energy. Permittivity of free space, 蔚鈧, is a constant in physics that measures the resistance to the electric field in a vacuum, a necessary factor when calculating forces and energies on a macroscopic scale.
Ionic compound
An ionic compound is a chemical compound composed of ions held together by electrostatic forces termed ionic bonding. The ions are positively charged cations and negatively charged anions which can be elements or polyatomic ions.

For example, sodium chloride (NaCl) is made up of Na鈦 and Cl鈦 ions. Ionic compounds normally form crystalline solids with high melting points due to the strong attraction between the oppositely charged ions. The structure of these solids is called a crystal lattice, where ions pack together in a pattern that balances the attraction and repulsion between them to maximize overall stability.

The properties of an ionic compound, like solubility and electrical conductivity in the molten state or when dissolved in water, are inherent due to the nature of its ionic bonds. Educators should emphasize to students that the fundamental understanding of these compounds involves recognition of their electrostatic-driven properties and their predictable formation based on the gain and loss of electrons to achieve stable electron configurations.
Electrostatic attraction
Electrostatic attraction is the force drawing oppositely charged particles toward one another. It plays a crucial role in chemistry, especially in the formation of ionic compounds like table salt (NaCl).

This attraction is inversely proportional to the square of the distance between the charges鈥攎eaning that as ions get closer to each other, the attraction force increases dramatically. It's also directly proportional to the product of the magnitudes of the charges; higher charged ions exert a stronger electrostatic attraction.

Application in Lattice Energy Calculation

In the context of lattice energy, the electrostatic attraction asserts itself when ions bond to form a solid. The energy change associated with this process, either absorbed or released, is a direct consequence of the electrostatic forces at play.

As seen in the comparative analysis of Na鈧侽 and Al鈧侽鈧, the difference in electrostatic attraction due to varying charges and distances greatly influences the lattice energies. Such concepts offer students a foundational understanding of ionic solids and the forces that govern their formation and stability.

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